6 · characteristic classes

classification of vector bundles by characteristic classes

Two vector bundles over a common base $M$ are isomorphic if and only if their classifying maps into a suitable classifying space are homotopic. Characteristic classes are the cohomology classes pulled back from the universal classes that generate the cohomology of those classifying spaces. Consequently they are functorial under bundle morphisms and stable under addition of trivial bundles. Once a complete set of characteristic classes is known, the isomorphism type of a bundle is largely determined (exactly determined for many structure groups after stabilization).

definition (classifying map). a complex rank-$k$ vector bundle $E\to M$ determines, up to homotopy, a continuous map $f_E:M\to\mathrm{Gr}_k(\mathbb{C}^\infty)=BU(k)$ such that $E\simeq f_E^*\gamma^k$ for the tautological bundle $\gamma^k$ on the Grassmannian. real and oriented real bundles are classified likewise by maps into $BO(k)$ and $BSO(k)$.

definition (characteristic class). a characteristic class for a family of bundles (real, complex, oriented, \ldots) is a natural assignment $E\mapsto c(E)\in H^\bullet(M;R)$ such that for every continuous map $f:N\to M$ one has $c(f^*E)=f^*c(E)$ in $H^\bullet(N;R)$. equivalently, it is the pullback of a fixed universal class on the classifying space.

theorem (naturality and stability). every characteristic class $c$ satisfies $c(f^*E)=f^*c(E)$. if $c$ is stable for complex $K$-theory in the sense that $c(E\oplus\underline{ \mathbb{C}})=c(E)$, then $c$ descends to a well-defined invariant of classes in $K^0(M)$.

proof. naturality is built into the definition via classifying maps: the classifying map of $f^*E$ is homotopic to $f_E\circ f$, so $$ c(f^*E)=(f_E\circ f)^*c(\gamma)=f^*(f_E^*c(\gamma))=f^*c(E). $$ stability under adding a trivial line means that $c$ factors through the stabilization maps $BU(k)\to BU(k+1)$, hence through $BU$, and therefore through the representing space of complex $K$-theory. $\square$

remark (how much is determined). for complex line bundles, $c_1$ is a complete invariant: $\operatorname{Vect}_1^\mathbb{C}(M)\simeq H^2(M;\mathbb{Z})$. for higher rank, Chern classes are complete after rationalization and stabilization in a range of dimensions (Atiyah-Hirzebruch), but integral isomorphism types may require additional data.

the principal characteristic classes

Four families dominate applications to field theory and geometry.

Stiefel-Whitney classes $w_i(E)\in H^i(M;\mathbb{Z}/2)$ are defined for any real vector bundle. The first class $w_1(E)$ vanishes if and only if $E$ is orientable; the second class $w_2(E)$ is the obstruction to the existence of a spin structure. Higher classes refine the $\mathbb{Z}/2$ cohomological information.

Chern classes $c_i(E)\in H^{2i}(M;\mathbb{Z})$ are defined for complex vector bundles. The first Chern class $c_1(L)$ of a complex line bundle is the curvature class of any Hermitian connection (normalized by $2\pi i$); its integral over a closed surface yields the magnetic monopole charge. The second Chern class $c_2(E)$ evaluated on a four-manifold gives the instanton number of an $\mathrm{SU}(N)$ gauge field.

Pontryagin classes $p_i(E)\in H^{4i}(M;\mathbb{Z})$ are the real analogs of Chern classes, obtained by complexification: $p_i(E)=(-1)^i c_{2i}(E\otimes\mathbb{C})$. They are fundamental invariants of the oriented cobordism ring and appear in the Hirzebruch $L$-polynomial that computes the signature.

The Euler class $e(E)\in H^n(M;\mathbb{Z})$ is defined for an oriented real bundle of rank $n$; it coincides with the top Chern class when the bundle admits a complex structure. Its integral over a closed oriented manifold recovers the Euler characteristic (Gauss-Bonnet theorem). Vanishing of $e(E)$ is necessary for the existence of a nowhere-zero section.

