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Chern class

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The ordinary Chern classes are the integral characteristic classes

c i:BUB 2i c_i : B U \to B^{2 i} \mathbb{Z}

of the classifying space BUB U of the unitary group.

Accordingly these are characteristic classes in ordinary cohomology of U-principal bundles and hence of complex vector bundle

The first Chern class is the unique characteristic class of circle group-principal bundles.

The analogous classes for the orthogonal group are the Pontryagin classes.

More generally, there are generalized Chern classes for any complex oriented cohomology theory (Adams 74, Lurie 10).

Definition

Definition

For n1n \geq 1 the Chern universal characteristic classes c iH 2i(BU(n),)c_i \in H^{2i}(B U(n), \mathbb{Z}) of the classifying space BU(n)B U(n) of the unitary group are characterized as follows:

  1. c 0=1c_0 = 1 and c i=0c_i = 0 if i>ni \gt n;

  2. for n=1n = 1, c 1c_1 is the canonical generator of H 2(BU(1),)H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z};

  3. under pullback along the inclusion i:BU(n)BU(n+1)i : B U(n) \to B U(n+1) we have i *c i (n+1)=c i (n)i^* c_i^{(n+1)} = c_i^{(n)};

  4. under the inclusion BU(k)×BU(l)BU(k+l)B U(k) \times B U(l) \to B U(k+l) we have i *c i= j=0 ic ic jii^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}.

Properties

Existence

Proposition

The cohomology ring of the classifying space BU(n)B U(n) (for the unitary group U(n)U(n)) is the polynomial ring on generators {c k} k=1 n\{c_k\}_{k = 1}^{n} of degree 2, called the Chern classes

H (BU(n),)[c 1,,c n]. H^\bullet(B U(n), \mathbb{Z}) \simeq \mathbb{Z}[c_1, \cdots, c_n] \,.

Moreover, for Bi:BU(n 1)BU(n 2)B i \colon B U(n_1) \longrightarrow BU(n_2) the canonical inclusion for n 1n 2n_1 \leq n_2 \in \mathbb{N}, then the induced pullback map on cohomology

(Bi) *:H (BU(n 2))H (BU(n 1)) (B i)^\ast \;\colon\; H^\bullet(B U(n_2)) \longrightarrow H^\bullet(B U(n_1))

is given by

(Bi) *(c k)={c k for1kn 1 0 otherwise. (B i)^\ast(c_k) \;=\; \left\{ \array{ c_k & for \; 1 \leq k \leq n_1 \\ 0 & otherwise } \right. \,.

(e.g. Kochmann 96, theorem 2.3.1)

Proof

For n=1n = 1, in which case BU(1)P B U(1) \simeq \mathbb{C}P^\infty is the infinite complex projective space, we have (prop)

H (BU(1))[c 1], H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,,

where c 1c_1 is the first Chern class. From here we proceed by induction. So assume that the statement has been shown for n1n-1.

Observe that the canonical map BU(n1)BU(n)B U(n-1) \to B U(n) has as homotopy fiber the (2n-1)sphere (prop.) hence there is a homotopy fiber sequence of the form

S 2n1BU(n1)BU(n). S^{2n-1} \longrightarrow B U(n-1) \longrightarrow B U(n) \,.

Consider the induced Thom-Gysin sequence.

In odd degrees 2k+1<2n2k+1 \lt 2n it gives the exact sequence

H 2k(BU(n1))H 2k+12n(BU(n))0H 2k+1(BU(n))(Bi) *H 2k+1(BU(n1))0, \cdots \to H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k+1-2n}(B U(n))}} \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,

where the right term vanishes by induction assumption, and the middle term since ordinary cohomology vanishes in negative degrees. Hence

