5 · algebraic topology and global obstructions

cech-de rham cohomology

The de Rham cohomology groups $H^k_{\mathrm{dR}}(M)$ arise from the complex of global differential forms. An equivalent combinatorial description is provided by Cech cohomology with values in the constant sheaf $\underline{\mathbb{R}}$. Given an open cover $\mathfrak{U}=\{U_i\}$ of $M$, a Cech $k$-cochain assigns to each ordered intersection $U_{i_0}\cap\cdots\cap U_{i_k}$ a locally constant real function. The Cech differential is the alternating sum of restrictions, and the resulting cohomology $H^k(\mathfrak{U};\underline{\mathbb{R}})$ is independent of the cover once a refinement limit is taken.

definition (de rham cohomology). $$ H^k_{\mathrm{dR}}(M) =\frac{\ker\bigl(d:\Omega^k(M)\to\Omega^{k+1}(M)\bigr)} {\operatorname{im}\bigl(d:\Omega^{k-1}(M)\to\Omega^k(M)\bigr)}. $$ closed forms that are not exact represent nontrivial classes.

definition (cech cochains and differential). let $\mathfrak{U}=\{U_i\}_{i\in I}$ be an open cover of $M$ and let $\mathcal{F}$ be a sheaf of abelian groups on $M$. a Cech $k$-cochain with values in $\mathcal{F}$ assigns to every ordered $(k+1)$-tuple $(i_0,\dots,i_k)$ a section $$ \sigma_{i_0\dots i_k} \in\mathcal{F}\bigl(U_{i_0}\cap\cdots\cap U_{i_k}\bigr) $$ (vanishing when the intersection is empty). the Cech differential $\delta:C^k(\mathfrak{U};\mathcal{F})\to C^{k+1}(\mathfrak{U};\mathcal{F})$ is $$ (\delta\sigma)_{i_0\dots i_{k+1}} =\sum_{j=0}^{k+1}(-1)^j \sigma_{i_0\dots\widehat{i_j}\dots i_{k+1}} \big|_{U_{i_0}\cap\cdots\cap U_{i_{k+1}}}. $$ one has $\delta^2=0$; the cohomology of $(C^\bullet,\delta)$ is the Cech cohomology of the cover.

definition (cech cohomology of $M$). $$ H^k(M;\mathcal{F}) :=\varinjlim_{\mathfrak{U}} H^k(\mathfrak{U};\mathcal{F}), $$ the direct limit over refinements of open covers. for $\mathcal{F}=\underline{\mathbb{R}}$ (locally constant real functions) one writes $H^k(M;\underline{\mathbb{R}})$ or simply $H^k(M;\mathbb{R})$ when singular and sheaf cohomology have been identified.

The Cech-de Rham theorem asserts a canonical isomorphism $$ H^k_{\mathrm{dR}}(M)\simeq H^k(M;\underline{\mathbb{R}}). $$ The proof proceeds by constructing the double complex of Cech cochains with values in differential forms; its total cohomology computes both theories. Presheaves of differential forms encode the local-to-global passage, while monodromy representations of $\pi_1(M)$ appear as the obstruction to extending local closed forms to global ones. In field theory this equivalence translates the analytic question "Does a closed field strength admit a global potential?" into the purely topological question of the vanishing of a Cech cohomology class.

theorem (cech-de rham). for a smooth manifold $M$ there is a natural isomorphism $$ H^k_{\mathrm{dR}}(M)\xrightarrow{\simeq} H^k(M;\underline{\mathbb{R}}) $$ for every $k\ge 0$.

proof. form the Cech-de Rham double complex associated to a good open cover $\mathfrak{U}$ (all finite intersections diffeomorphic to convex open sets of $\mathbb{R}^n$, hence contractible): $$ C^{p,q}=C^p\bigl(\mathfrak{U};\Omega^q\bigr), $$ with differentials $\delta$ (Cech) of bidegree $(1,0)$ and $(-1)^p d$ (exterior derivative) of bidegree $(0,1)$. the total complex $(C^\bullet_{\mathrm{tot}},D=\delta+d)$ has cohomology written $H^\bullet_{\mathrm{tot}}$. two spectral sequences abut to $H_{\mathrm{tot}}$:

filtering by Cech degree, the $E_1$-page is $C^p(\mathfrak{U};H^q_{\mathrm{dR}})$. on a good cover the Poincare lemma gives $H^q_{\mathrm{dR}}(U_{i_0}\cap\cdots\cap U_{i_p})=0$ for $q\ge 1$ and $\mathbb{R}$ for $q=0$, so $E_2^{p,0}=H^p(\mathfrak{U};\underline{\mathbb{R}})$ and $E_2^{p,q}=0$ for $q>0$. thus $H^k_{\mathrm{tot}}\simeq H^k(\mathfrak{U};\underline{\mathbb{R}})$.

