• Last updated: 01/15/2012
• Math projects
• Projects
• Formal schemes associated to K-theory spaces
• Aut Ga equivariance of bu^H --> C^k
• There is an action of Aut Ga as a spec F2-scheme on spec F2 x C^k(Ga; Gm), which we managed to compute on the PD part of the algebra using the obstruction (1 + x)(1 + y) = 1 + (x + y) for xy = 0.
• We don't have a calculation for the free part.
• Do we know the action of the Steenrod algebra on H_* BU<2k>? Is that buried in Singer's work somehow?
• Is the map BU<2k>^H --> C^k(Ga; Gm) equivariant w/r/t the Aut Ga action? Does this help rigidify it in any way?
• Constructing a representing space for C^k
• There should exist a space Xk such that HZ_(p)^* Xk = C^k(Ga; Gm)(Z_(p)) = HZ_(p)^* BU<2k> / < the Steenrod subalgebra generated by the odd dimensional cohomology classes >. How do we produce such a thing? And, since BU<2k> was the old choice of representing space, we should start there by understanding the image of the classifying map constructed by A-H-S.
• Odds and ends about C^k
• Relation between the C^k schemes and Weil pairings: There's a map delta_1: C^k(Ga; Gm) --> C^(k+1)(Ga; Gm). What's its cokernel? For k < 3, the cokernel of this map was described in terms of Weil pairings; we should try to make a similar identification. It was also identified in topology with the covering fibration; what does the covering fibration represent in general?
• Cartier duality: Strickland showed that a group scheme C_k G exists if C^k(G; Gm) is sufficiently polite (namely, has a good basis / is coalgebraic), satisfying Hom(C_k G, Gm) = C^k(G; Gm). Can we use our calculation of C^k(Ga; Gm) x spec F2 to show this is the case for large k? Is this useful? Does it belong in a paper?
• Hopf ring structure: The sum of the C^k rings should assemble into something with either an extra product or coproduct, depending upon a variance check. What does this coproduct look like? Away from the prime 2, what information does it tell you about classes that extend freely? On the level of C_k, this should be like investigating the maps V^{(x) k} / Sigma_k x V^{(x) l} / Sigma_l --> V^{(x) (k+l)} / Sigma_{k+l}...
• The word 'orientation': Classically, a vector bundle is A-oriented if its classifying map to BO factors through the fiber of the map BO --> bglS --> bglA. This space is supposed to be partially described by e.g. the orientation theory of Atiyah, Bott, and Shapiro for A = ko and Ando, Hopkins, and Strickland for A = tmf --- in what sense is this true? How is the C^k stuff is supposed to help describe this space for the higher real K-theories, or, rather, what cohomology theory can be constructed where C^k does tell us something about orientations? (For example, each BP<n> has an associated Thom spectrum MBP<n>...)
• The Adams splitting: We do have a computation of C^k(Ga; Ga) at every prime, which does display a kind of (p-1)-fold stratification. Can we deduce information about the structure of C^k(Ga; Ga) implied by the Adams splitting of bu into a wedge of suspensions of BP<1>? (Sinkinson wrote a paper computing the homologies of the BP<1>s. This might help.)
• Existing literature
• What is the Atiyah-Bott-Shapiro map? What is the Atiyah-Singer index theorem? Why Spin bundles? Why Spin^c bundles? What does the Hirzebruch series actually tell you about orientations? The Miller invariant?
• Now that you're an adult, you should also reread the Ando-Strickland paper on K(n)_* BU<6>
• Read: The sigma-orientation is an H_infty map
• Image of tau
• Our gathering construction in the computation of C^k(Ga; Ga) has an enlightening interpretation on the level of the Cartier dual groups C_k. Starting with a group G, we build the group ring Z[G], which has an augmentation ideal, defined to be C_1, spanned by elements of the form [g] := <g> - <0>. The groups C_k are defined to be the kth symmetric powers of C_1. Then, there is a map C_k --> C_{k+1} sending [g1; ...; gk] to [g1; ...; gk; -(g1+...+gk)]. This induces a map C^(k+1)(Ga; Ga) = Hom(C_(k+1) Ga, Ga) --> Hom(C_k Ga, Ga) = C^k(Ga; Ga) given by f(x1, ..., xk, x(k+1)) |-> f(x1, ..., xk, -(x1+...+xk)). This is very similar to our gathering construction, which we defined combinatorially rather than algebraically.
• Strickland's programme for understanding ko_(2k)^E involves studying the cokernel of the map tau. This is a computation we should be capable of, at least on the level of C^k(Ga; Ga), if not something fancier like C^k(Ga; Gm). or C^k(CP^infty_E; Gm). And we should do it.
• Real K-theory
• Collaboration with Strickland has enlarged this project substantially. He describes a sequence of guesses for what (ko_n)_E should look like for K(n)-local theories E, and he has proofs for n even and p odd. The road involved is winding, and the main ingredient is to feed off the preexisting work of Kitchloo, Laures, and Wilson, but I'll attempt to summarize:
• The first step is to analyze E_* BSp. Writing G = BU(1)_E, BSp(1)_E turns out to be the fiber of the map G^{x 2} / Sigma_2 --> G, called bar G; this is a classical calculation rephased for formal schemes. Then, BSp_E is the scheme of divisors on BSp(1)_E. This comes with an important map q: G --> bar G given by q(a) = (a, -a), an isogeny of degree 2. Further, there is a map j: Div(bar G) --> C_2 given by sum_i q(a_i) = sum_i [a_i, -a_i], where [a, b] = (<a> - <0>)(<b> - <0>) and <x> denotes the unnormalized point x on Div G. In all, the sequence BU_E --> BSp_E --> BSU_E is identified with C_1 = Div_0 G --q^*-> Div(bar G) --j-> C_2.
• The space BO is known by Wilson to have BO_E = Div_0(E)^chi = {x in Div_0(E) | x = chi x}, and hence (BO x Z)_E = Div(E)^chi =: C_0^+.
• Provided G has finite height, the isogeny 2: G --> G has finite kernel, and so induces an isogeny 2: bar G --> bar G with the same rank. Writing bar G[2] for the kernel of this isogeny, Strickland claims that Div bar G --q^*-> C_0^+ restricts to Div bar G[2] --q^*-> Div G[2], and further this square is co/Cartesian, to be checked on the level of homology coalgebras. Restricting to Div_0 gives a similar story for BSO_E.
• There is a fibration BSpin --> BSO --omega-> K(Z/2, 2) which induces a sexseq of group schemes. Ravenel and Wilson identify K(Z/2, 2)_E = G[2]^{^ 2}, and Strickland has a guess for a formula for the action of omega on group schemes. He also points out an accidental isomorphism Div_0 bar G --> BSpin_E at height 2, which is not induced by a map of spaces.
• We also have a pair of fibrations K(Z, 3) --> BU<6> --> BSU and K(Z, 3) --> BO<8> --> BSpin, along with a map of fibrations between them. Both are known to give sexseqs of group schemes by AHS/KLW, and then we're supposed to get that BO<8>_E = ker(tau: C_2 --> C_3).
• The isogeny q should also give rise to interesting data about power operations, but this looks nearly unexplored.
• There are also some spaces, like Sp/U and O/U, that he has guesses for but which branch off this path to BO<8>, the biggest goal.
• When 2 is inverted, the complex conjugation map ku --xi-> ku let us write id_ku as the sum of two idempotents: (1 + xi)/2 and (1 - xi)/2. This splits ku into ku^+ v ku^-, and his lambda cofiber sequence gives straightforward descriptions of (ko_2n)_E.
• The p-divisible cohomology of Eilenberg-Mac Lane spaces
• The Morava E-homology calculation is complete; see e-thy.pdf.
• The transchromatic case
• The finished computation is of the completed Johnson-Wilson theory of Eilenberg-Mac Lane spaces, so pi_* L_K(n) (E(n)^ ^ K(Z/p^infty, q)).
• A natural source of p-divisible groups are the K(t)-localizations of E_n for t < n. Rezk knew and Stapleton wrote down that CP^infty_{L_K(t) E_n}[p^infty] is a p-divisible group of height n with formal height t, and (L_K(t) E_n)_* = W_{F^(p^n)} [[u1, ..., u(n-1)]][ut^-1]^_{u1, ..., u(t-1)}.
• So, one thing we might do is try to study pi_* L_K(t) (E(n)^ ^ K(Z/p^infty, q)), which would hopefully again be the qth exterior power of the p-divisble group CP^infty_{L_K(t) E_n}[p^infty].
• It is not clear to me at all how to do this. There is a chromatic tower relating these layers, but the E(t)-localizations of E(n)^ ^ K(Z/p^infty, q) are no less mysterious I think. But, I'm sure someone has computed something transchromatically, and so I should go read about that. Nat's paper is extremely cool, and he does some very smart transchromatic stuff, but it looks like he largely avoids computing things (other than (E_n,t)_* BZ/n, which is known from HKR or before.
• Hovey's _vn-elements of ring spectra and applications to bordism theory_ gives a description of L_K(t) R for R a ring spectrum satisfying the telescope conjecture and admitting a vn-element (in his sense, rather than in the Hopkins-Smith sense). This is what lets you compute pi_* L_K(t) E_n, since BP satisfies the telescope conjecture and E_n = BP (x) E_n^*. Matt claimed in private that this worked for any BP-module (so, e.g., E(n)^ ^ K(Z/p^infty, q)!), but I don't think that that's true, or else Hovey-Strickland's proof of the Milnor-type sequence for E_n^vee would be empty and there would be no preconditions. It's possible, however, that the Hovey-Milnor-Strickland sequence can be reworked to say something about this localization too?
• The p-divisible case
• Lurie has a theorem which states (at minimum): If G is a p-divisible group on a separated locally Noetherian D-M stack X over a local ring Spec A with cover X~ --pi-> X by a scheme such that for each x in X~^_{mA} the induced map (pi^* G)_{X~^_x} --> Def(pi^* G)_x is an isomorphism, then there is a locally fibrant sheaf E_G on the etale site of X^_{mA} such that for each etale affine open gamma: Spec U --> X^_{mA} we have S_{E_G(gamma)} = Spec U and CP^infty_{E_G(gamma)} = (gamma^* G)^0.
• The way computations get done with sheaves of ring spectra is to cover the base, apply the sheaf to the covers and the resulting ring spectra to your target space, and from this (co)simplicial object build a spectral sequence of Bousfield-Kan type.