These classes are related by Whitney sum formulas, naturality under pullback, and the splitting principle described below.

definition (total stiefel-whitney, chern, pontryagin, and euler classes). write $$ w(E)=1+w_1(E)+w_2(E)+\cdots,\qquad c(E)=1+c_1(E)+c_2(E)+\cdots, $$ $$ p(E)=1+p_1(E)+p_2(E)+\cdots,\qquad e(E)\in H^{\operatorname{rank}E}(M;\mathbb{Z})\ \text{(oriented real case)}. $$ conventions: $p_i(E)=(-1)^i c_{2i}(E\otimes\mathbb{C})$ for real $E$.

theorem (whitney sum formulas). for real (resp. complex) vector bundles $E,F$ over $M$, $$ w(E\oplus F)=w(E)w(F),\qquad c(E\oplus F)=c(E)c(F). $$ for oriented real bundles of even rank one has $e(E\oplus F)=e(E)e(F)$ when both factors are evenly ranked in the sense that the Euler classes multiply in complementary degrees.

proof. on classifying spaces the Whitney sum corresponds to the $H$-space product $BO\times BO\to BO$ (resp. $BU\times BU\to BU$). the total Stiefel-Whitney (Chern) class is by definition the pullback of the universal generator $w(\gamma)$ (resp. $c(\gamma)$), and the cohomology rings $H^*(BO;\mathbb{Z}/2)=\mathbb{Z}/2[w_i]$ and $H^*(BU;\mathbb{Z})=\mathbb{Z}[c_i]$ are free polynomial, with the product classifying map inducing the cup product of total classes. pulling back by $(f_E,f_F)$ yields $w(E\oplus F)=w(E)w(F)$ and likewise for $c$. for Euler classes use the Thom/Euler construction of chapter 5, or the identity $e(E)=c_n(E)$ for complex bundles of rank $n$ together with Whitney for Chern classes. $\square$

theorem (chern-weil for $c_1$ and $c_2$). let $E\to M$ be a complex vector bundle with Hermitian connection $\nabla$ and curvature $\Omega\in\Omega^2(M,\operatorname{End}(E))$. then the closed forms $$ c_1(\nabla)=\frac{i}{2\pi}\operatorname{tr}\Omega,\qquad c_2(\nabla)=\frac{1}{8\pi^2}\bigl(\operatorname{tr}(\Omega\wedge\Omega)-(\operatorname{tr}\Omega)^2\bigr) $$ (up to the standard polynomial normalization of $\det(I+i\Omega/2\pi)$) represent $c_1(E)$ and $c_2(E)$ in de Rham cohomology and are independent of $\nabla$.

proof. the Chern-Weil invariant polynomials are $\operatorname{Ad}$-invariant, so $P(\Omega)$ is basic and closed by the Bianchi identity $d_\nabla\Omega=0$. if $\nabla_t=(1-t)\nabla_0+t\nabla_1$ is the affine path of connections, the secondary (transgression) form $\mathrm{TP}(\nabla_t)$ satisfies $d\,\mathrm{TP}=P(\Omega_1)-P(\Omega_0)$, so de Rham classes are independent of connection. agreement with integral Chern classes follows by naturality from the universal bundle on $BU(k)$ equipped with its tautological connection, or by the axiomatic characterization (Whitney, normalization on line bundles, and naturality). for a line bundle one has $\Omega=F\cdot\mathrm{id}$ and $c_1=[iF/2\pi]$. $\square$

theorem (orientability and spin). a real vector bundle $E$ is orientable if and only if $w_1(E)=0$. an oriented real vector bundle admits a spin structure if and only if $w_2(E)=0$.

proof. orientation characters are continuous maps $M\to\mathbb{RP}^\infty=BO(1)$, corresponding to $w_1\in H^1(M;\mathbb{Z}/2)$. the structure group reduces to $\mathrm{SO}$ precisely when this character is trivial. lifting further along $1\to\mathbb{Z}/2\to\operatorname{Spin}\to\mathrm{SO}\to 1$ is obstructed by a class in $H^2(M;\mathbb{Z}/2)$, the universal $w_2$ of the tautological bundle on $BSO$; naturality yields the claim for arbitrary oriented $E$. $\square$

theorem (gauss-bonnet-chern). if $M$ is a closed oriented smooth $n$-manifold then $$ \langle e(TM),[M]\rangle=\chi(M). $$ in particular, for $n=2$ and a Riemannian metric $g$, $$ \frac{1}{2\pi}\int_M K\,dA=\chi(M) $$ for Gaussian curvature $K$.

