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multiplicative cohomology theory

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Idea

A cohomology theory EE is called multiplicative if each graded abelian EE-cohomology group E (X)E^\bullet(X) is compatibly equippd with the structure of a graded ring.

Definition

Definition

Let E 1,E 2,E 3E_1, E_2, E_3 be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories

μ:E 1E 2E 3 \mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3

is a natural transformation (of functors on (Top CW ×Top CW ) op(Top_{CW}^{\hookrightarrow}\times Top_{CW}^{\hookrightarrow})^{op} ) of the form

μ n 1,n 2:E 1 n 1(X,A)E 2 n 2(Y,B)E 3 n 1+n 2(X×Y,A×YX×B) \mu_{n_1,n_2} \;\colon\; E_1^{n_1}(X,A) \otimes E_2^{n_2}(Y,B) \longrightarrow E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)

such that this is compatible with the connecting homomorphisms δ i\delta_i of E iE_i, in that the following are commuting squares

E 1 n 1(A)E 2 n 2(Y,B) δ 1id 2 E 1 n 1+1(X,A)E 2 n 2(Y,B) μ n 1,n 2 μ n 1+1,n 2 E 3 n 1+n 2(A×YX×B,X×B)E 3 n 1+n 2(A×Y,A×B) δ 3 E 3 n 1+n 2+1(X×Y,A×B) \array{ E_1^{n_1}(A) \otimes E_2^{n_2}(Y,B) &\overset{\delta_1 \otimes id_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , X \times B)} {E_3^{n_1 + n_2}(A \times Y, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) }

and

E 1 n 1(X,A)E 2 n 2(B) (1) n 1id 1δ 2 E 1 n 1+1(X,A)E 2 n 2(Y,B) μ n 1,n 2 μ n 1,n 2+1 E 3 n 1+n 2(A×YX×B,A×Y)E 3 n 1+n 2(X×B,A×B) δ 3 E 3 n 1+n 2+1(X×Y,A×B), \array{ E_1^{n_1}(X,A) \otimes E_2^{n_2}(B) &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)} {E_3^{n_1 + n_2}(X \times B, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) } \,,

where the isomorphisms in the bottom left are the excision isomorphisms.

Definition

An (unreduced) multiplicative cohomology theory is an unreduced generalized cohomology theory theory EE (def.) equipped with

  1. (external multiplication) a pairing (def. 1) of the form μ:EEE\mu \;\colon\; E \Box E \longrightarrow E;

  2. (unit) an element 1E 0(*)1 \in E^0(\ast)

such that

  1. (associativity) μ(idμ)=μ(μid)\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id);

  2. (unitality) μ(1x)=μ(x1)=x\mu(1\otimes x) = \mu(x \otimes 1) = x for all xE n(X,A)x \in E^n(X,A).

The mulitplicative cohomology theory is called commutative (often considered by default) if in addition

  • (graded commutativity)

    E n 1(X,A)E n 2(Y,B) (uv)(1) n 1n 2(vu) E n 2(Y,B)E X,A n 1 μ n 1,n 2 μ n 1,n 2 E n 1+n 2(X×Y,A×YX×B) (switch (X,A),(Y,B)) * E n 1+n 2(Y×X,B×XY×A). \array{ E^{n_1}(X,A) \otimes E^{n_2}(Y,B) &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}& E^{n_2}(Y,B) \otimes E^{n_1}_{X,A} \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}} \\ E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B) &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}& E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A) } \,.

Given a multiplicative cohomology theory (E,μ,1)(E, \mu, 1), its cup product is the composite of the above external multiplication with pullback along the diagonal maps Δ (X,A):(X,A)(X×X,A×XX×A)\Delta_{(X,A)} \colon (X,A) \longrightarrow (X\times X, A \times X \cup X \times A);

()():E n 1(X,A)E n 2(X,A)μ n 1,n 2E n 1+n 2(X×X,A×XX×A)Δ (X,A) *E n 1+n 2(X,AB). (-) \cup (-) \;\colon\; E^{n_1}(X,A) \otimes E^{n_2}(X,A) \overset{\mu_{n_1,n_2}}{\longrightarrow} E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A) \overset{\Delta^\ast_{(X,A)}}{\longrightarrow} E^{n_1 + n_2}(X, \; A \cup B) \,.

e.g. (Tamaki-Kono 06, II.6)

Properties

Ring and module structure on cohomology groups

Proposition

Let (E,μ,1)(E,\mu,1) be a multiplicative cohomology theory, def. 2. Then

  1. For every space XX the cup product gives E (X)E^\bullet(X) the structure of a \mathbb{Z}-graded ring, which is graded-commutative if (E,μ,1)(E,\mu,1) is commutative.