• To start, then, we need to identify a cover with good properties, so that we can compute the individual cohomology rings E(U)^* K(Z/p^j, q). This means finding faithfully flat maps Spec U --> X so that the p-divisible group G pulls back to Spec U to be something sane. For example, I think it's known that we can base-extend so that G pulls back to G^0 x G^et. It's possible that we could further extend so that G^0 pulls back to some Honda formal group? I think at this point the most likely thing we'll see is a cover whose pullbacks look like CP^infty_{L_K(t) E_n}[p^infty].
• We want to take this functor and post-compose with either the functor Spectra --F(Y, -)-> Spectra or the functor Spectra --(Y ^ -)-> Spectra. This construction is natural enough so that, say, in the cohomological case we will get a multiplication in the spectral sequence converging to the multiplication on the totalization of the cosimplicial object. This multiplication will mix the multiplication on E(U) with the Cech multiplication coming from the cover.
• However, it's worth noting that this postcomposition isn't going to be a sheaf unless Y satisfies some finiteness properties. But, this may already be built into the definition of the formal scheme Y_E by colimiting along compact subsets of Y. So, Y_E is built from the colimits of the Specs of the results of the spectral sequences for each compact subspace. But, we may not find these spectral sequences themselves to be well-behaved compared to the limits of the spectral sequences themselves... so this may require some study to understand the interplay, and it's something to be kept in mind.
• Other ideas
• There's a possibility of producing a spectral sequence in algebraic geometry reflecting the bar spectral sequence in topology, which may be enlightening. Namely, what does the simplex dimension filtration mean in algebraic geometry?
• What does the space of possible extensions look like, when we consider that these have to assemble to form a ring scheme?
• There are product maps K(Z/p^infty, q)_K(n) x K(Z/p^infty, q')_K(n) --> K(Z/p^infty, q+q')_K(n) induced by the cup product. Collectively we have a graded-commutative ring scheme K(Z/p^infty, *)_K(n).
• For a square-zero deformation R --> R_0 of ring spectra R, R_0 sufficiently amenable to chromatic homotopy, we get a deformation K(Z/p^infty, q)_(R_0) --> K(Z/p^infty, q)_R of formal groups. The multiplication map K(Z/p^infty, q)_R x K(Z/p^infty, q)_R --> K(Z/p^infty, q)_R (and similarly for R_0) corresponds to an N-tuple of elements in the power series ring R[[x_1, ..., x_N, y_1, ..., y_N]] which images to the R_0-multiplication power series under the pushforward along pi_* R --> pi_* R_0.
• The alternating algebra CP^infty_R[p^j]^(^ *) should give a ring scheme of the same graded dimension as K(Z/p^j, *)_R, also a deformation of the alternating algebra CP^infty_(R_0)[p^j]^(^ *).
• Beginning with a selected deformation of ring schemes, a certain subgroup of Ext^1(CP^infty_(R_0)[p^j]^{^ *}, M) should control the available ring scheme deformations. Not all Ext^1-elements should deform the formal groups to another ring scheme --- ideally there'd just be one, and so the free ring scheme is the only option.
• -----
• However, Ravenel and Wilson do not compute the ring maps for K(Z/p^infty, *)_K(n). They compute the K(n)-homology of K(Z/p^infty, q) using a direct limit along the group homomorphisms K(Z/p^j, q) --p-> K(Z/p^(j+1), q), but since multiplication by p is not a ring homomorphism we lose this information. This defect is not repairable; using the ring homomorphism Z/p^(j+1) --> Z/p^j means calculating the homology of an inverse system which is not Mittag-Leffler, and using cohomology means completely recomputing their argument. However, at each finite stage K(Z/p^j, *)_K(n) is described as the alternating tensor algebra on CP^infty_K(n)[p^j]; it's inconceivable that this wouldn't be the case in the limit as well.
• Can the deformation theory of ring schemes be viewed more innately? Is there a Hochschild homology of ring schemes or a deformation theory of ring schemes that controls it? Would this organization be useful in demonstrating that these objects are Hochschild-acyclic?
• Now that we know the answer, it may be possible to perform the Ravenel-Wilson computation again and produce an analysis of the bar spectral sequence. (However, I suspect that this will not tell us anything interesting.)
• Check that E/m^r is a sufficiently well-structured spectrum so that a Kunneth spectral sequence exists and so that the reduction morphism E/m^{r+1} --> E/m^r acts like tensoring with pi_* E/m^r on even-concentrated spaces. Lawson suggested the paper Vigleik Angeltveit's Topological Hochschild homology and cohomology of A-infinity ring spectra ( http://www.msp.warwick.ac.uk/gt/2008/12-02/p022.xhtml ). It may also follow formally from E_* K(Z/p^j, q) being even-concentrated, so that the reduction to E/m^r (which still needs to be verified comes from a ring spectrum) preserves things well enough that the spectral sequence for E, which is in fact A_infty (or E_n, even, which is E_infty!) just pushes down.
• Check that the homology of the algebra (E/m^r)_* K(Z/p, q-1) terminates at the second stage, using Tate's construction.
• Justify the existence of a Verschiebung on (E/m^r)_* K(Z/p, q-1)
• Check that a homology class in the homology of (E/m^r)_* K(Z/p, q-1) can always be extended to a homology class after tensoring with pi_* E/m^r. Moreover, produce a formula for the class that makes apparent that it is divided power.
• Writing gamma_I for that class, prove that gamma_{tilde I} circ a_{(I_1+1)} is gamma_I, up to a unit. You have a partial proof of this fact currently, but it uses the decomposition a_{ap^b} = a_{p^b} star a_{(a-1)p^b}, which is probably not quite true. --- In fact, it certainly isn't true for r > 1, but it probably is true for r = 1. You might finish writing this out anyway, just to demonstrate that the R-W lemma can be expressly computed.
• Work on the extension of this proof to (E/m^r)_* K(Z/p^j, q-1) by induction on j. Start by understanding the R-W argument for K(n).
• Figure out how the limiting case of j=infty works, and make explicit the statement that K(Z, q)^E = Hom(Lambda^q CP^infty_E[p^infty], Gm hat)
• We might translate this specific result into the language of Dieudonne modules, where the question of lifting from F_p-algebras to Z_p-algebras becomes one of finding a Hodge structure on the resulting Dieudonne crystal. We got this lift for free --- so why were we guaranteed a Hodge structure? What is this piece of data? (Note: Saramago has a paper studying Dieudonne rings, which marries Dieudonne modules with Hopf rings.)
• Another distant goal is to understand this in the language of derived algebraic geometry. Surely we can construct a derived deformation of rings --- surely that's what we're doing! --- and find reasons for our results internal to the category of spectra, without pushing down to homotopy/homology/cohomology groups.
• Loopspaces of spheres
• In attempting to form an improved understanding of the proof of the nilpotence conjectures, I got to asking about relationships between loopspaces of spheres and formal groups. Not much about this is known, and Neil says he feels this is a hard question, that a good answer ought to exist, and that it'll be a big day when someone finds it. A start might be just collecting a bunch of known things about these spaces.
• Ordinary (co)homology
• The integral cohomology of LS^(2n+1) is accessible via the Serre spectral sequence: Gamma_Z[x], with diagonal Delta x^[n] = sum_{i=0}^n x^[i] (x) x^[n-i].
• The modular cohomology of LLS^(2n+1) is also accessible by the Serre spectral sequence: at k = Z/2, it's k[x_i, i >= 0], and at k = Z/p, p > 2, it's Lambda[x_i : i >= 0] (x) k[y_i : i > 0].
• Ravenel conjectures that for BP we will get BP_* LLS^(2n+1) = Lambda[x_0] (x) BP_*[y_i : i > 0] / L, where L is generated by the homogeneous components of the formal group law sum expression sum^F v_j y_{i-j}^{p^j}.
• The James filtration and spectral sequence
• The quotients of the James filtration are quite simple (they are smash powers of the original space), and so give a nice spectral sequence converging to the cohomology of LX. Ravenel noticed this too, and showed that there are convergence issues, since K(n)_* K(Z/p^infty, n+1) vanishes but K(n)_* LK(Z/p^infty, n+1) = K(n)_* K(Z/p^infty, n) does not.
• Ravenel's basic idea, then, is to make one of these calculations mirror his K(n)_* K(Z/p^infty, *) calculations as closely as possible. In particular, we should seek out some kind of cup product pairing interrelating the spectral sequences. For example, there's a pairing L^mS^n x L^nS^l --> L^mS^l, to start. This story should really be *dual* to the one for E-M spaces, since we're dissecting L rather than B, and so we likely actually want a copairing.
• However, the filtration used for this spectral sequence does not descend well to iterated applications of Loops. The Loops functor does preserve fiber sequences, but it does not preserve cofibers.
• The spectrum (E smash Susp^infty_+ L^2 S^3) is the free E_2-algebra over E generated by (E smash S^1).
• This is another way of saying that the extended power functor acts by smashing against Susp^infty_+ L^2 S^3.
• The role of Loops^2 S^3 in Nishida's theorem
• I received an email from Rezk a while back, detailing Hopkins' improved proof of Nishida's theorem, which supposedly spurred his proof of the Nilpotence conjectures. A key point is that L^2 S^3 --> S^1 --> BO classifying the generator of pi_1 BO has as its Thom spectrum HZ/2. More generally, L^2 S^3 --> S^1 --> BGL_1 L_p S has Thom spectrum HZ/p. Supposedly, this has not been written down, and the steps that follow after this in his email do not treat the prime power case.
• This uses in some nontrivial way the E_2-action on double loopspaces.
• Some of this stuff about Thom spectra is discussed in http://arxiv.org/abs/0811.0553 .
• The role of Loops^2 S^(2n+1) in the Nilpotence theorem
• This also uses in some nontrivial way the E_2-action on double loopspaces.
• Write G(n) = LSU(n) and X(n) for the associated Thom spectrum for G(n) = LSU(n) --> LSU = BU. Then X(n+1) is filtered by X(n)-module spectra by restricting X(n+1) to Jr S^(2n), which filters J S^(2n) = LS^(2n+1). Each of these fibrations carries an action of LJrS^(2n), and so we have additionally a limiting action of L^2 S^(2n+1).
• Indeed, there's a sequence of fibrations L^2 S^(2n+1) --> LSU(n) --> LSU(n+1) --> LS^(2n+1) --> SU(n) --> SU(n+1) --> S^(2n+1).
• The spectra X(n) serve as moduli of formal groups with coordinates determined up to the nth power. These spectra are not complex oriented, but they have a 'partial' orientation in from CP^n. Relatedly, there is a homotopy-commutative square with edges CP^n --> LU(n+1) --> LS^(2n+1) and CP^n --> S^(2n) = J_1 S^(2n) --> JS^(2n) = LS^(2n+1).
