nLab Conner-Floyd Chern class

Idea

The concept of Chern classes of complex vector bundles, as universal characteristic classes in ordinary cohomology, generalizes to any complex oriented generalized cohomology theory: the Conner-Floyd Chern classes (Conner-Floyd 66, Section 7, Adams 74, Section I.4, review in Kochman 96, Section 4.3, Lurie 10, Lectures 4 and 5):

For $E$ a generalized cohomology theory, the analog of the first Chern class in $E$-cohomology is what appears in the very definition of complex oriented cohomology. The higher generalized Chern classes are induced from this by the splitting principle. See also at complex oriented cohomology – the cohomology ring of BU(n).

The Conner-Floyd Chern classes in top degree serve as the generalized Thom classes that make every complex vector bundle have orientation in generalized cohomology with respect to any complex oriented cohomology theory (Lurie 10, lecture 5, prop. 6).

Definition

The definition of the $E$-Chern classes according to Conner-Floyd 66, Theorem 7.6 proceeds as follows.

Let $n \in \mathbb{N}$. Let $c_1^E$ in

$$\array{ \mathbb{C}P^1 & \overset { \Sigma^2 (1^E) } {\longrightarrow} & E_2 \\ \big\downarrow & \nearrow_{ \mathrlap{ c_1^E } } \\ \mathbb{C}P^\infty }$$

be a complex orientation in $E$-cohomology.

For $\mathcal{V} \longrightarrow X$ a complex vector bundle of rank $n + 1$ consider the diagram

(1)$$\array{ && \mathcal{V}' \oplus \mathcal{L}_{{}_{P(\mathcal{V})}} &\longrightarrow& \mathcal{V} \\ && \big\downarrow & {}^{_{(pb)}} & \big\downarrow \\ \mathbb{C}P^n & \underset{ fib_x }{ \longrightarrow } & P(\mathcal{V}) & \underset {\pi} {\longrightarrow} & X }$$

where

$$P(\mathcal{V}) \;\coloneqq\; \big( \mathcal{V} \setminus \{ X \times \{0\} \} \big) / \mathbb{C}^\times \;\simeq\; S(\mathcal{V})/\mathrm{U}(1)$$

is the projective bundle of $\mathcal{V}$, with typical fiber the complex projective spaces $\mathbb{C}P^n$, and where in the middle we are displaying the splitting (here) of the pullback bundle of $E$ into a direct sum of a line bundle $\mathcal{L}$ with a remaining vector bundle $\mathcal{V}'$ of rank just $n$.

Using the defining conditions on the total Conner-Floyd Chern class

$$c^E(\mathcal{V}) \;\coloneqq\; 1 + c_1^E(\mathcal{V}) + c_2^E(\mathcal{V}) + c_3^E(\mathcal{V}) + \cdots$$

that

1. on a complex line bundle we have $c^E(\mathcal{L}) = 1 + c_1^E(\mathcal{L})$ for the given complex orientation $c_1^E$;

2. on a direct sum of vector bundles we have the Whitney sum formula $c^E(\mathcal{V} \oplus \mathcal{W}) \;=\; c^E(\mathcal{V}) \cdot c^E(\mathcal{W})$

we see from (1) that

$$\pi^\ast \big( {\color{blue} c^E(\mathcal{V}) } \big) \;=\; c^E(\mathcal{V}') \cdot \big( 1 + {\color{orange} c^E_1(\mathcal{L}) } \big) \;\;\; \in \; E^\bullet\big( P(\mathcal{V}) \big)$$

and hence

$$\begin{aligned} c^E(\mathcal{V}') & =\; \pi^\ast \big( {\color{blue} c^E(\mathcal{V}) } \big) \cdot \big( 1 + { \color{orange} c_1^E(\mathcal{L}) } \big)^{-1} \\ & \coloneqq\; \pi^\ast \big( {\color{blue} c^E(\mathcal{V}) } \big) \cdot \underset{k}{\sum} (-1)^k \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^k \,. \end{aligned}$$

But since $\mathcal{V}'$ is just of rank $n$, we must have $c^E_{n+1}(\mathcal{V}') = 0$, and hence in degree $2(n+1)$ this condition reads as follows:

(2)$$\begin{aligned} 0 & =\; \pi^\ast \big( {\color{blue} c^E_{n+1}(\mathcal{V}) } \big) - \pi^\ast \big( {\color{blue} c^E_{n}(\mathcal{V}) } \big) \cdot {\color{orange} c_1^E(\mathcal{L}) } + \pi^\ast \big( {\color{blue} c^E_{n-1}(\mathcal{V}) } \big) \cdot \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^2 - \cdots + (-1)^n \pi^\ast \big( {\color{blue} c^E_{1}(\mathcal{V}) } \big) \cdot \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^n \\ & \phantom{=} + (-1)^{n+1} \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^{n+1} \,. \end{aligned}$$

It is now sufficient to observe that

1. the $E$-cohomology of $P(\mathcal{V})$ is a free $E^\bullet(X)$-module spanned by the first cup powers of $c_1^E(\mathcal{L})$:

   (3)$$E^\bullet\big( P(\mathcal{V}) \big) \;\simeq\; E^\bullet(X) \Big\langle 1, {\color{orange} c_1^E(\mathcal{L}) } , \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^2 , \cdots , \big( {\color{orange} c_1^E(\mathcal{L}) } \big)^n \Big\rangle \,;$$

2. and in particular

   $$\pi^\ast \;\colon\; E^\bullet( X ) \overset{\;\;\;\;\;}{\hookrightarrow} E^\bullet\big( P(\mathcal{V}) \big)$$

   is an injective function,

because this means that (2) has a unique solution for the classes ${\color{blue} c_k^E(\mathcal{V})} \,\in\, E^{2k}(X)$ – these are the Conner-Floyd Chern classes of $\mathcal{V}$.

But (3) holds on $\mathbb{C}P^n$ after pullback along $fib_x$ (by standard arguments in complex oriented cohomology theory, e.g. Lurie 10, Lecture 4, Example 8) and hence holds on $P(\mathcal{V})$ by the generalized-cohomology version of the Leray-Hirsch theorem (Conner-Floyd 66, Thm. 7.4).

Since this construction is natural, one finds the following universal characteristic classes in $E$-cohomology:

Proposition

Given a complex oriented cohomology theory $E$ with complex orientation $c_1^E$, then the $E$-generalized cohomology of the classifying space $B U(n)$ is freely generated over the graded commutative ring $\pi_\bullet(E)$ (prop.) by classes $c_k^E$ for $0 \leq k \leq n$ of degree $2k$, these are called the Conner-Floyd-Chern classes

$$E^\bullet(B U(n)) \;\simeq\; \pi_\bullet(E)[ [ c_1^E, c_2^E, \cdots, c_n^E ] ] \,.$$

Moreover, pullback along the canonical inclusion $B U(n) \to B U(n+1)$ is the identity on $c_k^E$ for $k \leq n$ and sends $c_{n+1}^E$ to zero.

For $E =$ HZ this reduces to the standard Chern classes.

References

The concept was introduced for:

• Pierre Conner, Edwin Floyd, Theorems 7.5, 7.6 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

  (with the example of MSp- and of MU-cohomology theory in Section 8)

An early account in a broader context of complex oriented cohomology theory:

• Frank Adams, part I.4, part II.2, part III.10 of Stable homotopy and generalised homology, 1974

More recent accounts:

• Stanley Kochman, Section 4.3 of: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Dai Tamaki, Akira Kono, Section 3.4 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Jacob Lurie, Chromatic Homotopy Theory, 2010, lecture 4 (pdf) and lecture 5 (pdf)