H 2k+1(BU(n))0for2k+1<2n H^{2k+1}(B U(n)) \simeq 0 \;\;\; for \; 2k+1 \lt 2n

Then for 2k+1>2n2k+1 \gt 2n the Thom-Gysin sequence gives

H 2k+12n(BU(n))H 2k+1(BU(n))(Bi) *H 2k+1(BU(n1))0, \cdots \to H^{2k+1-2n}(B U(n)) \longrightarrow H^{2k+1}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} \underset{\simeq 0}{\underbrace{H^{2k+1}(B U(n-1))}} \to \cdots \,,

where again the right term vanishes by the induction assumption. Hence exactness now gives that

H 2k+12n(BU(n))H 2k+1(BU(n)) H^{2k+1-2n}(B U(n)) \overset{}{\longrightarrow} H^{2k+1}(B U(n))

is an epimorphism, and so with the previous statement it follows that

H 2k+1(BU(n))0 H^{2k+1}(B U(n)) \simeq 0

for all kk.

Next consider the Thom Gysin sequence in degrees 2k2k

H 2k1(BU(n1))0H 2k2n(BU(n))H 2k(BU(n))(Bi) *H 2k(BU(n1))H 2k+12n(BU(n))0. \cdots \to \underset{\simeq 0}{\underbrace{H^{2k-1}(B U(n-1))}} \longrightarrow H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \longrightarrow \underset{\simeq 0}{\underbrace{H^{2k +1 - 2n}(B U(n))}} \to \cdots \,.

Here the left term vanishes by the induction assumption, while the right term vanishes by the previous statement. Hence we have a short exact sequence

0H 2k2n(BU(n))H 2k(BU(n))(Bi) *H 2k(BU(n1))0 0 \to H^{2k-2n}(B U(n)) \longrightarrow H^{2k}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} H^{2k}(B U(n-1)) \to 0

for all kk. In degrees 2n\bullet\leq 2n this says

0c n()H 2n(BU(n))(Bi) *([c 1,,c n1]) 2n0 0 \to \mathbb{Z} \overset{c_n \cup (-)}{\longrightarrow} H^{\bullet \leq 2n}(B U(n)) \overset{(B i)^\ast}{\longrightarrow} (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \to 0

for some Thom class c nH 2n(BU(n))c_n \in H^{2n}(B U(n)), which we identify with the next Chern class.

Since free abelian groups are projective objects in Ab, their extensions are all split (the Ext-group out of them vanishes), hence the above gives a direct sum decomposition

H 2n(BU(n)) ([c 1,,c n1]) 2n2n ([c 1,,c n]) 2n. \begin{aligned} H^{\bullet \leq 2n}(B U(n)) & \simeq (\mathbb{Z}[c_1, \cdots, c_{n-1}])_{\bullet \leq 2n} \oplus \mathbb{Z}\langle 2n\rangle \\ & \simeq (\mathbb{Z}[c_1, \cdots, c_{n}])_{\bullet \leq 2n} \end{aligned} \,.

Now by another induction over these short exact sequences, the claim follows.

First Chern class

Splitting principle and Chern roots

Under the splitting principle all Chern classes are determnined by first Chern classes:

Write i:TU(1) nU(n)i \colon T \simeq U(1)^n \hookrightarrow U(n) for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then

H (BT,)H (BU(1) n,) H^\bullet(B T, \mathbb{Z}) \simeq H^\bullet(B U(1)^n, \mathbb{Z})

is the polynomial ring in nn generators (to be thought of as the universal first Chern classes c ic_i of each copy of BU(1)B U(1); equivalently as the weights of the group characters of U(n)U(n)) which are traditionally written x ix_i:

H (BU(1) n,)[x 1,,x n]. H^\bullet(B U(1)^n, \mathbb{Z}) \simeq \mathbb{Z}[x_1, \cdots, x_n] \,.

Write

Bi:BU(1) nBU(n) B i \;\colon\; B U(1)^n \to B U(n)

for the induced map of deloopings/classifying spaces, then the kk-universal Chern class c kH 2k(BU(n),)c_k \in H^{2k}(B U(n), \mathbb{Z}) is uniquely characterized by the fact that its pullback to BU(1) nB U(1)^n is the kkth elementary symmetric polynomial σ k\sigma_k applied to these first Chern classes:

(Bi) *(c k)=σ k(x 1,,x n). (B i)^\ast (c_k) = \sigma_k(x_1, \cdots, x_n) \,.