• The James and Snaith splittings
• Classically, we have a James splitting SLS^(2n+1) = wedge_{i>0} S^(2ni+1).
• This is a special case of Snaith's splitting of L^(m+1) S^(2n+1), which is bound up somehow with the Goodwillie calculus. It states that there's an equivalence S^infty L^(m+1) S^(2n+1) = wedge_{i>0} S^{i(2n-m)} D^i_{m, n} for certain spaces D^i_{m, n}.
• These spaces have D^i_{0, n} = S^0 (this is a statement of James splitting) and D^1_{m, n} = S^0, a normalization. After localizing at p, we have D^{ip+e}_{1, n} = S^(i(p-2)) B_i if e = 0 or 1 mod p and = pt otherwise. Here, B_i is the ith Brown-Gitler spectrum.
• There are pairings D^i_{m, n} smash D^j_{m, n} --> D^{i+j}_{m, n} with degree 1 on the bottom cell.
• Generally, we have the problem of understanding the homotopy type of D^infty_{m, n} = colim_i D^i_{m, n}. We know D^infty_{0, n} = S^0 and D^infty_{1, n} = HZ/p, but D^infty_{2, n} is some kind of module spectrum over MU. Generally, it is given by the Thom spectrum of the composite L^{2m+1}_0 S^{2n+1} --> Q_0 S^0 = (BSigma_infty)_+ --((s-m)rho)-> BU.
• It suffices to consider odd spheres, since we have a counit L^2 Susp^2 --> id to compare D^infty_{m, n} with D^infty_{m-2, n-2}. This gives a map D^i_{1, s-k} --> D^i_{2k+1, s}, turning D^infty_{2k+1,s} into a module spectrum over D^infty_{1, s-k} = HZ/p. It also follows that MU is a module spectrum over each D^infty_{2m, n}.
• Ravenel's spectral sequence
• The p-local homotopy of D^i_{m,2s} for fixed m and i is only dependent on the congruence class of s modulo p^f(m) for some arithmetic function f. When the congruence class is 0, we write D^i_m, C^i_m for some normalized suspension, and L_m = wedge_{i > 0} C^i_m for their wedge.
• The Hopf map induces a map C^{pi}_{m,2s} --> C^i_{m, 2ps}, which for s highly divisible by p induces a map C^{pi}_m --> C^i_m. Taking Spanier-Whitehead duals, we can take the homotopy colimit of the associated system, called K~_m, which has bottom cell in dimension m.
• Ravenel then conjectures that K~_m has (a nontrivial retract of) the suspension spectrum of the E-M space K_m as a retract. He further conjectures that there is a pairing K~_m smash K~_n --> K~_{m+n}, lifting the cup product on E-M spaces. One could then hope to produce a spectral sequence computing the K(n)-local cohomology of K~_m, lifting the known spectral sequence computation of the K(n)-local cohomology of K_m.
• This is all conjectural, but assuming this pairing exists, one can actually *do* the computation blindly. The answers you get agree with the two known answers for m = 1 (Yamaguchi) and m = 2 (Tamaki).
• Super Dieudonne theory
• Brown-Gitler spectra enjoy certain properties that greatly enhance the usefulness of the Snaith splitting.
• Namely, H^* B_n = A/A{chi(Sq^i) : 2i > n} at p=2 and H^* B_n = A/A{chi(bock^e P^i) : 2pi+2e > n} at p>2. If B_n --> HZ/p classifies the generator of H^0 B_n, then the induced map (B_n)_n Z --> H_n Z is surjective for all CW complexes Z.
• Goerss constructs a graded Dieudonne module to deal with supercommutativity in the Hopf rings E_* F_*. The above result shows that there is a natural surjection (B_n)_n X --> D_n H_* L^infty X, which is an isomorphism if n is not +/- 1 mod 2p. Furthermore, the Verschiebung V: D_{2pn} H_* L^infty X --> D_{2n} H_* L^infty X and Frobenius F: D_{2n} H_* L^infty X --> D_{2pn} H_* L^infty X are induced by topological maps B_{2pn} --> S^(2n(p-1)) B_n and S^(2n(p-1)) B_n --> B_{2pn} respectively. The former is familiar from the Mahowald exact sequence; the latter is bizarre.
• Then, provided that E_* B_{2n} is even-concentrated, a lot of mess collapses and we can make good sense of the Hopf ring E_* H_* = H_* E_*. Because L^2 S^3 (or, rather, its universal cover) contains all the spaces B_{2n} as wedge summands with even suspension shifts, it's sufficient to show that it is even-concentrated.
• This is hugely satisfying, but there are some confusing points. For instance, this relied very heavily on choosing a prime --- indeed, even the splitting of the sphere depended heavily on this, together with the fancy regrading. This is one suggestion that we don't have the right conceptual picture yet.
• Goerss's Witt schemes also occur in another topological context: pi_0^{Z/n} THH(R) = Witt_n(pi_0 R), i.e., the Witt schemes occur as certain cyclic homologies of ring schemes.
• One of the crucial features of loops of spheres that Strickland's improved proof of the nilpotence theorem uses is that these James and Snaith guys give filtrations by finite spectra, so one can apply the duality theory of Dwyer-Greenlees-Iyengar.
• Strickland's bestiary has a section on doubly looped spheres. Might be worth picking through.
• The Picard group of the K(n)-local stable category
• Some old remarks
• In the late 80s / early 90s, Strickland and Hopkins studied the group of spectra which were smash-invertible in the inherited 'symmetric monoidal structure' on the K(n)-local stable category. They used this to give a conceptual description of pi_* L_K(1) S, using primarily that the homotopy group functor, when extended to all smash-invertible spectra, became a continuous function on the Picard group, which is a p-adic Lie group.
• There was some trouble extending this to the K(n)-local category for n >= 2, in part because we don't know what the Picard group is, and in part because there was some genuine trouble in making the continuity idea work out. The main problem is that the K(1)-local Picard group has a variety of number theoretic interpretations that all coincide in this small dimension, but become distinct in higher dimensions, so it's difficult to write down a conjecture for what ought to happen in the interplay of these structures.
• But, Behrens has been revisiting Shimomura's calculation **and has found mistakes**, which means some of these objections from before may now be reconcilable!
• Strickland himself writes: In my opinion, the most interesting open question about Picard groups is as follows. Let R denote the ring spectrum F(K(Z,n+1)_+,L_{K(n)}S^0), and let I be the fibre of the augmentation R -> L_{K(n)}S^0. Let Q be the cofibre of the multiplication map I \Smash_R I -> I (or more heuristically, Q = I/I^2). I conjecture that Q is invertible, and that it is closely related to K(n)-local Brown-Comenetz duality as in the Hopkins-Gross duality theorem. An ideal proof of this would rederive the Hopkins-Gross theorem along the way. There is an interesting heuristic argument for the conjecture, but it cannot currently be made rigorous, essentially because of the difference between K(n) \Smash F(K(Z,n+1)_+,L_{K(n)}S^0) and F(K(Z,n+1)_+,K(n)).
• At any rate, I should start by completely understanding the _On the p-adic interpolation of stable homotopy groups_ paper, much of which is kind of a mystery to me. For instance, Strickland attributes his interpolation scheme to Iwasawa and Mazur, without specific reference; track this down. (Having read this more carefully recently, the reason it didn't make much sense is that a lot of statements are just guesses at how things ought to work, by noticing similar patterns. We don't assert that the geometry of the formal varities behaves this way, we assert that if it did then we'd get the answer we wanted.)
• There is an extremely interesting paper by Strickland concerning the group of smash-invertible objects in the K(n)-localized stable category. The central idea is that whereas the Picard group of the ordinary stable category is Z, consisting of the spheres, Pic_1 is known to be Z^_p x Z/(2p-2), generated by the standard spheres and also certain colimits of Moore spectra. This group carries an interesting topology, induced from one of several places, so: is this reflected somehow in a continuity of the generalized homotopy groups [S^lambda, X], S^lambda in Pic_n?
• There are a few places this topology can be seen to come from: in the case of Pic_1, where the Hopkins-Mahowald-Strickland calculation is performed by computing p-completed Adams operations, the Adams operations themselves are topologized. In every case, we have a map Pic_n --> AlgPic_n, where AlgPic_n is the group of invertible E_n-S_n-modules, i.e., the input to the Morava stabilizer spectral sequence. Neither of these is a very good source, since the cohomology operations of K(n) for n > 1 are less well-understood and since there's no reason for the map Pic_n --> AlgPic_n to be injective (indeed it often isn't). So, there is a need to produce a topology without these facts and to check that it is compatible with them afterward.
• One idea I had was to study the stabilizers of the action of the Picard group on the K(n)-localized category itself: for each spectrum X, we can smash through with the elements of the Picard group to get a map a_X: Pic_n --[(X, S^lambda) |-> X ^ S^lambda]-> SHC_(K(n)). This map can have interesting kernels, and it would be cool to find a sequence of spaces X_n such that ker a_{X_n} form a linear basis for the topology on Pic_n.
• The specific spectra I tried were the Moore spectra M^0(p^j). These have the property that M^0(p^j) ^ M^0(p^(j+k)) = M^0(p^j) ^ M^0(p^j) (a fact reminisicent of Z/p^j (x) Z/p^(j+k) = Z/p^j (x) Z/p^j), and since the known elements of Pic_1 are constructed by the homotopy colimit of the sequence M^0(p) --v1^l1-> M^{-|v1|a1}(p) --> M^{-|v1|a1}(p^2) --v1^l2-> M^{-|v1|v2}(p^2) --> ... for li the digits of a p-adic integer a in Zp^, 'stabilizing' this sequence seemed like a fine idea.
• Indeed, the map a_{M^0(p^j)} has a kernel, but unfortunately it's not very interesting: it's always (the same copy of) Zp^, independent of j. The essential problem is that smashing through one of these sequences with M^0(p^j) does truncate the system of spaces, and so all we're left with to distinguish the colimits by are their suspension shifts. This can necessarily only remember the information in Z/(2p-2) <= Pic_1. I have no idea as of this moment on how to circumnavigate this problem. I'm also not even sure this is a valid complaint --- Pic_1 can tell the difference between S^0 and S^(2p-2), after all, so...?
• In any case, producing a good model for this topology seems essential to trying to understand what 'continuity' could possibly mean in the general setting.
