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Conner-Floyd Chern class

Contents

Contents

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 EE a generalized cohomology theory, the analog of the first Chern class in EE-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 EE-Chern classes according to Conner-Floyd 66, Theorem 7.6 proceeds as follows.

Let nn \in \mathbb{N}. Let c 1 Ec_1^E in

P 1 Σ 2(1 E) E 2 c 1 E P \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 EE-cohomology.

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

(1) 𝒱 P(𝒱) 𝒱 (pb) P n fib x P(𝒱) π X \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(𝒱)(𝒱{X×{0}})/ ×S(𝒱)/U(1) 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 𝒱\mathcal{V}, with typical fiber the complex projective spaces P n\mathbb{C}P^n, and where in the middle we are displaying the splitting (here) of the pullback bundle of EE into a direct sum of a line bundle \mathcal{L} with a remaining vector bundle 𝒱\mathcal{V}' of rank just nn.

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

c E(𝒱)1+c 1 E(𝒱)+c 2 E(𝒱)+c 3 E(𝒱)+ 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()=1+c 1 E()c^E(\mathcal{L}) = 1 + c_1^E(\mathcal{L}) for the given complex orientation c 1 Ec_1^E;

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

we see from (1) that

π *(c E(𝒱))=c E(𝒱)(1+c 1 E())E (P(𝒱)) \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

c E(𝒱) =π *(c E(𝒱))(1+c 1 E()) 1 π *(c E(𝒱))k(1) k(c 1 E()) k. \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 𝒱\mathcal{V}' is just of rank nn, we must have c n+1 E(𝒱)=0c^E_{n+1}(\mathcal{V}') = 0, and hence in degree 2(n+1)2(n+1) this condition reads as follows:

(2)0 =π *(c n+1 E(𝒱))π *(c n E(𝒱))c 1 E()+π *(c n1 E(𝒱))(c 1 E()) 2+(1) nπ *(c 1 E(𝒱))(c 1 E()) n =+(1) n+1(c 1 E()) n+1. \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 EE-cohomology of P(𝒱)P(\mathcal{V}) is a free E (X)E^\bullet(X)-module spanned by the first cup powers of c 1 E()c_1^E(\mathcal{L}):

    (3)E (P(𝒱))E (X)1,c 1 E(),(c 1 E()) 2,,(c 1 E()) n; 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

    π *:E (X)E (P(𝒱)) \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 c k E(𝒱)E 2k(X){\color{blue} c_k^E(\mathcal{V})} \,\in\, E^{2k}(X) – these are the Conner-Floyd Chern classes of 𝒱\mathcal{V}.

But (3) holds on P n\mathbb{C}P^n after pullback along fib xfib_x (by standard arguments in complex oriented cohomology theory, e.g. Lurie 10, Lecture 4, Example 8) and hence holds on P(𝒱)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 EE-cohomology:

Proposition

Given a complex oriented cohomology theory EE with complex orientation c 1 Ec_1^E, then the EE-generalized cohomology of the classifying space BU(n)B U(n) is freely generated over the graded commutative ring π (E)\pi_\bullet(E) (prop.) by classes c k Ec_k^E for 0kn0 \leq k \leq n of degree 2k2k, these are called the Conner-Floyd-Chern classes

E (BU(n))π (E)[[c 1 E,c 2 E,,c n E]]. 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 BU(n)BU(n+1)B U(n) \to B U(n+1) is the identity on c k Ec_k^E for knk \leq n and sends c n+1 Ec_{n+1}^E to zero.

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

References

The concept was introduced for:

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

More recent accounts:

Last revised on January 25, 2021 at 10:29:12. See the history of this page for a list of all contributions to it.