Equivalently, for c= i=1 nc kc = \sum_{i = 1}^n c_k the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials σ k(x 1,,k n)\sigma_k(x_1, \cdots, k_n) are the degree-kk piece in (1+x 1)(1+x n)(1+x_1) \cdots (1+x_n), this means that

(Bi) *(c)=(1+x 1)(1+x 2)(1+x n). (B i)^\ast (c) = (1+x_1) (1+ x_2) \cdots (1+ x_n) \,.

Since here on the right the first Chern classes x ix_i appear as the roots of the Chern polynomial, they are also called Chern roots.

See also at splitting principle – Examples – Complex vector bundles and their Chern roots.

(e.g. Kochmann 96, theorem 2.3.2, tom Dieck 08, theorem 19.3.2)

Lemma

For nn \in \mathbb{N} let μ n:B(U(1) n)BU(n)\mu_n \;\colon\; B (U(1)^n) \longrightarrow B U(n) be the canonical map. Then the induced pullback operation on ordinary cohomology

μ n *:H (BU(n);)H (BU(1) n;) \mu^\ast_n \;\colon\; H^\bullet( B U(n); \mathbb{Z} ) \longrightarrow H^\bullet( B U(1)^n; \mathbb{Z} )

is a monomorphism.

A proof of lemma 1 via analysis of the Serre spectral sequence of U(n)/U(1) nBU(1) nBU(n)U(n)/U(1)^n \to B U(1)^n \to B U(n) is indicated in (Kochmann 96, p. 40). A proof via transfer of the Euler class of U(n)/U(1) nU(n)/U(1)^n, following (Dupont 78, (8.28)), is indicated at splitting principle (here).

Proposition

For knk \leq n \in \mathbb{N} let Bi n:B(U(1) n)BU(n)B i_n \;\colon\; B (U(1)^n) \longrightarrow B U(n) be the canonical map. Then the induced pullback operation on ordinary cohomology is of the form

(Bi n) *:[c 1,,c k][(c 1) 1,(c 1) n] (B i_n)^\ast \;\colon\; \mathbb{Z}[c_1, \cdots, c_k] \longrightarrow \mathbb{Z}[(c_1)_1,\cdots (c_1)_n]

and sends the kkth Chern class c kc_k (def. 1) to the kkth elementary symmetric polynomial in the nn copies of the first Chern class:

(Bi n) *:c kσ k((c 1) 1,,(c 1) n). (B i_n)^\ast \;\colon\; c_k \mapsto \sigma_k( (c_1)_1, \cdots, (c_1)_n ) \,.
Proof

First consider the case n=1n = 1.

The classifying space BU(1)B U(1) is equivalently the infinite complex projective space P \mathbb{C}P^\infty. Its ordinary cohomology is the polynomial ring on a single generator c 1c_1, the first Chern class (prop.)

H (BU(1))[c 1]. H^\bullet(B U(1)) \simeq \mathbb{Z}[ c_1 ] \,.

Moreover, Bi 1B i_1 is the identity and the statement follows.

Now by the Künneth theorem for ordinary cohomology (prop.) the cohomology of the Cartesian product of nn copies of BU(1)B U(1) is the polynomial ring in nn generators

H (BU(1) n)[(c 1) 1,,(c 1) n]. H^\bullet(B U(1)^n) \simeq \mathbb{Z}[(c_1)_1, \cdots, (c_1)_n] \,.

By prop. 1 the domain of (Bi n) *(B i_n)^\ast is the polynomial ring in the Chern classes {c i}\{c_i\}, and by the previous statement the codomain is the polynomial ring on nn copies of the first Chern class

(Bi n) *:[c 1,,c n][(c 1) 1,,(c 1) n]. (B i_n)^\ast \;\colon\; \mathbb{Z}[ c_1, \cdots, c_n ] \longrightarrow \mathbb{Z}[ (c_1)_1, \cdots, (c_1)_n ] \,.