• The twisted equivariant Chern character
• Matt suggested beginning to learn about equivariant twisted K-theory, which is Constantin's kind of thing, by considering the twisted equivariant Chern character.
• Namely, there is a kind of K-theory resulting from elliptic cohomology by taking a j-invariant formal neighborhood of infinity. This object has formal height 1, so behaves like K-theory, with this extra q-parameter running along the curve; it's called Tate K-theory, and the curve is the Tate-curve.
• Grojnowski defines a twisted equivariant elliptic cohomology by effectively building the ellipticity into the group of equivariance, then covering the elliptic curve by small neighborhoods and building a globally nontrivially twisted sheaf out of ordinary equivariant cohomology applied to these neighborhoods.
• It's somewhat reasonable to expect that there is then a Chern character from Tate K-theory to Grojnowski's twisted cohomology, constructed locally from the standard Chern character and then getting twisted up from there.
• (Actually, Matt's email seems to indicate that Tate K-theory is not the object of interest, but whatever it is that Freed-Hopkins-Teleman use, since that's the thing we're comparing. Or, rather, the story is more complicated than I know, with so many versions of things around.)
• TQFTs valued in finite homotopy types
• Constantin has a machine for produce a TQFT from a fixed space X which has bounded homotopy groups, all of which are finite. The specific construction is something of a mystery to me, but it's in his paper with Freed, Hopkins, Lurie, and Teleman; the rough idea is to consider the mapping spaces Maps(Loops^j X, M), where M is the manifold we want to evaluate the TQFT on. These spaces have pi_0s which organize into an E_n-algebra (called 'quantization'), and that's the actual TQFT.
• However, this quantization process is not a very 'good' one, in the sense that a lot of information about X is lost. Constantin thinks? knows? has shown? has written down? that we can mix a ring spectrum into this construction, sort of as coefficients. When we mix in an E-M spectrum, nothing happens. When we mix in (completed, at least, but maybe we would rather use equivariant? also, twisted??) K-theory, we recover 'one more level' of information. His idea, then, is that mixing in the Morava E-theory spectra should recover steadily more and more information about the algebra. In the limit, one might expect to mix in all of MU, which should perhaps obviously capture everything available, since MU corepresents bordism and TQFTs are all about bordism.
• Lurie also has a separate perspective on this construction toward the back of his Survey, where he discusses n-equivariant theories. This seems extremely important to read carefully if I even get started on this.
• Questions and ideas and whatever that are already known
• Stable homotopy theory
• Snaith's theorem
• Tyler Lawson said that somehow the Hopkins-Miller(-Goerss)(-Lurie) result was a vast generalization of Snaith's theorem. In what sense is this true?
• In fact, Lurie's _Survey of Elliptic Cohomology_ explicitly mentions Snaith's theorem (thm 3.1), but only as a sidenote that his moduli-theoretic description of orientations of the multiplicative group recovers, via Snaith's theorem, complex K-theory. How does this address the point above?
• Dualities in stable homotopy theory
• Here's a partial list of some of the dualities found in stable homotopy theory: Spanier-Whitehead, Brown-Comenetz, Anderson, Gross-Hopkins, Morita/Koszul(, Langlands?). Understanding their applications and interrelations would be a pretty cool project.
• Chromatic homotopy theory
• Supposedly there's a program relating THH, EO(n), and the Singer transfer. What's that?
• The J homomorphism
• L_(K(1)) S ~= j_(p) is absolutely central -- how is this shown?
• Look at Ravenel, American Journal of Mathematics 1983
• Look at Bousfield on K-theory localization
• What does the statement 'The image of the J-homomorphism is the Adams 1-line' mean? In general, how are ANSS height and chromatic height related?
• Ext_(A(n))(k, k)
• H^* MO<8> looks like this for n = 2 through a range, what's the deal?
• What about arbitrary n? This is one way to compute pi_* L_(2) ko and apparently some other things, why?
• The Lawson overview of taf talks about how Ext_A(1)(k, k) is really the cohomology of the stack parametrizing isomorphism classes of quadratics, which are like forms of the multiplicative group over a nonalgebraically closed field. (Section 7.) Why does this construction make homotopical sense?
• This is related to a splitting tmf^_p = wedge Susp^{p(k)} BP<2>, just like the Adams splitting L_p ko = wedge Susp^* BP<1>. Where are these splittings coming from? --- NOTE: I found a remark by Matt that says L_K(2) TMF = prod_{supersingular elliptic curves in char p} EO_2, EO_2 = E_2^{h(max finite supgp of S_2)}.
• The definition of higher real K-theory is: take the Morava E-theory spectrum E_n, which carries a strict action of the Morava stabilizer group. The fixed points with respect to a maximal finite subgroup gives EO_n, and the 0-truncation gives eo_n; we're supposed to have eo_1 modelling ko and eo_2 modelling tmf. What properties do the eo_n / EO_n enjoy in the abstract?
• Related to this last point, there's a paper of Davis and Mahowald guaranteeing that there is no spectrum with mod-2 cohomology A//A(2) [whereas HZ and ko realize A//A(0) and A//A(1) respectively, and A//A(r) is known to be nonrealizable by a decomposition of Sq^(2^(r+1)) due to Adams].
• What chromatic significance does BP<n> have? What moduli problem does the associated derived stack address? What is the associated FGL? (This last bit should be obvious.)
• Spectral localization and completion
• Adams defines spectral completion, Bousfield and Kan (and Sullivan) define completion and localization at primes, but what does localization at, say, K(n) mean? I'm sure this is defined in Hirschorn! (This is studied in great detail by Hovey-Strickland. I have heard someone claim that for a BP-module E, L_K(n) E has homotopy pi_* E with vn inverted, then completed at the ideal <v0, ..., v(n-1)>. I've tried pretty hard to reproduce this and haven't yet figured it out!)
• The telescope conjecture! What is this, exactly? What is the story about its various proofs and disproofs? Supposedly Ravenel and company have a paper giving a disproof at chromatic height 2, but very few people have managed to read it, and many of those who have feel that there are gaps.
• Read Deligne-Tate in the Antwerp Proceedings
• Algebraic geometry
• Breen's theorem of the cube
• You've used cubical structures before; it's alarming that you don't know precisely what the theorem of the cube is.
• p-divisible groups
• Serre-Tate local moduli
• Tate - p-Divisible Groups
• Demazure - Lectures on p-divisible groups
• Crystals, the Dieudonne crystal, Cartier modules
• It would do you good to understand 'Pierre Berthelot and Arthur Ogus. Notes on crystalline cohomology. Princeton Univ. Press, 1978' for reading about crystals and so forth. In particular, the relationship between modules with integrable connections seems extremely relevant. Ando's notes from Hopkins' class will also be good, but they just cite BO78 over and over...
• Devinatz wrote an introductory paper to the theory of Cartier modules and Dieudonne crystals called 'The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts', which is very readable up until it suddenly isn't.
• Ando's course notes 'Dieudonne Crystals associated to Lubin-Tate Formal Groups'
• Abstract homotopy theory
• Quillen's +-construction
• The Handbook of K-theory contains a nice description by Carlsson of all the various constructions of K-theory spectra from amenable categories. Quillen's +-construction isn't covered, but every other conceivable approach is --- including his Q-construction, which is what gave him all the formal properties of his higher K-functors anyway.
• The stable Pi_infty
• Learning about the comparison between oo-groupoids and spaces changed my life. Stable categories are a thing; is there a connection to a sort of Pi_infty of spectra?
• Thom spectra
• Strickland has a preprint concerning the theory of Thom spectra from a uniquely categorical point of view. In particular, he produces the relationship to the space of units and describes the space of maps [X^xi, E] for a ring spectrum E and stable virtual bundle xi over X without really getting his hands dirty. Read this!
• Equivariant stable homotopy theory
• There's a theory of S_n-equivariant spectra set up by May et al in order to understand power operations in extraordinary cohomology theories, i.e., H_infty ring spectra. This may be easier to digest than a general description of G-equivariant spectra.
• But, at some point, you should understand both. Equivariant stable homotopy theory is remarkably rigid, which makes it remarkably prolific.
• The Kervaire invariant
• You've been to at least three series of talks on this topic; it's time to actually learn some things about it.
• HHR is clearly the real target, but you don't really know much about equivariant (stable) homotopy, so maybe one of the many May books would be a better place to start.
• Hu-Kriz was mentioned several times during the MSRI talks as a sort of source of inspiration for the equivariant techniques used in the paper, so would also be a good reference. In particular, they discuss a theory called MR, which HHR called MUR or MU_R.
• Hopkins also said that he didn't really set out to prove this theorem, that it came out of thinking about higher real K-theories. It would be good to have enough intuition about these objects to see why that's true as you read HHR.
• Rehka Santhanam wrote a paper called _Units of equivariant ring spectra_, where he reviews an equivariant version of Gamma-spaces (recall that such rings are supposed to be Quillen equivalent to equivariant connective E_infty rings a la May) and defines a gl_1 for them, in the presence of a finite group of equivariance. This would be worth reading.
• Santhanam's paper is doubly worth reading, because Andrew Blumberg is said to have a nose for how this can be generalized to general compact Lie groups of equivariance.
• Lurie's survey discusses equivariant spectra in some detail; the point seems to be to design E_G(*) to be a smarter version of the completion E(* // G) = E(BG). This is not how I'm accustomed to thinking about equivariant objects, so it's probably worth meditating on.
• Grojnowski is now famous for introducing a version of equivariant elliptic cohomology which dodges the problem that O(E) = k for an elliptic curve E, by making Ell_{S^1}(X) output a sheaf of functions rather than just a module. This is terribly smart, and it's the viewpoint that Lurie chases after in the Survey.
• On the other hand, I've heard the complaint that we don't really know what's going on with the equivariant SHC because we have no 'guiding light', like we do with chromatic homotopy in the ordinary stable category. It would be interesting to try to understand this more carefully, since definitely there are equivariant spectra like MUG floating around. I recently went to a talk about Schwede where he considers a construction built on top of orthogonal spectra which are equivariant w/r/t all finite groups (in fact, he says you can make it work for all compact Lie groups) simultaneously. This causes certain constructions of, like, KU to express distinct 'global' homotopy types from one another. It would be neat to see which constructions of MU do what in this context.
• Geometric topology
• How does Baas-Sullivan theory work? How does one use it to construct truncated Brown-Peterson spectra?
• What is the Conner-Floyd conjecture? This is what Ravenel-Wilson settles, but it's not clear why anyone would care to start.