  2. For every pair (X,A)(X,A) the external multiplication μ\mu gives E (X,A)E^\bullet(X,A) the structure of a left and right module over the graded ring E (*)E^\bullet(\ast).

  3. All pullback morphisms respect the left and right action of E (*)E^\bullet(\ast) and the connecting homomorphisms respect the right action and the left action up to multiplication by (1) n 1(-1)^{n_1}

Proof

Regarding the third point:

For pullback maps this is the naturality of the external product: let f:(X,A)(Y,B)f \colon (X,A) \longrightarrow (Y,B) be a morphism in Top CW Top_{CW}^{\hookrightarrow} then naturality says that the following square commutes:

E n 1(*)E n 2(Y,B) μ n 1,n 2 E n 1+n 2(Y,B) (id,f *) f * E n 1(*)E n 2(X,A) μ n 1,n 2 E n 1+n 2(Y,B). \array{ E^{n_1}(\ast) \otimes E^{n_2}(Y,B) &\overset{\mu_{n_1,n_2}}{\longrightarrow}& E^{n_1 + n_2}(Y, B) \\ {}^{\mathllap{(id,f^\ast)}}\downarrow && \downarrow^{\mathrlap{f^\ast}} \\ E^{n_1}(\ast) \otimes E^{n_2}(X,A) &\overset{\mu_{n_1,n_2}}{\longrightarrow}& E^{n_1 + n_2}(Y,B) } \,.

For connecting homomorphisms this is the (graded) commutativity of the squares in def. 2:

E n 1(*)E n 2(A) (1) n 1(id,δ) E n 1(*)E n 2+2(X) μ n 1,n 2 μ n 1,n 2 E n 1+n 2(A) δ E 3 n 1+n 2+1(X,B). \array{ E^{n_1}(\ast)\otimes E^{n_2}(A) &\overset{(-1)^{n_1} (id, \delta)}{\longrightarrow}& E^{n_1}(\ast) \otimes E^{n_2 + 2}(X) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}} \\ E^{n_1 + n_2}(A) &\overset{\delta}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X,B) } \,.

Multiplicative Atiyah-Hirzebruch spectral sequences

Proposition

Given a multiplicative cohomology theory (A,μ,1)(A,\mu,1) (def. 2), then for every Serre fibration XBX \to B the corresponding Atiyah-Hirzebruch spectral sequence inherits the structure of a multiplicative spectral sequence.

A proof of prop. 2 via Cartan-Eilenberg systems is given at multiplicative spectral sequence. A proof arguing via representing ring spectra is in (Kochman 96, prop. 4.2.9).

Proposition

Given a multiplicative cohomology theory (A,μ,1)(A,\mu,1) (def. 2), then for every Serre fibration XBX \to B all the differentials in the corresponding Atiyah-Hirzebruch spectral sequence

H (B,A (F))A (X) H^\bullet(B,A^\bullet(F)) \;\Rightarrow\; A^\bullet(X)

are linear over A (*)A^\bullet(\ast).

Proof

By construction (here) the differentials are those induced by the exact couple

s,tA s+t(X s) s,tA s+t(X s) s,tA s+t(X s,X s1)(A s+t(X s) A s+t(X s1) δ A s+t(X s,X s1) A s+t+1(X s,X s1)). \array{ \underset{s,t}{\prod} A^{s+t}(X_{s}) && \stackrel{}{\longrightarrow} && \underset{s,t}{\prod} A^{s+t}(X_{s}) \\ & \nwarrow && \swarrow \\ && \underset{s,t}{\prod} A^{s+t}(X_{s}, X_{s-1}) } \;\;\;\;\;\;\; \left( \array{ A^{s+t}(X_s) & \longrightarrow & A^{s+t}(X_{s-1}) \\ \uparrow && \downarrow_{\mathrlap{\delta}} \\ A^{s+t}(X_s, X_{s-1}) && A^{s+t+1}(X_{s}, X_{s-1}) } \right) \,.

consisting of the pullback homomorphisms and the connecting homomorphisms of AA.

By the nature of spectral sequences induced from exact couples (this prop.) its differentials on page rr are the composites of one pullback homomorphism, the preimage of (r1)(r-1) pullback homomorphisms, and one connecting homomorphism of AA. Hence the statement follows with prop. 1.

Brown representability by ring spectra

A multiplicative structure on a generalized (Eilenberg-Steenrod) cohomology theory is the structure of a ring spectrum on the spectrum that represents it.

e.g. (Tamaki-Kono 06, appendix C, Lurie 10, lecture 4)

In particular every E-∞ ring is a ring spectrum, hence represents a multiplicative cohomology theory, but the converse is in general false.

Examples

See also

References

See also

Revised on June 3, 2016 04:46:49 by Urs Schreiber (131.220.184.222)