proof. the Euler class of $TM$ is $s_0^*\Phi(TM)$ for the Thom class of the tangent bundle (chapter 5). the Hopf index theorem computes the same integer as the sum of indices of a vector field with isolated zeros, which equals $\chi(M)$ by Poincare-Hopf. the two-dimensional Riemannian expression is Chern-Weil for the Euler class of an oriented rank-$2$ bundle: the curvature form of the Levi-Civita connection is $K\,dA$ times the generator of $\mathfrak{so}(2)$, and $e=[K\,dA/2\pi]$. $\square$

remark (instanton number). for a principal $\mathrm{SU}(N)$-bundle on a closed oriented four-manifold the second Chern number is $c_2[E]=\frac{1}{8\pi^2}\int\operatorname{tr}(F\wedge F)$ (sign conventions as in chapter 5), matching the instanton charge of a connection $A$ with curvature $F$.

grassmannians, universal bundles, and the splitting principle

The Grassmann manifold $\mathrm{Gr}_k(\mathbb{C}^\infty)$ (respectively $\mathrm{Gr}_k(\mathbb{R}^\infty)$) classifies complex (real) $k$-plane bundles. It carries a tautological universal bundle $\gamma^k$ whose characteristic classes generate the entire cohomology ring of the Grassmannian. Every rank-$k$ bundle $E\to M$ is the pullback of $\gamma^k$ by a classifying map $f:M\to\mathrm{Gr}_k$, and the characteristic classes of $E$ are simply $f^*$ of the universal classes.

definition (tautological bundle). on $\mathrm{Gr}_k(\mathbb{C}^N)$ the tautological bundle $\gamma^k$ has fiber over a $k$-plane $V$ the plane $V$ itself. the stable Grassmannian is $\mathrm{Gr}_k(\mathbb{C}^\infty)=\varinjlim_N\mathrm{Gr}_k(\mathbb{C}^N)$, with tautological bundle still written $\gamma^k$.

theorem (cohomology of $BU(k)$). $$ H^*\bigl(BU(k);\mathbb{Z}\bigr)=\mathbb{Z}[c_1,\dots,c_k], $$ where $c_i=c_i(\gamma^k)$ is the universal Chern class of degree $2i$. similarly $H^*(BO(k);\mathbb{Z}/2)=\mathbb{Z}/2[w_1,\dots,w_k]$.

proof (outline). the fibration $U(k-1)\to U(k)\to S^{2k-1}$ and the associated Serre spectral sequence for $BU(k-1)\to BU(k)\to\mathbb{CP}^\infty$ (or induction on the Schubert-cell cellulation of Grassmannians) show that $H^*(BU(k))$ is free polynomial on the Chern classes of the tautological bundle, with no relations below the stable range and the standard truncated polynomial presentation for finite $k$. the real mod $2$ calculation is analogous with Stiefel-Whitney classes and $\mathbb{RP}^\infty$. $\square$

The splitting principle asserts that, for the purpose of computing characteristic classes, any complex vector bundle may be treated as a direct sum of line bundles. More precisely, there exists a flag manifold $F(E)\to M$ such that the pullback of $E$ splits into a sum of complex line bundles; the Chern classes of $E$ become the elementary symmetric polynomials in the first Chern classes of those line bundles. An analogous statement holds for real bundles after complexification. The principle reduces all formal manipulations of characteristic classes to calculations with Chern roots.

definition (flag bundle and chern roots). the full flag bundle $\pi:F(E)\to M$ of a complex rank-$k$ bundle $E$ is the bundle of complete flags in the fibers of $E$. the pullback $\pi^*E$ admits a filtration by subbundles with successive line-bundle quotients $L_1,\dots,L_k$. the Chern roots are $x_i:=c_1(L_i)\in H^2(F(E);\mathbb{Z})$.