filtering by form degree, $E_1^{p,q}=H^q_{\mathrm{dR}}$ of the Cech sheafification; exactness of the Cech resolution for the fine sheaves $\Omega^q$ implies $E_1^{0,q}=\Omega^q(M)$ and $E_1^{p,q}=0$ for $p>0$, with $d$-differential producing $E_2^{0,q}=H^q_{\mathrm{dR}}(M)$. hence $H^k_{\mathrm{tot}}\simeq H^k_{\mathrm{dR}}(M)$. identifying the two computations of $H_{\mathrm{tot}}$ yields the isomorphism. passage to the refinement limit removes dependence on a single good cover. $\square$

theorem (existence of global potentials). a closed $k$-form $F$ on $M$ is exact if and only if its de Rham class $[F]$ vanishes in $H^k_{\mathrm{dR}}(M)$. under the Cech-de Rham isomorphism, this is equivalent to the vanishing of the corresponding Cech class $[\{A_{i_0\dots i_{k-1}}\}]$ built from local primitives on a good cover.

proof. if $F=dA$ globally then $[F]=0$ by definition of de Rham cohomology. conversely, if $[F]=0$ then $F=dA$ for some global $A$ by definition of the quotient. on a good cover the Poincare lemma supplies local primitives $A_i$ with $F|_{U_i}=dA_i$; the differences $A_i-A_j$ are closed on overlaps and, after further primitives if $k>1$, produce a Cech cocycle whose class is the image of $[F]$. vanishing of that class is equivalent to the existence of adjusted local primitives that glue to a global potential. $\square$

remark (monodromy). for $k=1$, a closed one-form $\alpha$ defines the periods $\int_\gamma\alpha$ along loops. the class $[\alpha]$ vanishes if and only if all periods vanish. the monodromy representation $\pi_1(M)\to\mathbb{R}$ given by those periods is the obstruction to a global single-valued primitive, the classical Abel theorem in elementary form.

sphere bundles, the euler class, and the thom isomorphism

Let $\pi:E\to M$ be an oriented real vector bundle of rank $n$. The associated sphere bundle $S(E)$ carries a global angular form $\psi$, a differential form of degree $n-1$ whose restriction to each fiber generates the cohomology of the sphere and whose exterior derivative satisfies $$ d\psi=\pi^*e(E), $$ where $e(E)\in H^n(M;\mathbb{Z})$ is the Euler class of the bundle. The Euler class vanishes if and only if $E$ admits a nowhere-zero section. When an isolated zero of a section occurs, the Hopf index theorem identifies the local degree of that zero with the pairing of the Euler class against the fundamental class of a small surrounding sphere.

definition (sphere bundle and angular form). for a Riemannian metric on an oriented rank-$n$ vector bundle $E\to M$, let $S(E)=\{v\in E:\|v\|=1\}$. an angular form $\psi\in \Omega^{n-1}(S(E))$ is a form that restricts on each fiber $S(E_p)\simeq S^{n-1}$ to a generator of $H^{n-1}(S^{n-1})$ with total integral $1$ (normalized volume form), and that is globally defined on $S(E)$.

definition (euler class via angular form). for any angular form $\psi$, the form $d\psi$ is basic (constant on fibers and horizontal), hence $d\psi=\pi_S^*\alpha$ for a unique closed $n$-form $\alpha$ on $M$. the de Rham class $[\alpha]=e(E)_{\mathbb{R}}$ is the real image of the integral Euler class $e(E)\in H^n(M;\mathbb{Z})$.

theorem (euler class obstructs nowhere-zero sections). an oriented real vector bundle $E\to M$ of rank $n$ admits a nowhere-vanishing continuous section if and only if $e(E)=0$ in $H^n(M;\mathbb{Z})$ (when $M$ is a closed oriented $n$-manifold, equivalently $\langle e(E),[M]\rangle=0$).