This allows to compute (Bi n) *(c k)(B i_n)^\ast(c_k) by induction:

Consider n2n \geq 2 and assume that (Bi n1) n1 *(c k)=σ k((c 1) 1,,(c 1) (n1))(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)}). We need to show that then also (Bi n) *(c k)=σ k((c 1) 1,,(c 1) n)(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n).

Consider then the commuting diagram

BU(1) n1 Bi n1 BU(n1) Bj t^ Bi t^ BU(1) n Bi n BU(n) \array{ B U(1)^{n-1} &\overset{ B i_{n-1} }{\longrightarrow}& B U(n-1) \\ {}^{\mathllap{B j_{\hat t}}}\downarrow && \downarrow^{\mathrlap{B i_{\hat t}}} \\ B U(1)^n &\underset{B i_n}{\longrightarrow}& B U(n) }

where both vertical morphisms are induced from the inclusion

n1 n \mathbb{C}^{n-1} \hookrightarrow \mathbb{C}^n

which omits the ttth coordinate.

Since two embeddings i t^ 1,i t^ 2:U(n1)U(n)i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n) differ by conjugation with an element in U(n)U(n), hence by an inner automorphism, the maps Bi t^ 1B i_{\hat t_1} and B i^ t 2B_{\hat i_{t_2}} are homotopic, and hence (Bi t^) *=(Bi n^) *(B i_{\hat t})^\ast = (B i_{\hat n})^\ast, which is the morphism from prop. 1.

By that proposition, (Bi t^) *(B i_{\hat t})^\ast is the identity on c k<nc_{k \lt n} and hence by induction assumption

(Bi n1) *(Bi t^) *c k<n =(Bi n1) *c k<n =σ k((c 1) 1,,(c 1) t^,,(c 1) n). \begin{aligned} (B i_{n-1})^\ast (B i_{\hat t})^\ast c_{k \lt n} &= (B i_{n-1})^\ast c_{k \lt n} \\ = \sigma_k( (c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n ) \end{aligned} \,.

Since pullback along the left vertical morphism sends (c 1) t(c_1)_t to zero and is the identity on the other generators, this shows that

(Bi n) *(c k<n)σ k<n((c 1) 1,,(c 1) t^,,(c 1) n)mod(c 1) t. (B i_n)^\ast(c_{k \lt n}) \simeq \sigma_{k\lt n}((c_1)_1, \cdots, \widehat{(c_1)_t}, \cdots, (c_1)_n) \;\; mod (c_1)_t \,.

This implies the claim for k<nk \lt n.

For the case k=nk = n the commutativity of the diagram and the fact that the right map is zero on c nc_n by prop. 1 shows that the element (Bj t^) *(Bi n) *c n=0(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0 for all 1tn1 \leq t \leq n. But by lemma 1 the morphism (Bi n) *(B i_n)^\ast, is injective, and hence (Bi n) *(c n)(B i_n)^\ast(c_n) is non-zero. Therefore for this to be annihilated by the morphisms that send (c 1) t(c_1)_t to zero, for all tt, the element must be proportional to all the (c 1) t(c_1)_t. By degree reasons this means that it has to be the product of all of them

(Bi n) *(c n) =(c 1) 1(c 1) 2(c 1) n =σ n((c 1) 1,,(c 1) n). \begin{aligned} (B i_n)^{\ast}(c_n) & = (c_1)_1 \otimes (c_1)_2 \otimes \cdots \otimes (c_1)_n \\ & = \sigma_n( (c_1)_1, \cdots, (c_1)_n ) \end{aligned} \,.

This completes the induction step.