theorem (splitting principle). the pullback $\pi^*:H^*(M;\mathbb{Z})\to H^*(F(E);\mathbb{Z})$ is injective. moreover $$ c(\pi^*E)=\prod_{i=1}^k(1+x_i), $$ so $c_j(E)$ pulls back to the $j$-th elementary symmetric polynomial $e_j(x_1,\dots,x_k)$. any polynomial identity among Chern classes that holds for split bundles therefore holds for all bundles.

proof. $F(E)$ is obtained by iterated projectivizations $\mathbb{P}(E)\to M$, then $\mathbb{P}$ of the quotient bundles, etc. the projective-bundle theorem gives $$ H^*\bigl(\mathbb{P}(E)\bigr) =H^*(M)[\zeta]\big/\bigl(\zeta^k+c_1(E)\zeta^{k-1}+\cdots+c_k(E)\bigr), $$ so restriction $H^*(M)\to H^*(\mathbb{P}(E))$ is split injective (basis $1,\zeta,\dots,\zeta^{k-1}$). induction on rank yields injectivity for the full flag bundle. over $F(E)$ the bundle splits by construction of the tautological line filtration, so Whitney sum yields $c(\pi^*E)=\prod(1+x_i)$. injectivity of $\pi^*$ transfers any universal polynomial relation from the split case back to $M$. $\square$

remark (chern character). writing $c(E)=\prod(1+x_i)$ formally, the Chern character is $\operatorname{ch}(E)=\sum e^{x_i}=\operatorname{rank}E+c_1+\frac12(c_1^2-2c_2)+\cdots$, a ring homomorphism $K^0(M)\to H^{\mathrm{even}}(M;\mathbb{Q})$ used in index theory (chapters 7 and 8).

cobordism and the hirzebruch signature theorem

Two closed manifolds are cobordant if their disjoint union bounds a compact manifold of one higher dimension. The resulting cobordism rings $\Omega_*^{\mathrm{SO}}$ and $\Omega_*^{\mathrm{O}}$ are determined by Pontryagin numbers and Stiefel-Whitney numbers respectively. Thom's construction realizes these rings as the homotopy groups of the Thom spectra $\mathrm{MSO}$ and $\mathrm{MO}$; transversality identifies cobordism classes with homotopy classes of maps into those spectra.

definition (cobordism). closed $n$-manifolds $M_0,M_1$ are oriented cobordant if there exists a compact oriented $(n+1)$-manifold $W$ with $\partial W\simeq M_0\sqcup(-M_1)$. isomorphism classes under this relation form the abelian group $\Omega_n^{\mathrm{SO}}$; the disjoint union and Cartesian product make $\Omega_*^{\mathrm{SO}}$ a graded ring.

definition (characteristic numbers). for $M$ closed oriented of dimension $4k$, and a monomial $p_{i_1}\cdots p_{i_r}$ of total degree $4k$ in Pontryagin classes, the Pontryagin number is $\langle p_{i_1}(TM)\cdots p_{i_r}(TM),[M]\rangle$. Stiefel-Whitney numbers are the analogous pairings with $w_I(TM)$ for unoriented manifolds.

theorem (thom: characteristic numbers determine cobordism). two closed oriented manifolds are oriented-cobordant if and only if all corresponding Pontryagin numbers (and Stiefel-Whitney numbers for the unoriented theory) agree. equivalently, $\Omega_*^{\mathrm{SO}}\otimes\mathbb{Q}$ is a polynomial ring on generators detected by Pontryagin numbers.

proof (outline). Thom's transversal representing maps $M\to MSO$ convert cobordism classes into stable homotopy classes $\pi_*(MSO)$. the rational homotopy of $MSO$ is computed by the spectral sequence of the Thom spectrum and matches the polynomial generators dual to Pontryagin numbers. if all characteristic numbers vanish, the corresponding map is rationally nullhomotopic and bounds after killing torsion by a standard surgery/transversality argument in the Thom spectrum. $\square$

