Chapter 1 - p8 (1.1.11) should be $(1+mm^*)(\alpha^+-m\alpha^-)$. Chapter 2 - p36 I think the displayed equation between (2.1.17) and (2.1.18) should be $T_{ij}(\delta)=1-F_{ij}\delta^2+O(\delta^3)$ as $\delta\to0$. - p57 Last equation, second term on RHS should be $ud_A^*a$. - p59 Theorem 2.3.8, we might need the stronger assumption $||F_{\tilde A}||_{L^2(B^4)}<\epilson_2$, and it seems we can only prove a nonlinear bound $||\tilde A||_{L_\ell^2(D)}\le f_{\ell,D}(||F_A||_{L^2(B^4)})$ for some function $f_{\ell,D}$ with $f(0)=0$. These changes will not affect later developments. - p61 Lemma 2.3.11 the bound in the condition should be $||B||_{L^4}<\eta'$. - p61 The second to last displayed equation should add a term $C||F||_{L^2}$ to the RHS. - p62 The three-line displayed equation should be $$\delta(a)=\delta(\psi\tilde A)=\phi\delta(\tilde A)+(d\psi\wedge\tilde A)^+-(d\psi,\tilde A)=-\psi(\tilde A\wedge\tilde A)^++(d\psi\wedge\tilde A)^+-(d\psi,\tilde A)=-(\tilde A\wedge\alpha)^++(d\psi\wedge\tilde A)^+-(d\psi,\tilde A).$$ The last displayed equation should add $||(d\psi,\tilde A)||_{L_1^2}$ to the RHS. The next sentence on the next page should be "the last two terms are bounded by a multiple of $||\tilde A||_{L_1^2}$." - p68 Displayed equation missing a parenthesis. Chapter 3 - p82 Seventh line $Ker(\beta_x)$ should be $Ker(\beta_{x_0})$. - p82 Proof of Lemma 3.1.20, $\theta$ should be $\alpha$ and $\phi$ should be $\beta$. Chapter 4 - p132 Fourth line is not true. A non-simply-connected example is given by a flat connection on a complex line bundle over $S^1$ whose parallel transport around $S^1$ is an irrational fraction of $U(1)$. See also https://math.stackexchange.com/questions/2612912. - p132 Proposition 4.2.9 third line $h(a)$ should be $A+a$. - p132 The second to last line is not true. For example take the structure group to be abelian and the connection to be reducible. - p134 Proposition 4.2.15 $c$ should be a nonzero class in $H^-(X;\mathbb{Z})$ (this depends on the conformal structure on $X$). - p136 The paragraph above Proposition 4.2.19, second line, $\phi'\colon U\to V_0$. - p144 Displayed equation after (4.3.7), missing a subscript, $\psi_1(t)$. - p148 (4.3.16) and the equation above it should both be $F^+(A)-mF^-(A)$ instead of $+$. - p149 Corollary 4.3.19(i) second line, should be contain no reducible connections instead of no irreducible ones. - p153 Fourth displayed equation, LHS $(d\alpha/dt)$ should be $(d\alpha_t/dt)$. Next line the derivative of $*_t$ should be $-2m$. Two lines below that, missing a parenthesis. - p154 Second displayed equation, RHS second term should be $m(F_{A_0}^-)$. - p159~160 Lemma 4.4.5, 4.4.6, 4.4.7 are incorrect as stated. There may be topological obstruction to extending $\tilde u_\alpha$ (as in Lemma 4.4.5) to global gauge transformations. Although ultimately the gauge transformations we are patching can be chosen to be extendable, it is not clear (as in Lemma 4.4.7) why patching two extendable gauge transfromations gives another extendable one. An alternative approach for Proposition 4.4.9 is to use elliptic regularity for ASD connections (Theorem 2.3.8) to obtain $L^p (p>2)$ bound on curvature away from singularities and follow Uhlenbeck's original proof. Chapter 5 - p174 Three lines above Proposition 5.1.4, we should at least assume $Y\subset Z$ is a cofibration. - p175 (5.1.5) first line extra underline on $X$. - p176 (5.1.9) should be $X$ instead of script $X$. - p180 should a give different name to the universal covering space of $\Omega^2S^3$ (appeared 4 times). - p187 Second to last displayed equation first line, RHS missing a minus sign. - p188 Second to last displayed equation, first and last $\phi_t$ should be $D_{A_t}$. - p189 Third displayed equation first line both $\slashed\partial_\Sigma^*$ should be $\slashed\partial_\Sigma$. - p190 Last paragraph fourth line wrong font for $\mathbb{C}$. - p192 The paragraph above Lemma 5.2.9, first line "to many" should be "too many". - p193 The paragraph below (5.2.11), last line $\mathcal{B}^*|_{v(\Sigma)}$ should be $M_k^*$. - p205 Last displayed equation RHS missing a direct summand $\mathbb{R}$. Chapter 7 - p282 Second to last paragraph fifth line "Grassmannian" misspelled. Chapter 8 - p318 Two lines below (8.1.2), the reference should be to Section 4.4 instead of 4.3.