• There's a lot of geometric topology bound up in the positive parts of the Kervaire conjecture. For example, John Jones' thesis contains a construction of a Kervaire 30-fold, and the construction of the 14-fold is nontrivial.
• Structured ring spectra
• H_infty ring spectra
• (Namely, the H_infty structure makes the quotients of the formal group by finite subgroups interact well with level structures. This is discussed in Matt's thesis, both the MIT form and the published form, and then again in his various other works.)
• A standard reference for this stuff is a set of lecture notes by Bruner, May, McClure, Steinberger, where they also talk about applications to Nishida's nilpotence theorems in a language that smacks strongly of the techniques used by Hopkins to prove the general MU nilpotence theorems. This was the part of the argument that you had the biggest trouble understanding --- maybe reading this book would shed some light on it.
• A_infty / E_infty ring spectra
• What is an A_infty ring spectrum in the sense of Lurie? (I've read this definition, but it's just a sea of indices to me.)
• What is an an A_infty ring spectrum in the operadic sense? How does this compare to EKMM's S-algebras? How does this compare to Lurie's approach?
• What structure does an A_infty ring spectrum come equipped with? EKMM suggests that there are Kunneth and UCT spectral sequences associated to such cohomology theories. What operation information does the A_infty structure provide?
• All these questions are also important for E_infty ring spectra.
• How does E_infty rings and H_infty rings compare? If H_infty rings are all one needs to build a nice theory of power operations, what extra niceities do E_infty rings provide?
• Examples of such spectra, computations
• How do we construct truncated Brown-Peterson spectra as E_infty ring spectra using Lurie-like methods? Lawson says that multiple multiplicative structures exist on the unstructured spectrum BP<n>, which the Baas-Sullivan construction provides evidence for. (WARNING: trying to build BP itself as a structured ring spectrum has been a long-standing open problem. Matt says that he and Mike Hopkins have had discussions about resolving this by using Matt's work on power operations and showing that the cohomology operations on BP do not compose accordingly.)
• The H_infty structure is apparently what you need to construct power operations in cohomology, and the H_infty structure on MU is what Quillen uses to prove his theorem, on KU it's what Adams used to prove his theorem on the Hopf invariant, and on KZ/2 you recover Steenrod operations. How do we demonstrate that KU, MU, and HR are H_infty rings, and how do we compute the power operations associated to them?
• Actually, I've never seen a proof of Quillen's theorem. It would be nice to actually read the original papers, titled 'On the formal group laws of unoriented and complex cobordism theory', an outline paper, and 'Elementary proofs of some results of cobordism theory using Steenrod operations', the real deal. Then, it would be nice to recast these ideas in the modern language of E_infty ring spectra, or find a source that does exactly that.
• The nilpotence theorem
• What statements can be made about the homotopy of X(n), or the (moduli problem associated to the) stack associated to X(n)? Lawson gave an answer at http://mathoverflow.net/questions/40432/identifying-the-stacks-in-devinatz-hopkins-smith explaining in part the relationship between the Hopf algebroid attached to X(n) and M_fg; in particular he highlights a description of MU_* X(n) appearing in the paper. What does the ANSS converging to pi_* X(n) look like? Is it at all computable?
• What statements can be made about the geometry classified by the bottom approximating stacks in Part II of DHS?
• How is part III related to extended smash powers? Where, exactly, is the H_infty structure of KU used, and how?
• There's a remark in part II stating that the X(n) spectra are generalizations / cousins of some spectra also called X(n) previously used by Ravenel. What were the originals? Where and how were they used?
• (Really I have lots of questions about part III, but I haven't gone back to see what they are. :) )
• Rezk asserts that Mahowald showed that the Thom spectrum of the canonical bundle over Loops^2 S^3 is equivalent to HZ/2. This is neat, and important in Hopkins' proof of Nishida's theorem. Lawson mentioned where I can actually find a legible account of this at http://mathoverflow.net/questions/36345/construction-of-morava-e-theory ; look there for more clues.
• Strickland's take on improving the nilpotence proof
• Strickland's take on improving the nilpotence proof is to use Dwyer-Greenlees-Iyengar duality, which is supposed to give roughly a correspondence between Susp^infty_+ L^2 S^{2m+1} modules and spectra with coaction maps X --> X smash Susp^infty_+ Loops S^(2m+1), but there are finiteness problems.
• What it actually gives is a correspondence between R[n, k] = F(Susp^infty_+ J_{p^k-1} S^{2n}, S^0) modules and E[n, k] = F_{R[n,k]}(S^0, S^0) = Susp^infty_+ Loops J_{p^k-1} S^{2n} modules.
• Writing X(n, k) for the pullback of X(n) to J_k S^2n and X[n, k] for the p-localization of X(n, p^k-1), we get X[n, k] = F_R[np^k, 1](S^0, X[n, k+1]). This shows that X[n, k] is a module over E[np^k, 1], and this extends canonically over E[np^k, infty] = Susp^infty_+ L^2 S^{2np^k+1}.
• The crystalline period map
• Ando's course notes from Hopkins' course
• Hopkins-Gross
• Gross-Hopkins
• Devinatz-Hopkins
• Where does Brown-Cominetz duality turn up originally?
• Sullivan's Geometric Topology notes (started, unfinished)
• Bousfield and Kan - Homotopy Limits, Completions, and Localizations
• Hirschorn
• Grothendieck - Pursuing Stacks
• Lurie - DAG I - VI, HTT, Survey of Elliptic Cohomology
• Behrens - Constructing tmf, taf, buildings & elliptic curves
• Goerss - A history of tmf
• Ando's thesis on cohomology operations
• Miller, Ravenel, and Wilson
• Hovey and Ravenel's paper on how BO<2k> can't orient chromatic homotopy after a while (What might this even mean??)
• Goerss, Jardine - simplicial something or other
• Barwick - Applications of Derived Algebraic Geometry to Homotopy Theory
• Borceux, Janelidze - Galois Theories
• Kamps, Porter - Abstract Homotopy and Simple Homotopy Theory
• EKMM
• Stong - Notes on Cobordism Theory
• Conner, Floyd - The relation of cobordism to K-theories
• Buoncristiano, Rourke, Sanderson - A geometric approach to homology theory
• Thom's thesis
• Strickland asserts that part 3 of Thom's thesis is equivalent to showing that Aut Ga acts freely on spec H_* MO, so that the associated ASS has good properties. Is this the case? Work this out. (Cary basically did this in the final xkcd seminar; it's definitely in Thom's thesis.)
• What good properties does a free action induce on the ANSS, or any other relevant SS?
• Dugger - An Atiyah-Hirzebruch spectral sequence for KR-theory
• Voevodsky, Hopkins-Morel contain the slice spectral sequence for motivic homotopy theory
• Beware! This setting has a triangulated category to work with, which they close their Dror nullifying category under. HHR doesn't, which makes things subtler.
• Salch - Grothendieck Duality under Spec Z
• Deligne Rapoport - Artin's algebrization thm
• Project ideas
• The telescope conjecture and tmf
• Behrens believes that there's a counterexample to the telescope conjecture in tmf ^ tmf.
• Monodromy on the modified stack of elliptic curves
• Nick Katz took the stack M_Ell and deleted the points of height 2, then used monodromy to prove a variety of facts about the resulting reduced moduli space of height 1 (and 0?) elliptic curves. Translate these results to topology!
• http://press.princeton.edu/titles/7234.html
• Strickland's view of the image of J
• One should be able to write down a homotopy-cartesian cube of strictly commutative ring spectra, one corner of which is the K-local sphere, which we denote by J.
• The square J --> K --> KQ = J --> SQ --> KQ should relate well to the cube.
• The cube should provide a proof that the corner is indeed J.
• The cube should provide a calculation of pi_* J. The Bernoulli numbers should enter the calculation in a natural way as coefficients of the series log(t/(1-e^-t)), rather than as gcds of expressions like k^N(k^n-1)
• If we let j denote the connective cover J[0, infty) and write Ij for its Brown-Comenetz dual, then we have J/j = S^-2 Ij. Ideally, this should be visible from the cube. Note also that (L_K kO)/kO = S^-6 IkO, and (L_K kU)/kU = S^-4 IkU.
• The stable J-homomorphism k --> bglp_1 (S; K) should enter naturally into the picture.
• Everything should be done rationally or integrally, without separation of primes.
• Ideally, the quadratic and symmetric L-theory spectra should fit into the picture, as well as M ST op and possibly KZ.
• For a number field F it is known (by work of Borel) that K_* sheafO_F (x) Q is abstractly isomorphic to Q (+) S((K_*(R (x) F)/Z) (x) Q). Similar things seem to work if we complete at an odd regular prime rather than rationalizing. It would be good to have a more structured approach to all this. If I understand correctly, one can deduce that the map K(sheafO_F) --> kU actually factors through j (integrally!)
• Orientations of spectra
• Writing / Talks
• Haynes Miller's notes
• These need to be proofread!
• These need spectral sequences drawn out!
• These need diagrams done properly! Mahowald's diagrams are great, everyone else was a disappointment.
• Michael Donovan at MIT has offered to help clean these up. It would be nice to keep him in the loop, since you forced GIT onto him.
• XKCD talks
• 9/21 - K-theory
• Kinds of K-theory
• Vector bundles over spaces (topological K-theory)
• Projective modules over sheaves (algebraic K-theory)
• Prime fields in the p-local stable homotopy category (Morava K-theory)
• (Something is a field when E_* X is a free E_*-module for all spectra X.)
• If E is a field, then E has the homotopy type of a wedge of suspensions of K(n) for some fixed n. (!!)
• Connective K-theory?
• Twisted K-theory, twists by K-theory?
• Definitions of a vector bundle
• Projection-style definition with transition maps specified by coherent isomorphisms of vector spaces
• Principle U(n)-bundle
• Singular maps Pi_infty X --> VectorSpaces
• Building classifying spaces
• The classifying space of rank n bundles
• CP^infty as a K(Z, 2)
• Their colimit, BU, which classifies stable isomorphism classes of vector bundles
• The splitting principle
• Take U(n)-bundle, consider it as a C^n-bundle, convert to a CP^(n-1)-bundle by taking the sphere bundle and quotienting by the action of U(1). The cohomology of the old base injects into the cohomology of the total space of the CP^(n-1)-bundle.
• Pull back the old vector bundle along the CP^(n-1)-bundle. There's a new bundle over CP^(n-1) given by {(l, y) in CP^(n-1)-bundle x old bundle | y is in l}, and this bundle includes into the pullback bundle.
• Hence, the pullback bundle splits as a direct sum of this cross-bundle and its orthogonal complement, which is a U(n-1)-bundle.