proof. a nowhere-zero section $s$ normalizes to $s/\|s\|:M\to S(E)$, a global section of the sphere bundle. the pullback of the fiberwise volume yields a global primitive relationship: $s^*\psi$ satisfies $d(s^*\psi)=e(E)_{\mathbb{R}}$, and on a closed oriented base $\int_M e(E)=\int_M d(s^*\psi)=0$, so the Euler number vanishes; integrally, the obstruction class defining $e(E)$ is zero once a section of $S(E)$ exists. conversely, primary obstruction theory for sections of $S(E)$ (fibers $(n-1)$-connected through the range of the first obstruction) places the primary obstruction in $H^n(M;\pi_{n-1}(S^{n-1}))\simeq H^n(M;\mathbb{Z})$, and that class is $e(E)$. vanishing allows a section of $S(E)$, hence a nowhere-zero section of $E$. $\square$

theorem (hopf index). let $M$ be a closed oriented manifold of dimension $n$ and let $X$ be a smooth vector field with isolated zeros $p_1,\dots,p_N$. then $$ \sum_{i=1}^N \operatorname{index}_{p_i}(X) =\langle e(TM),[M]\rangle=\chi(M), $$ where $\operatorname{index}_{p_i}(X)$ is the degree of the normalized map $S^{n-1}_\varepsilon(p_i)\to S^{n-1}$ and $\chi(M)$ is the Euler characteristic.

proof. remove small balls $B_i$ about each zero. on the compact manifold with boundary $M\setminus\bigcup\mathrm{int}(B_i)$, the field $X$ has no zeros and may be normalized to a section of $S(TM)$. Stokes and the angular-form identity $d\psi=\pi^*e(TM)$ convert $\int_M e(TM)$ into a sum of fiber integrals of $\psi$ over $\partial B_i$, each of which equals the local degree of $X/\|X\|$ on $\partial B_i$. the Poincare-Hopf theorem identifies the resulting integer with $\chi(M)$. $\square$

The Thom isomorphism relates the cohomology of the base to the compactly supported cohomology of the total space: $$ H^k(M)\xrightarrow{\simeq} H^{k+n}_{\mathrm{c}}(E),\qquad \alpha\mapsto\pi^*\alpha\wedge\Phi(E), $$ where $\Phi(E)$ is the Thom class, a compactly supported form that restricts to a generator of the cohomology of each fiber. In classical field theory the Euler and Thom classes control the existence of global frame fields, the localization of topological charge, and the intersection theory of submanifolds defined by vanishing loci of sections.

definition (thom class). for an oriented rank-$n$ vector bundle $E\to M$, a Thom class is a cohomology class $\Phi(E)\in H^n_{\mathrm{c}}(E)$ (or $H^n(D(E),S(E))$ in the disk-sphere pair picture) whose restriction to each fiber is the positive generator of $H^n_{\mathrm{c}}(\mathbb{R}^n)\simeq\mathbb{Z}$.

theorem (thom isomorphism). cup product (or wedge) with $\Phi(E)$ induces isomorphisms $$ H^k(M;\mathbb{Z})\xrightarrow{\simeq} H^{k+n}_{\mathrm{c}}(E;\mathbb{Z}) $$ for every $k$. the Euler class is recovered as $e(E)=s_0^*\Phi(E)$ for the zero section $s_0:M\to E$.

proof. for a trivial bundle $E=M\times\mathbb{R}^n$ the claim is the Kunneth theorem with compact supports in the fiber, with $\Phi=1\otimes u$ for the standard volume class $u$ of $\mathbb{R}^n$. for general $E$, cover $M$ by opens that trivialize $E$ and apply a Mayer-Vietoris induction: the Thom classes on restrictions glue because orientations of the fibers match, and the five lemma upgrades local Thom isomorphisms to a global isomorphism. naturality of the construction under pullback identifies $s_0^*\Phi(E)$ with the Euler class defined by the angular form. $\square$

spectral sequences and rational homotopy

When a manifold or a fiber bundle is filtered, or when a double complex appears (as in the Cech-de Rham construction), the associated spectral sequence converges to the cohomology of the total complex. The Serre spectral sequence of a fibration $F\to E\to B$ has $E_2$-page $$ E_2^{p,q}=H^p(B;H^q(F)) $$ and converges to $H^{p+q}(E)$. It computes the cohomology of path fibrations, loop spaces, and configuration spaces of fields.

definition (spectral sequence of a filtered complex). a decreasing filtration $F^\bullet C$ of a cochain complex $(C,d)$ produces pages $E_r^{p,q}$ with differentials $d_r:E_r^{p,q}\to E_r^{p+r,q-r+1}$ such that $E_{r+1}=H(E_r,d_r)$. under mild exhaustiveness and completeness hypotheses, $E_r\Rightarrow H^{p+q}(C)$.

