group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The ordinary Chern classes are the integral characteristic classes
of the classifying space $B U$ 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).
For $n \geq 1$ the Chern universal characteristic classes $c_i \in H^{2i}(B U(n), \mathbb{Z})$ of the classifying space $B U(n)$ of the unitary group are characterized as follows:
$c_0 = 1$ and $c_i = 0$ if $i \gt n$;
for $n = 1$, $c_1$ is the canonical generator of $H^2(B U(1), \mathbb{Z})\simeq \mathbb{Z}$;
under pullback along the inclusion $i : B U(n) \to B U(n+1)$ we have $i^* c_i^{(n+1)} = c_i^{(n)}$;
under the inclusion $B U(k) \times B U(l) \to B U(k+l)$ we have $i^* c_i = \sum_{j = 0}^i c_i \cup c_{j-i}$.
The cohomology ring of the classifying space $B U(n)$ (for the unitary group $U(n)$) is the polynomial ring on generators $\{c_k\}_{k = 1}^{n}$ of degree 2, called the Chern classes
Moreover, for $B i \colon B U(n_1) \longrightarrow BU(n_2)$ the canonical inclusion for $n_1 \leq n_2 \in \mathbb{N}$, then the induced pullback map on cohomology
is given by
(e.g. Kochmann 96, theorem 2.3.1)
For $n = 1$, in which case $B U(1) \simeq \mathbb{C}P^\infty$ is the infinite complex projective space, we have (prop)
where $c_1$ is the first Chern class. From here we proceed by induction. So assume that the statement has been shown for $n-1$.
Observe that the canonical map $B U(n-1) \to B U(n)$ has as homotopy fiber the (2n-1)sphere (prop.) hence there is a homotopy fiber sequence of the form
Consider the induced Thom-Gysin sequence.
In odd degrees $2k+1 \lt 2n$ it gives the exact sequence
where the right term vanishes by induction assumption, and the middle term since ordinary cohomology vanishes in negative degrees. Hence
Then for $2k+1 \gt 2n$ the Thom-Gysin sequence gives
where again the right term vanishes by the induction assumption. Hence exactness now gives that
is an epimorphism, and so with the previous statement it follows that
for all $k$.
Next consider the Thom Gysin sequence in degrees $2k$
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
for all $k$. In degrees $\bullet\leq 2n$ this says
for some Thom class $c_n \in H^{2n}(B U(n))$, 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
Now by another induction over these short exact sequences, the claim follows.
The first Chern class of a bundle $P$ is the class of its determinant line bundle $det P$
See determinant line bundle for more.
Under the splitting principle all Chern classes are determnined by first Chern classes:
Write $i \colon T \simeq U(1)^n \hookrightarrow U(n)$ for the maximal torus inside the unitary group, which is the subgroup of diagonal unitary matrices. Then
is the polynomial ring in $n$ generators (to be thought of as the universal first Chern classes $c_i$ of each copy of $B U(1)$; equivalently as the weights of the group characters of $U(n)$) which are traditionally written $x_i$:
Write
for the induced map of deloopings/classifying spaces, then the $k$-universal Chern class $c_k \in H^{2k}(B U(n), \mathbb{Z})$ is uniquely characterized by the fact that its pullback to $B U(1)^n$ is the $k$th elementary symmetric polynomial $\sigma_k$ applied to these first Chern classes:
Equivalently, for $c = \sum_{i = 1}^n c_k$ the formal sum of all the Chern classes, and using the fact that the elementary symmetric polynomials $\sigma_k(x_1, \cdots, k_n)$ are the degree-$k$ piece in $(1+x_1) \cdots (1+x_n)$, this means that
Since here on the right the first Chern classes $x_i$ 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)
For $n \in \mathbb{N}$ let $\mu_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on ordinary cohomology
is a monomorphism.
A proof of lemma 1 via analysis of the Serre spectral sequence of $U(n)/U(1)^n \to B U(1)^n \to B U(n)$ is indicated in (Kochmann 96, p. 40). A proof via transfer of the Euler class of $U(n)/U(1)^n$, following (Dupont 78, (8.28)), is indicated at splitting principle (here).