• Calculation of H^* BU(n) and H^* BU
• To calculate H^* CP^infty, use the motivic decomposition CP^0 u CP^1 u ... and cellular cohomology. To get the ring structure, use the spherical fibration EU(1) --> BU(1) and the SSS.
• U(n-1) acts freely on EU(n), and we can show EU(n) / U(n-1) --> EU(n) x_{U(n)} S^{2n-1} has an inverse. The map is given by sending e in EU(n) to the pair (e, (0, ..., 1)). This has a U(n-1)-action inherited from EU(n), where we calculate g(eA) = (eA, s0) = (e, As0) = (e, s0) = g(e). Hence, EU(n) x_{U(n)} S^{2n-1} / U(n-1) ~~ EU(n) x_{U(n)} S^{2n-1}, and g descends along the quotient off EU(n).
• This target space is defined to be the sphere-bundle of the universal vector bundle over EU(n), so we get a fiber sequence S^{2n-1} --> EU(n) x_{U(n)} S^{2n-1} --> EU(n) / U(n) ~~~ S^{2n-1} --> BU(n-1) --> BU(n).
• So, we get a Serre spectral sequence of type (draw a sketch), and analysis of this spectral sequence shows that our new H^* BU(n) looks like H^* BU(n-1), possibly with a new class corresponding to the total Chern class of CP^infty x ... n times ... x CP^infty. And, sure enough, symmetric function theory says that this class exists and is non-torsion.
• In the end, H_* BU = Sym(H_* CP^infty). This works for any even periodic ring spectrum.
• The homotopy of BU (Harris)
• First, BGr ~~ U by the exponential twisting map.
• Then, define the following three categories:
• E: objects inclusions C^n --> C^infty, arrows should be unitary automorphisms of C^infty commuting with the inclusions.
• F: objects integers, arrows should be unitary automorphisms of C^n
• D: objects n-dim'l subspaces of C^infty, arrows should be identities
• We have maps D <-- E --> F which are equivalences of categories (E --> D forgets the 'frame' structure, noting that for any two frames the unitary automorphism taking one to the other is unique, so we're throwing out a contractible subcategory. E --> F forgets the immersion structure, so we're throwing out the contractible space of n-frames in C^infty)
• Moreover, we have simplicial objects E_*, F_*, D_* where E_n consists of n-tuples of orthogonal objects in E, similarly for F and D, and the equivalence of categories induces a homotopy equivalence on birealization. Hence, we have |D_*| weakly equivalent to BGr, |F_*| weakly equivalent to B(u_n BU(n)).
• Then, B(u_n BU(n)) ~~ B(BU x Z) by group completion. (Generalization of Loops B Monoid = Group.)
• This gives the K-groups of a point: Z[u^{pm 1}]
• Snaith's theorem
• So, BBU ~~ U, or equivalently Loops^2 BU ~~ BU. Tinkering with adjoints gives a selected virtual line bundle S^2 --> BU which we've inverted, corresponding to the Hopf fibration.
• There is a map Suspend^infty_+ CP^infty --> KU classifying the tautological line bundle (which restricts to the Hopf fibration), and hence a map from the localization Suspend^infty_+ CP^infty[beta^-1] --> KU. Moreover, the homotopy of homology of CP^infty looks like the homotopy of BU, and Snaith played around with this fact until he found that this localization map is a homotopy equivalence.
• Gepner has since given a more conceptual proof, which relies on the following fact: R_* CP^infty ~~ Add(BU, Loops^infty R), where R is an even periodic ring spectrum. This isomorphism on cohomology groups lifts to a map on cochain spectra map(Suspend^infty_+ CP^infty, R) --> map(Suspend^infty BU x Z, R), after which point the proof is fairly easy.
• 10/12 - The Geometry of Formal Varieties in Algebraic Topology
• Notes posted on UC-B website.
• 11/30 - Hopf rings in algebraic topology, Ravenel-Wilson on K(n)_* K(Z, *)
• Pick two ring spectra E and F so that E has Kunneth isomorphisms over the representing spaces for F. Then E_* F_* is a coalgebraic ring, i.e., a Hopf ring.
• We have the following maps:
• Addition: E_k F_n x E_k F_n --> E_k F_n (addition on E)
• *-product: E_k F_n x E_j F_n --> E_{k+j} F_n (addition on F)
• o-product: E_k F_n x E_j F_m --> E_{k+j} F_{n+m} (multiplication on F)
• Diagonal: E_k F_n --> E_* F_n (x) E_* F_n (diagonal map on F)
• We also have the following axioms:
• * and o are coalgebra maps, so are +-linear and distribute over the diagonal
• * and o are related by a distribution law: if psi a = sum a' | a'', then a o (b * c) = sum (a' o b) * (a'' o c) (up to a -1?)
• So, we can produce huge quantities of elements of E_* F_* using only a few elements to start, then applying * and o products.
• Morava K-theory has a Kunneth isomorphism, a coefficient ring Fp[vn^{pm 1}], and a p-series [p](x) = vn x^{p^n}. Let's compute K(n)_* K(Z, q) for all q.
• p-locally, we have an equivalence K(Z, q) ~~ K(Z/p^infty, q) ~~ lim K(Z/p^j, q), so we'll calculate K(n)_* K(Z/p^j, q) individually inductively.
• Start with j = 1, q = 1. BZ/p occurs as the fiber of CP^infty --p-> CP^infty, a map whose action we've understood by construction of K(n). Namely, delooping gives a spherical fibration S^1 --> BZ/p --> CP^infty (or, this is the homotopy fiber of the above fiber). This gives a Serre spectral sequence / Gysin sequence K(n)_* BZ/p --> K(n)_* CP^infty --Phi-> K(n)_* CP^infty --partial-> K(n)_* BZ/p. Phi(y) = y cap [p](x), x the orientation class of K(n)^* CP^infty, whose action we calculate: Phi(beta_k) = beta_{k-p^n}, so it's surjective, so partial vanishes, and K(n)_* BZ/p is the kernel, i.e., beta_k with 0 <= k < p^n.
• Take the model for BG which has as its 0-cells the elements of G. Then BG admits a filtration BG_k consisting of the m-simplices, m <= k. The filtration quotients are Suspend^k G^{smash k}, and d_1 of the associated spectral sequence is the bar resolution of K(n)_* BZ/p in the category of K(n)_*-comodules, so E_2 of this spectral sequence is H_{*, *} K(n)_* BZ/p = Tor^{K(n)_* BZ/p)}_{*, *}(K(n)_*, K(n)_*). The cup product of K(Z/p, *) respects this filtration, and hence descends to a product on the spectral sequence of type E_{*, *}^s K(Z/p^j, q) (x) K(n)_* K(Z/p^j, q') --o-> E_{*, *}^s K(Z/p^j, q+q'), which respects the differentials! So, this gives us an immense amount of control over the regions of the spectral sequence which can be described using o-products.
• Compute H_{*, *} K(n)_* K(Z/p^j, 0) = H_{*, *} K(n)_*[Z/p^j] = Lambda[(x)] (X) Gamma[(x^{p^{j-1}(p-1) | x^{p^j})]. The only differential is d^{2p^{nj}-1}(x^{p-1} | x)^{[p^{nj}]} = c(x).
• The general structure of the bar spectral sequence is similar. E^2 = ((X)_{0 < I_1 < .. < I_{q-1} < n-1} Lambda[(a_I)]) (x) ((X)_{0 <= I_1 < ... < I_{q-1} < n-1} Gamma[(a_I^{p-1} | a_I)]). a_I is represented in E^infty by (a_tilde I^{p-1} | a_tilde I)^{[p^{I_1}]}. Let 0 <= I_1 < ... < I_{q-1} < n-1 and set J = (I_2 - I_1 - 1, .., I_{q-1} - I_1 - 1, n - 1 - I_1 - 1); the differentials are described by d (a_I^{p-1} | a_I)^{[p^{I_1+1}]} = r_I (a_{I+1}) for various units r_I. The E^infty page is a free K(n)_*-module and is even a free K(n)_*-algebra on the a_I, modulo the action of the Frobenius. The map -^p acts as the Frobenius, and the Verschiebung is determined by V(a_I) = a_{I-1}.
• The computation of the E^2 is not hard. What's tricky is the computation of the action of the circle product on it; have (a_tilde I^{p-1} | a_tilde I) o a_{(I_1+1)} = (-1)^{q-2} a_I. This is tough! But, once we have that, we can uncover the action of the o-product on divided powers and then use the differentials in the previous spectral sequence to describe the differentials here. Then, the fact that a Verschiebung exists means that the Frobenius is determined by the Frobenius on K(n)_* BZ/p by F(a_I) = F(V(a_{(I_1, ..., I_{q-1})+1)} o a_{(I_q)}) = a_{(I_1, ..., I_{q-1})} o a_{(I_q)}^p, and so we get to solve the multiplicative extension problem immediately.
• The case of K(Z/p^j, q) is related to the case of K(Z/p^{j-1}, q) by the map Z/p^j --> Z/p^{j-1}. Its action on the K(n)-homology of classifying spaces, one can compute, is the same as the Frobenius. :) So these same inductive techniques work to compute K(n)_* K(Z/p^j, q), and hence the limit K(n)_* K(Z/p^infty, q), and by duality K(n)^* K(Z/p^infty, q).
• In the language of formal groups, we get a really nice expression: K(Z, q)_K(n) = Lambda^q CP^infty_K(n)[p^infty]. This also gives a description of the formal scheme represented by the homology ring: K(Z, q)^K(n) = hom(Lambda^q CP^infty_K(n)[p^infty], Gm), which corresponds to the space of q-variate alternating, multiexponential power serie taking values on the points of nilpotent subscheme of the formal group associated to K(n).
• It's NUTS that this is actually computable!!
• 2/19 - Introduction to the Goodwillie Calculus of Functors
• These notes basically turned into a blog post: http://chromotopy.org/?p=614
• The mod-2 homology of connective covers of BU
• What is BU<2k>? MU<2k>?
• The Postnikov-Whitehead tower: BU<6> --> BU<4> --> BU<2> --> BU<0> as a sequence of fibers of maps BU<2k> --> K(pi_2k BU<0>, 2k) = K(Z, 2k). These are selecting certain integral cohomology classes of BU, and so when a classifying map X --> BU sends these characteristic classes to 0, it lifts through the connective covers.
• Identifications of the bottom spaces: BU<0> = BU x Z. BU<2> = BU. BU<4> = BSU.