The Hirzebruch signature theorem expresses the topological signature of a closed oriented $4k$-manifold as the evaluation of a universal polynomial $L_k(p_1,\dots,p_k)$ in the Pontryagin classes: $$ \mathrm{Sign}(M)=\langle L_k\bigl(p(TM)\bigr),[M]\rangle. $$ The same $L$-polynomial appears in the Atiyah-Singer index theorem for the signature operator, linking the global topology of the tangent bundle to the analytic index of an elliptic differential operator.

definition (signature and $L$-class). for $M$ closed oriented of dimension $4k$, the signature $\mathrm{Sign}(M)$ is the signature of the intersection form $H^{2k}(M;\mathbb{R})\times H^{2k}(M;\mathbb{R})\to\mathbb{R}$. the Hirzebruch $L$-class is the multiplicative sequence associated to the power series $x/\tanh x$, expanded as a polynomial in Pontryagin roots; its degree-$4k$ component is $L_k$.

theorem (hirzebruch signature theorem). for every closed oriented smooth manifold of dimension $4k$, $$ \mathrm{Sign}(M)=\langle L_k(p(TM)),[M]\rangle. $$ in dimension $4$, $L_1=p_1/3$, so $\mathrm{Sign}(M)=\frac13\langle p_1(TM),[M]\rangle$.

proof. both $\mathrm{Sign}$ and $\langle L(p),[\,]\rangle$ are oriented cobordism invariants: signature is a cobordism invariant by Novikov's algebraic argument (or by the Atiyah-Singer index of the signature operator, whose analytic index is $\mathrm{Sign}$ and vanishes on boundaries), while $L$-numbers are Pontryagin numbers. they therefore define rational ring homomorphisms $\Omega_*^{\mathrm{SO}}\otimes\mathbb{Q}\to\mathbb{Q}$. the multiplicative sequence $x/\tanh x$ is uniquely characterized as the one that matches the signatures of the complex projective spaces $\mathbb{CP}^{2k}$ (which generate the rational oriented cobordism ring together with known relations): $\mathrm{Sign}(\mathbb{CP}^{2k})=1$, and a direct Chern/Pontryagin computation gives $\langle L(\mathbb{CP}^{2k}),[\mathbb{CP}^{2k}]\rangle=1$. agreement on generators yields the theorem for all rational cobordism classes, hence for all closed oriented smooth manifolds. $\square$

Characteristic classes therefore convert the geometric question of bundle triviality into concrete cohomology classes that can be evaluated by integration or pairing. They label the topological sectors of classical gauge fields, control the existence of spinors, quantize charges, and furnish the characteristic numbers that enter every index theorem relating the spectrum of field operators to the topology of spacetime.

remark (to $K$-theory and index theory). chapter 7 reorganizes bundles into a ring $K^0(M)$ with Chern character landing in ordinary cohomology; chapter 8 identifies analytical indices with pairings of characteristic classes such as $\widehat{A}$ and $\operatorname{ch}$.

exercises

exercise 1 (whitney for lines). if $L,L'\to M$ are complex line bundles, prove $c_1(L\otimes L')=c_1(L)+c_1(L')$ using either Whitney for $L\oplus L'$ and the identity $c_1(\operatorname{det})=c_1$, or Chern-Weil for product connections.

exercise 2 (hopf bundle). for the Hopf line bundle $H\to\mathbb{CP}^1\simeq S^2$, show $\int_{S^2}c_1(H)=\pm 1$. deduce $H$ is nontrivial and that $w_2(TS^2)\neq 0$ after identifying real reductions as appropriate, or alternatively recompute $\chi(S^2)=2$ via Gauss-Bonnet.

exercise 3 (signature of $\mathbb{CP}^2$). compute $p_1(T\mathbb{CP}^2)$ using $c(T\mathbb{CP}^2)=(1+x)^3$ with $x=c_1(\gamma^*)$, and verify $\mathrm{Sign}(\mathbb{CP}^2)=\langle p_1/3,[\mathbb{CP}^2]\rangle=1$.

exercise 4 (whitney sum formula in low rank). for complex rank-two bundles $E,F$ over a space, expand $c(E\oplus F)=c(E)c(F)$ and write $c_2(E\oplus F)$ in terms of $c_i(E)$ and $c_i(F)$. check against $E=L_1\oplus L_2$ with line bundles in terms of $c_1(L_i)$.