Whitney sum formula

Proposition

For knk\leq n \in \mathbb{N}, consider the canonical map

μ k,nk:BU(k)×BU(nk)BU(n) \mu_{k,n-k} \;\colon\; B U(k) \times B U(n-k) \longrightarrow B U(n)

(which classifies the Whitney sum of complex vector bundles of rank kk with those of rank nkn-k). Under pullback along this map the universal Chern classes (prop. 1) are given by

(μ k,nk) *(c t)=i=0tc ic ti, (\mu_{k,n-k})^\ast(c_t) \;=\; \underoverset{i = 0}{t}{\sum} c_i \otimes c_{t-i} \,,

where we take c 0=1c_0 = 1 and c j=0H (BU(r))c_j = 0 \in H^\bullet(B U(r)) if j>rj \gt r.

So in particular

(μ k,nk) *(c n)=c kc nk. (\mu_{k,n-k})^\ast(c_n) \;=\; c_k \otimes c_{n-k} \,.

e.g. (Kochmann 96, corollary 2.3.4)

Proof

Consider the commuting diagram

H (BU(n)) μ k,nk * H (BU(k))H (BU(nk)) μ k * μ k *μ nk * H (BU(1) n) H (BU(1) k)H (BU(1) nk). \array{ H^\bullet( B U(n) ) &\overset{\mu_{k,n-k}^\ast}{\longrightarrow}& H^\bullet( B U(k) ) \otimes H^\bullet( B U(n-k) ) \\ {}^{\mathllap{\mu_k^\ast}}\downarrow && \downarrow^{\mathrlap{ \mu_{k}^\ast \otimes \mu_{n-k}^\ast }} \\ H^\bullet( B U(1)^n ) &\simeq& H^\bullet( B U(1)^k ) \otimes H^\bullet( B U(1)^{n-k} ) } \,.

This says that for all tt then

(μ k *μ nk *)μ k,nk *(c t) =μ n *(c t) =σ t((c 1) 1,,(c 1) n), \begin{aligned} (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) & = \mu^\ast_n(c_t) \\ & = \sigma_t((c_1)_1, \cdots, (c_1)_n) \end{aligned} \,,

where the last equation is by prop. 2.

Now the elementary symmetric polynomial on the right decomposes as required by the left hand side of this equation as follows:

σ t((c 1) 1,,(c 1) n)=r=0tσ r((c 1) 1,,(c 1) nk)σ tr((c 1) nk+1,,(c 1) n), \sigma_t((c_1)_1, \cdots, (c_1)_n) \;=\; \underoverset{r = 0}{t}{\sum} \sigma_r((c_1)_1, \cdots, (c_1)_{n-k}) \cdot \sigma_{t-r}( (c_1)_{n-k+1}, \cdots, (c_1)_n ) \,,

where we agree with σ q((c 1) 1,,(c 1) p)=0\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0 if q>pq \gt p. It follows that

(μ k *μ nk *)μ k,nk *(c t)=(μ k *μ nk *)(r=0tc rc tr). (\mu_k^\ast \otimes \mu_{n-k}^\ast) \mu_{k,n-k}^\ast(c_t) = (\mu_k^\ast \otimes \mu_{n-k}^\ast) \left( \underoverset{r=0}{t}{\sum} c_r \otimes c_{t-r} \right) \,.

Since (μ k *μ nk *)(\mu_k^\ast \otimes \mu_{n-k}^\ast) is a monomorphism by lemma 1, this implies the claim.

In Yang-Mills theory field configurations with non-vanishing second Chern-class (and minimal energy) are called instantons. The second Chern class is the instanton number .

References

Original articles include

  • A. Grothendieck, La théorie des classes de Chern, Bulletin de la Société Mathématique de France 86 (1958), p. 137–154, numdam

Textbook accounts include

A brief introduction is in chapter 23, section 7

For Conner-Floyd Chern classes in complex oriented cohomology theory:

See also

  • Dupont, Curvature and characteristic classes, Springer 1978

Revised on July 7, 2016 14:42:47 by Urs Schreiber (131.220.184.222)