theorem (serre spectral sequence, statement). for a Serre fibration $F\to E\to B$ of path-connected spaces with $B$ simply connected (or with local coefficients), there is a spectral sequence with $$ E_2^{p,q}=H^p\bigl(B;H^q(F)\bigr) \Rightarrow H^{p+q}(E). $$

proof (outline). filter the singular cochains of $E$ by the skeleta of a CW model of $B$ (or use the Cartan-Serre filtration of the cochain path fibration). the associated graded of the filtration at $E_1$ is the cochain complex of $B$ with coefficients in the cohomology of the fiber. passage to $E_2$ yields $H^p(B;H^q(F))$. standard convergence theorems for bounded filtrations give the abutment $H^\bullet(E)$. $\square$

Rational homotopy theory, via Sullivan's minimal models, replaces a space by a differential graded algebra of rational differential forms whose cohomology recovers the rational homotopy type. The minimal model encodes all higher-order Massey products and is especially effective for calculating the rational cohomology of mapping spaces and gauge-orbit spaces that arise as configuration spaces of classical fields.

definition (sullivan minimal model). a commutative differential graded algebra (cdga) $(\mathcal{M},d)$ over $\mathbb{Q}$ is minimal if $\mathcal{M}$ is free commutative on a graded vector space $V$ with generators decomposable under $d$ (in positive degrees, $dV\subset \mathcal{M}^+\cdot\mathcal{M}^+$). a minimal model of a space $X$ is a quasi-isomorphism $\mathcal{M}_X\to A_{\mathrm{PL}}(X)$ from a minimal cdga to the algebra of piecewise-linear rational forms on $X$.

theorem (existence and uniqueness of minimal models). every path-connected space $X$ of finite rational type admits a Sullivan minimal model, unique up to isomorphism. its cohomology is $H^\bullet(X;\mathbb{Q})$, and the indecomposables of $\mathcal{M}_X$ compute $\operatorname{Hom}(\pi_\bullet(X)\otimes\mathbb{Q},\mathbb{Q})$ under mild nilpotence hypotheses.

proof (outline). proceed inductively by killing successive cohomology classes: start with $\mathcal{M}(0)=\mathbb{Q}$, and at stage $n$ adjoin generators in degree $n$ with differentials recording a basis of $\ker(H^n(\mathcal{M}(n-1))\to H^n(X))$ and of the cokernel of that map, using relative lifting properties of free cdgas. the process converges to a quasi-isomorphism. uniqueness follows from the lifting theorem for minimal cdgas: a quasi-isomorphism between minimals is an isomorphism. the identification with rational homotopy groups is Sullivan's dual of the Whitehead tower at the level of indecomposables. $\square$

remark (configuration spaces). gauge-orbit spaces $\mathcal{A}/\mathcal{G}$ and mapping spaces $\operatorname{Map}(M,G)$ inherit rational models from those of $M$ and $BG$; spectral sequences and minimal models then compute their rational cohomology, which classifies infinitesimal deformations of topological sector labels.

topological sectors in the configuration space of classical fields

The space of smooth sections of a fiber bundle $E\to M$, modulo gauge transformations, decomposes into path components labeled by characteristic classes and homotopy invariants. For an abelian gauge field the first Chern class (or equivalently the de Rham class of the field strength) labels the magnetic flux sectors. For nonabelian fields the second Chern number distinguishes instanton sectors. On a spin manifold the existence of a spin structure itself is a $\mathbb{Z}/2$-valued topological datum. Each such sector supports its own copy of the classical phase space; continuous time evolution cannot interpolate between them. Consequently the global topology of spacetime and of the bundles that carry the fields imposes a discrete superselection structure on the classical theory, a structure that survives quantization and reappears as the topological selection rules of quantum field theory.

definition (configuration space and gauge orbits). for a principal $G$-bundle $P\to M$, let $\mathcal{A}$ be the affine space of connections on $P$ and $\mathcal{G}$ the group of gauge transformations (vertical automorphisms of $P$). the configuration space of classical gauge fields is the quotient stack $\mathcal{A}/\mathcal{G}$ (or a Sobolev completion thereof). path components of the space of based gauge transformations and of $\operatorname{Map}(M,G)$ label components of the quotient.

theorem (u(1) flux sectors). isomorphism classes of principal $\mathrm{U}(1)$-bundles on a manifold $M$ are in natural bijection with $H^2(M;\mathbb{Z})$. for a connection $A$ on such a bundle, the curvature $F=dA$ (locally) represents $2\pi c_1$ in de Rham cohomology, and the class $c_1\in H^2(M;\mathbb{Z})$ is independent of $A$.