For $k \leq n \in \mathbb{N}$ let $B i_n \;\colon\; B (U(1)^n) \longrightarrow B U(n)$ be the canonical map. Then the induced pullback operation on ordinary cohomology is of the form
and sends the $k$th Chern class $c_k$ (def. 1) to the $k$th elementary symmetric polynomial in the $n$ copies of the first Chern class:
First consider the case $n = 1$.
The classifying space $B U(1)$ is equivalently the infinite complex projective space $\mathbb{C}P^\infty$. Its ordinary cohomology is the polynomial ring on a single generator $c_1$, the first Chern class (prop.)
Moreover, $B i_1$ is the identity and the statement follows.
Now by the Künneth theorem for ordinary cohomology (prop.) the cohomology of the Cartesian product of $n$ copies of $B U(1)$ is the polynomial ring in $n$ generators
By prop. 1 the domain of $(B i_n)^\ast$ is the polynomial ring in the Chern classes $\{c_i\}$, and by the previous statement the codomain is the polynomial ring on $n$ copies of the first Chern class
This allows to compute $(B i_n)^\ast(c_k)$ by induction:
Consider $n \geq 2$ and assume that $(B i_{n-1})^\ast_{n-1}(c_k) = \sigma_k((c_1)_1, \cdots, (c_1)_{(n-1)})$. We need to show that then also $(B i_n)^\ast(c_k) = \sigma_k((c_1)_1,\cdots, (c_1)_n)$.
Consider then the commuting diagram
where both vertical morphisms are induced from the inclusion
which omits the $t$th coordinate.
Since two embeddings $i_{\hat t_1}, i_{\hat t_2} \colon U(n-1) \hookrightarrow U(n)$ differ by conjugation with an element in $U(n)$, hence by an inner automorphism, the maps $B i_{\hat t_1}$ and $B_{\hat i_{t_2}}$ are homotopic, and hence $(B i_{\hat t})^\ast = (B i_{\hat n})^\ast$, which is the morphism from prop. 1.
By that proposition, $(B i_{\hat t})^\ast$ is the identity on $c_{k \lt n}$ and hence by induction assumption
Since pullback along the left vertical morphism sends $(c_1)_t$ to zero and is the identity on the other generators, this shows that
This implies the claim for $k \lt n$.
For the case $k = n$ the commutativity of the diagram and the fact that the right map is zero on $c_n$ by prop. 1 shows that the element $(B j_{\hat t})^\ast (B i_n)^\ast c_n = 0$ for all $1 \leq t \leq n$. But by lemma 1 the morphism $(B i_n)^\ast$, is injective, and hence $(B i_n)^\ast(c_n)$ is non-zero. Therefore for this to be annihilated by the morphisms that send $(c_1)_t$ to zero, for all $t$, the element must be proportional to all the $(c_1)_t$. By degree reasons this means that it has to be the product of all of them
This completes the induction step.
For $k\leq n \in \mathbb{N}$, consider the canonical map
(which classifies the Whitney sum of complex vector bundles of rank $k$ with those of rank $n-k$). Under pullback along this map the universal Chern classes (prop. 1) are given by
where we take $c_0 = 1$ and $c_j = 0 \in H^\bullet(B U(r))$ if $j \gt r$.
So in particular
e.g. (Kochmann 96, corollary 2.3.4)
Consider the commuting diagram
This says that for all $t$ then
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:
where we agree with $\sigma_q((c_1)_1, \cdots, (c_1)_p) = 0$ if $q \gt p$. It follows that
Since $(\mu_k^\ast \otimes \mu_{n-k}^\ast)$ 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 . For more on this see at SU(2)-instantons from the correct maths to the traditional physics story.
Original articles include
Textbook accounts include
Werner Greub, Stephen Halperin, Ray Vanstone, chapter IX of volume II of Connections, Curvature, and Cohomology Academic Press (1973)
John Milnor, James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton University Press (1974).
Stanley Kochmann, section 2.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Tammo tom Dieck, Algebraic topology, EMS 2008
A brief introduction is in chapter 23, section 7
For Conner-Floyd Chern classes in complex oriented cohomology theory:
Frank Adams, part II.2 and part III.10 of Stable homotopy and generalised homology, 1974
Jacob Lurie, Chromatic Homotopy Theory, 2010, lecture 4 (pdf) and lecture 5 (pdf)
See also