• Computing the homologies of BSU and BU<6> using the Serre spectral sequence and the fibrations K(Z, 2k-1) --> BU<2k+2> --> BU<2k>
• MU<2k> the associated Thom spectrum, the cobordism theory of manifolds with a trivialization of the first k-1 Chern classes of the tangent bundle.
• Singer's calculation of H^*(BU<2k>; Z/p)
• H^*(BU<2k>; Z/p) = H^*(BU; Z/p) / (Z/p[theta_2i | sigma_p(i-1) < k-1]) (x) prod_{t=0}^{p-2} F[M_{2k-3-2t}]
• F[M_...] is some kind of Steenrod subalgebra of H^* K(Z/p, *)
• This is a pretty gross ring. But, at least when k <= 3, it is even-concentrated, and so we get a collapse of the Atiyah-Hirzebruch spectral sequence and whence the map RingSpectra(BU<2k>, E) --> Alg_{pi_0 E}(E_0 BU<2k>, pi_0 E) is an isomorphism.
• So, to understand multiplicative maps MU<2k> --> E of ring spectra, it suffices to understand what functor E_0 MU<2k> represents.
• Constructing C^k(Ga; Gm)
• What does the k-fold tensor product of line bundles look like? Use (L-1) over CP^infty, the universal case. Find that the ith Chern classes vanish for 0 <= i < k, and hence we produce a lift (CP^infty)^k --> BU<2k>. On homology, this induces a map E_* (CP^infty)^k --> E_* BU<2k>, which by the universal coefficient theorem is an element of E^*((CP^infty)^k; E_* BU<2k>) = E_* BU<2k>[[x1, ..., xk]]
• This power series satisfies various properties:
• Rigidity: f(x1, ..., 0, ..., xk) = 1. (Tensoring against 0 is zero.)
• Symmetry: f(x) = f(sigma x). (Tensor product is symmetric.)
• 2-cocycle: f(x1, ..., xk) f(x0 +_E x1, ..., xk)^-1 f(x0, x1 +_E x2, ..., xk) f(x0, x1, x3, ..., xk)^-1 = 1. (Pull back along various projection and multiplication maps, note some isomorphisms of line bundles.)
• What is the group scheme C^k(CP^infty_E; Gm), corresponding to the set of such power series?
• Reduce to the case of rational and mod-p ordinary homology
• We want to study E_* BU<2k> and C^k(CP^infty_E; Gm), where E is complex orientable.
• MU is the universal complex oriented theory, so we reduce to the case of MU_* BU<2k> and C^k(CP^infty_MU; Gm).
• MU are BU<2k> are even-concentrated (for k <= 3), so the Atiyah-Hirzebruch spectral sequence H_*(BU<2k>; pi_* MU) ==> MU_* BU<2k> collapses.
• So, really, we want to understand H_* BU<2k> for various ordinary theories H. CP^infty_H is the formal additive group, so we're interested in the space of power series f(x1, ..., xk) satisfying the above 3 conditions, with x +_E y replaced with just x + y.
• The Hasse local-to-global principle states that it's enough to study H_* BU<2k> and C^k(Ga; Gm) with Q-coefficients and Z/p-coefficients
• Zariski tangent space and the reduction to additive cocycles
• Formal groups are kind of like Lie groups. If you want to understand a Lie group, you start by understanding its tangent space at the identity.
• TX = hom(spec k[eps]/eps^2, X) is the definition of the Zariski tangent bundle. The Zariski tangent space at the identity consists of diagrams {spec k --unit-> X, spec k --inclusion-> spec k[eps]/eps^2 --?-> X}.
• What is the Zariski tangent space of C^k(Ga; Gm) at the identity? It's the space of a multiplicative 2-cocycles over k[eps]/eps^2 whose reduction mod eps is 1. So, f = 1 + eps u(x1, ..., xk), and we check that our multiplicative cocycle condition becomes u(x1, ..., xk) - u(x0 + x1, ..., xk) + u(x0, x1 + x2, ..., xk) - u(x0, x1, x3, ..., xk) = 0.
• This is the same as the formal scheme C^k(Ga; Ga), which has been studied before --- C^2(Ga; Ga) appears in Lazard's symmetric cocycle lemma!
• Equivalently, we get surjective maps C^k(Ga; Ga) -->> TC^k(Ga; Gm) and H^*(Ga; Ga) -->> H^*(Ga; Gm).
• Calculation of C^k(Ga; Ga) (Lazard k=2 1955, AHS k=3 1995-2001, HLP k > 3 2008)
• The rational case: zeta^n_k = d^-1 sum_{nonempty X subset {x1, ..., xk}} (-1)^|X| (\sum_{x in X} x)^n = (gcd_{0 not in lambda} (n choose lambda))^-1 sum_{0 not in lambda} (n choose lambda) tau(lambda), where d is the largest positive integer that leaves the thing integral.
• The modular case: Frobenius 2-cocycles
• Gluing and the big picture. Use n = 12, 3 <= k <= 6.
• The big theorem: Select a power-of-p partition lambda of n with length k. Let T^m lambda denote the set of all possible partitions of the form G_{i1 j1}...G_{im jm} lambda. Then, if either m <= p - 2 or if lambda is the shortest power-of-p partition of n, then polynomial sum_{mu in T^m lambda} c_mu tau(mu) is a cocycle, where c_mu is the coefficient of tau(mu) in zeta^n_{k-m}. In addition, cocycles formed in this manner give a basis for the space of modular cocycles.
• All other rings: Hasse local-to-global principle.
• Multiplicative calculations (Mumford <=1969, Breen <=1983, AHS 1995-2001, HLP 2009)
• The rational case: use the exponential!
• The modular case... kind of use an exponential. The Artin-Hasse exponential gives a multiplicative extension for zeta^n_k when nu_p n <= nu_p phi(n, k).
• H^*(Ga; Ga) -->> TH^*(Ga; Gm) as a spectral sequence. Filter H^(Ga; Gm) by lowest degree, get H^*(Ga; Ga) quotients and a filtration spectral sequence starting from H^*(Ga; Ga) and converging to H^*(Ga; Gm). This corresponds to trying to iteratively extend an additive cocycle to a multiplicative one, one degree at a time.
• Cohomological calculations (Lubin-Tate 1966, Hopkins <=1998): H^*(Ga; Ga) = (x)_i F_2[a_i] over Z/2, (x)_i (R[b_i] (x) Lambda[a_i]) over Z/p, p odd.
• Differentials: we're interested in H^2(Ga; Ga). Over F2, classes are of the form ai aj for distinct i, j, representable as x^{2^i} y^{2^j}, get a nontrivial differential d_{2^i + 2^j}(k a_i a_j) = k^2 a_i^2 a_{j+1} - j^2 a_{i+1} a_j^2. Over Fp, same shit, get a nontrivial differential d_{p^i + p^j}(k a_i a_j) = k^p(a_{i+1} b_{j+1} - a_{j+1} b_{i+1}).
• Symmetry and graded commutativity in H^*(Ga; Ga)
• Biextension geometry and the half Weil pairing: Factor the multiplication-by-p map as L --i-> L^{(x) p} --left mult-> L. Then we get triangles L --> p_* L --> L^{(x) p} and L^{(x) p} --> (p x 1)^* L --> L. The universally supplied maps are isomorphisms, and this turns out to mean that e = prod_{i=1}^{p-1} u(ix1, x1, ..., x{k-1}) is a (k-1)-variate multiplicative 2-cocycle with delta_1 e = u^p.
• Obstruction framework, the 2-local calculation: Every additive cocycle u_+ over a ring S of positive characteristic can be written in the form u_+ = sum_{n, m, l(I) = k-3} r_{n, m, I} zeta_2^n x_3^m (x4, ..., xk)^I, where r_{n, m, I} is an element of S. If r_{p^n, p^m, I} != r_{p^m, p^n, I} for any n, m, I, then any multiplicative 2-cocycle 1 + b u_+ + o(|u_+|) must have b^p = 0. PROOF: The Weil half-pairing descends to the additive linear term and to cohomology classes. So, since e is a multiplicative cocycle, [e_+] shouldn't have any nonzero terms of the a_i a_j, i != j. e sends zeta_2^n to zeta_1^n when n is a power of p and to 0 otherwise and only operates on the first two variables.
• Example: use beta_{1, 12} = tau(10, 1, 1) - tau(9, 2, 1) and beta_{2, 12} = tau(6, 3, 3) to show that neither extends individually, but their sum does with the Artin-Hasse exp. However, no information about tau(1, 1, 1).
• But, in the 2-local case, when we're always next to a power-of-2 entry, we can use this to calculate C^k(Ga; Gm)(Z_(2)) = Z_(2)[zn | nu_2 n <= nu_2 phi(n, k)] (x) Gamma[b_{n, i} | 1 < i <= D_{n-1, k-1}] (x) Gamma[b_{n, 1} | nu_2 n > nu_2 phi(n, k), D_{n-1, k-1} != 0.]
• An action of the Steenrod algebra (Singer 1967, HP 2010-?), Topological considerations (AHS 1995-2001, HP 2009-?)
• There's an evident action C^k(Ga; Gm) x_S Aut Ga --> C^k(Ga; Gm), which dualizes to R_k --> R_k (x) A_*, where A_* is the dual of the Steenrod algebra
• The AHS map BU<2k>^HZP --> C^k(Ga; Gm) is a map of comodules over this algebra, which rigidifies it significantly if we understand the coaction on both sides
• The calculation of the divided power part is really not hard at all. (1+x)(1+y) = 1+x+y when xy=0. Free part: who knows!
• Computational observation: the number of indecomposibles in H^*(BU<2k>; F2) matches the number of primitives in H_*(BU<2k>; F2), which matches the number of basis elements in C^k(Ga; Gm)(F2) after deleting the Steenrod algebra generated by the odd dim'l classes from the cohomology of BU<2k>. This lends serious credence to the idea that the AHS map is an isomorphism onto its image of coalgebras over the dual of the Steenrod algebra.
• It would still be nice to attach a homotopy type X to the problem, so that X^E is C^k(CP^infty_E; Gm). Ideas about this: Wilson's BP<n>_k and the wedge sum splitting of ku away from 2 sounds relevant to Hopkins. Reduction towers due to Postnikov, Priddy, and Zabrodsky might be relevant.
• Connection to Weil pairings: E_* K(Z, *) is supposed to be like hom(Lambda^* CP^infty_E[p^infty], Gm), and K(Z, *) is the fiber of BU<2*+2) --> BU<2*>. In fact, K(Z, 3)^E is shown to be the cofiber of the map of group schemes BU<4>^E --delta1-> BU<6>^E.