proof. the exponential sequence of sheaves $0\to\mathbb{Z}\to\underline{\mathbb{R}}\to\underline{\mathrm{U}(1)}\to 0$ yields, because $H^1(M;\underline{\mathbb{R}})=H^2(M;\underline{\mathbb{R}})=0$ for the soft sheaf of smooth real functions, $$ H^1\bigl(M;\underline{\mathrm{U}(1)}\bigr)\simeq H^2(M;\mathbb{Z}). $$ Cech $1$-cocycles with values in $\mathrm{U}(1)$ are precisely the transition functions of principal $\mathrm{U}(1)$-bundles. given a connection, local curvature forms $F_i=dA_i$ glue because $A_i-A_j=g_{ij}^{-1}dg_{ij}$ is closed as a real form after taking $d\log$, and Chern-Weil (chapter 6) identifies $[F/2\pi]$ with the real image of $c_1$. independence of connection follows because the difference of two connection forms is globally defined and the curvatures differ by an exact form. $\square$

theorem (instanton number as a topological invariant). for a principal $\mathrm{SU}(n)$-bundle $P$ on a closed oriented four-manifold $M$, the integer $$ k =-\frac{1}{8\pi^2}\int_M\operatorname{tr}(F_A\wedge F_A) $$ equals the second Chern number $c_2(P)[M]$ and is independent of the connection $A$. connections with different $k$ lie in distinct path components of $\mathcal{A}/\mathcal{G}$.

proof. by Chern-Weil theory, $\operatorname{tr}(F\wedge F)$ is closed and its de Rham class is a characteristic class of $P$, independent of $A$ up to exact forms; on a closed manifold the integral is therefore constant on the affine space $\mathcal{A}$. integrality follows from the identification with $c_2\in H^4(M;\mathbb{Z})$. a continuous path of connections would produce a continuous (hence constant) integer $k$, so different $k$ cannot be joined by a continuous path of gauge fields even after gauge transformations (which preserve $c_2$). $\square$

remark (spin structures as $\mathbb{Z}/2$ data). as in chapter 3, spin structures exist iff $w_2(TM)=0$ and then form a torsor over $H^1(M;\mathbb{Z}/2)$. each choice yields a distinct spinor bundle and therefore a distinct configuration space for fermionic fields.

Algebraic topology therefore converts the global geometric questions left open by local differential equations into a precise system of computable invariants. These invariants label the distinct topological sectors of classical field configurations and prepare the ground for the characteristic classes, $K$-theory, and index theorems that refine the classification still further.

remark (to the next chapters). chapter 6 constructs characteristic classes from curvature by Chern-Weil theory; chapter 7 reorganizes bundles into $K$-theory; chapter 8 relates topology to analytical indices.

exercises

exercise 1 (magnetic flux). on $S^2$, let $F$ be a closed two-form with $\int_{S^2}F=2\pi n$ for $n\in\mathbb{Z}$. using the Cech-de Rham correspondence with the cover by northern and southern hemispheres, show that a $\mathrm{U}(1)$ potential exists globally if and only if $n=0$, and construct the transition function on the equator when $n\neq 0$.

exercise 2 (euler number of $TS^2$). compute $\langle e(TS^2),[S^2]\rangle$ two ways: (i) as the Euler characteristic $\chi(S^2)$; (ii) as the sum of indices of a vector field with isolated zeros (for example the gradient of height). conclude that $TS^2$ is nontrivial.

exercise 3 (path components of maps). identify $\pi_0\operatorname{Map}(S^3,\mathrm{SU}(2))$ with $\pi_3(\mathrm{SU}(2))\simeq\mathbb{Z}$ and interpret the integer as a winding / instanton number for pure gauge transformations on $S^3$ (or on $\mathbb{R}^3$ with decay at infinity).

exercise 4 (mayers-vietoris for $S^1$). cover $S^1$ by two open arcs $U,V$ with $U\cap V$ a disjoint union of two arcs. using Mayer-Vietoris (or a Cech computation with constant coefficients), compute $H^1(S^1)$ and relate the generator to the winding of the angle form pulled back from $\mathbb{R}^2\setminus\{0\}$.

exercise 5 (obstruction to trivializing a line bundle). if $L\to S^2$ is a complex line bundle with $c_1(L)\neq 0$, explain why $L$ admits no nowhere-vanishing continuous section. contrast with the trivial bundle $\varepsilon^1$.