• Application to elliptic spectra
• Elliptic ring spectra (spectrum with spf E^* CP^infty modeled by an elliptic FG)
• The theorem of the cube: every formal group E hat stemming from an elliptic group E has a unique element of C^3(E hat; Gm) attached to it.
• C^3(E hat; Gm) is the same as pi_0 spectra(BU<6>, E) is the same as pi_0 spectra(MU<6>, E) is the same as a homotopy class of maps MU<6> --> E, called the sigma-orientation.
• This induces a map MU<6>_2n(pt) --> E_2n(pt), a genus
• 11/15 - Extraordinary homotopy groups
• Morava K-theories
• Bousfield localization of spectra
• K(n)-localization as restriction to a quasiclosed substack
• Pic(hS) = Z; H_*(X) = Z in some dimension k, Z --> S^k a homology iso via the Postnikov tower, then Z = S^k v A. Z^-1 = S^-k v B, and S = Z ^ Z^-1 = S v A v B v (A ^ B), so A = B = pt.
• the Adams splitting K^_p = (+) Susp^* integral-K(1), plausibility
• Pic(L_K(1) hS) = Z_p x Z/(2p-2)
• Z_p^* --> Pic_n in general
• the algebraic Picard group and H^1(Sn; WFp^n[[u1, ..., u(n-1)]])
• the spectral sequence H^**(Sn; K_{n, *}(X)) ==> pi_* L_K(n) X
• ---------
• the Hopkins-Morava dictionary
• MU // moduli of formal groups, which is an infinite dimensional object with a unique subobject Sn for each codimension n
• --------
• p-local spectrum // qc sheaf or complex of qc sheaves
• finite p-local spectrum // finite cplx of qc sheaves
• smash product // tensor product
• homotopy groups // hypercohomology
• homotopy classes of maps // RHom
• function spectra // sheaf RHom(A^*, B^*)
• p-local sphere spectrum // O_S
• category C_n // subcat of c sheaves supported on S_n
• K(n) // total quotient field of Sn \ (Sn n U(n+1))
• functor L_n // functor Ri_n* . i_n^*
• functor L_K(n) // completion of Un along Un n S(n-1)
• chromatic tower // Cousin complex
• Snaith's theorem in Lurie's framework
• Formal varieties are supposed to be functors from adic R-algebras to products of copies of their set of topologically nilpotent elements. Formal schemes are supposed to just be general functors off adic R-algebras. Derived algebraic geometry sits in the middle of algebraic geometry and homotopy theory, namely oo-categories. The main idea of oo-categories is to replace the word 'set' everywhere with 'space' -- i.e., a model for oo-categories are ordinary categories enriched over the category of spaces. Similarly, we should expect derived algebraic geometry to be functors off adic R-algebra spectra to spaces.
• Similarly, a derived group X-scheme is a group-valued functor from the category of X-schemes to the category of spaces. (Often, we want the representing X-scheme G to be flat over X so that we can form well-behaved fiber products G x_X G.)
• Cohomology operations in K-theory and other generalized cohomology theories (Ando)
• Let F be an FGL over an m-adically complete Noetherian ring R. F(m) is a group with addition specified by F; let H be a finite subgroup. Then the quotient F(m) --> F(m)/H is realizable as F(m) --> (F/H)(m), set pi_H(x) = prod_{h in H} (h +_F x). This turns out to be a map of FGLs; F/H exists, and f_H is a map F --> F/H with kernel H. (Lubin)
• Moreover, if you have a tower of inclusions H c G c F of finite subgroups G, H of F, then the quotients work out as f_G = f_{G/H} f_H, or (F/H) / (G/H) = F/G.
• Generalized twisted cohomology theories, the Umkehr map, and chromatic interactions between analysis and topology. (What I would have talked about here is now a published paper of Ando, Blumberg, and maybe Gepner.)
• Spectral sequences flipbook
• Introduction: The filtration spectral sequence
• The Atiyah-Hirzebruch spectral sequence
• Cellular cohomology, cup product preservation, H^* CP^infty
• Exotic cohomologies of K(G, n)s?
• Example 7.4 in Strickland's bestiary concerns KU^* BZ/2, which collapses for degree reasons but has interesting filtration behavior. This is worth exploring fully to point out the relevance of filtrations.
• The Serre spectral sequence
• Computing the differentials for spherical fibrations, the Hopf fibration
• The cohomology of loopspaces and Serre's calculation for K(Z/2, *)
• Maps of spectral sequences and the Kudo transgression theorem
• The cohomology of U(n), BU(n), U, BU, SU(n), SU, BSU
• To get U(2), use the splitting U(1) --diagonal-> U(2) --det-> U(1), which is x |-> x^2, not the zero map.
• To get U(n), use degree arguments and d(y1 en) = d(y1) en + y1 d(en) = 0.
• To get SU(n), use S^1 --> U(n) --> SU(n).
• To get BU(n) and BSU(n), use the G --> EG --> BG fibration.
• Postnikov towers and the connective covers of BU
• Postnikov towers and some homotopy groups of spheres
• The Eilenberg-Moore / Rothenberg-Steenrod spectral sequences
• The Eilenberg-Moore spectral sequence: recomputing Serre spectral sequence examples with the square F --> E --> B, F --> pt --> B
• Fiber sequences involving groups, like U(n) --> 1 --> BU(n)
• The James spectral sequence: the James construction has a filtration whose E_2 page looks like a Tor
• The Rothenberg-Steenrod spectral sequence for G --> EG --> BG: Tor_{E_* G}(E_*, E_*) ==> E_* BG.
• Pullback square S^(2n+1) --> CP^n --> CP^infty and S^(2n+1) --> 1 --> CP^infty
• Spectral sequences associated to (co)simplicial objects
• The Mayer-Vietoris spectral sequence, and the simplicial object associated to a cover
• The bar spectral sequence: Hopf rings, Tor groups, and another perspective on the homology of E-M spaces
• The descent spectral sequence: sheaves of spaces, Cech cohomology
• Cohomology of Hopf algebroids, the stack of multiplicative formal groups, tilman bauer / charles rezk's computations of pi_* tmf
• The May / Bockstein spectral sequences
• pi_* ko from Ext_{A(1)}(k, k).
• pi_* ku from Ext_{A(1)}(k, k).
• The chromatic spectral sequence, continuous group cohomology and the Morava stabilizer group
• The Morava spectral sequence H^**(Sn; K_{n, *}(X)) ==> pi_* L_K(n) X
• The tangent spectral sequence: formal groups and tangent spaces, H^*(F; TG) ==> H^*(F; G)
• The homotopy fixed point spectral sequence
• The signature of this SS is H^p(G; pi_q X) ==> pi_{p-q} X^{hG}. It comes from filtering EG_+ in X^{hG} = Hom(EG_+, X). Alternatively, X can be written as a homotopy limit diagram over some category built from G (but I forget how this goes).
• There is a C_2-action on KU given by complex conjugation, and KU^{hC2} = KO.
• Related computational techniques
• Computing Tate resolutions, computing minimaler resolutions with Hopf algebra quotients, computing pi_* ko and pi_* ku
• Resolving by Hopf algebra quotients
• --- unsorted ---
• The EHP spectral sequence
• Compositions of derived functors. Some calculations in group cohomology, maybe?
• Massey products in spectral sequences
• Homotopy spectral sequences of spaces (details with pi_0 and pi_1, see _Homotopy limits, completions, and localizations_ by Bousfield and Kan)
• Unbased spectral sequences (choose basepoints iteratively, kind of. See Bousfield's _Homotopy spectral sequences and obstructions_, which does this for unbased cosimplicial spaces.)
• Dan Dugger has a paper titled _Multiplicative structures on homotopy spectral sequences_. Could be worth looking at.
• Edge homomorphisms in the Serre spectral sequence coming from (F --> F --> pt) ==> (F --> E --> B) ==> (pt --> B --> B).
• http://neil-strickland.staff.shef.ac.uk/courses/bestiary/ss.pdf
• A Primer in Stable Homotopy Theory
• TopTox talks
• 10/11 - Cohomology
• Hom(C^*, Z)
• Cohomology of RP^2 using duals to singular chains
• The simplicial cup product and S^1 x S^1
• Axiomatic cohomology and the Eilenberg-Steenrod axioms
• Cohomology is insensitive to homotopy.
• (X\U, A\U) --> (X, A) is an iso in cohom.
• H^*(coprod X_a) = (+) H^* X_a
• A --> X induces H^* A <-- H^* X <-- H^*(X, A) <-- ...
• Cech cohomology: f: U_I --> G, df(U_I) = sum_i (-1)^i f(U_I with ith slot removed)
• Covering S^1 by 1, 2, and 3 opens, their Cech cohoms
• Cech cohomology with twisted coeffs / coeffs in a sheaf
• 'Leray acyclicity' and a comparison <--> ordinary cohom
• The Cech product: (f.g)(U_I) = f(U_I') . g(U_I'')
• All maps S^2 --> T are cohomologically null
• The cap product?
• Poincare duality on oriented mflds
• Miscellany
• Prime sized sets
• What's the limiting value of sum_{p prime, p <= n} (n choose p) / 2^n ?
• Is there an analytic function f(x): R --> R such that f is asymptotically equivalent to this sum?
• Discarded things I don't want to delete
• Quasicategories as moduli stacks
• Demonstrate that there's a reasonable sense in which infty-Func(Pi_infty X, Spaces) ~~ Spaces_/X.
• Produce a good model of Spaces_/X
• Understand homotopy colimits of the form Pi_infty X --> Spaces
• EKMM uses the two-sided bar construction
• Shulman's paper should also have something relevant
• Lurie's appendix shows that homotopy colimits should compute quasicategorical colimits for quasicategories built from particular kinds of model categories -- you *need* to understand what this says to proceed
• From this produce a map of quasicategories SSetMaps(Pi_infty X, Spaces) --> N(Spaces \downarrow X)
• Change the current argument to demonstrate essential surjectivity
• Finish the proofs of fullness and faithfulness
• Can we say something like this for arbitrary categories instead of Spaces?
• Like colimits in G-torsors c--> free G-spaces should produce principal bundles with structure group G
• Or the colimit of the trivial functor into Spectra should produce the suspension spectrum of X
• How does the transfer map fit into this picture? Ando had some compelling argument that I've forgotten.
• The answer should be YES: for simplicially enriched model categories, the bar construction exists!
• What are Lurie's straightening and unstraightening constructions? How are they related to straightening and unstraightening of model categories?