nLab multiplicative cohomology theory

Idea

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

The corresponding representing spectra are ring spectra.

Definition

Definition

Let $E_1, E_2, E_3$ be three unreduced generalized cohomology theories (def.). A pairing of cohomology theories

$$\mu \;\colon\; E_1 \Box E_2 \longrightarrow E_3$$

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

$$\mu_{n_1,n_2} \;\colon\; E_1^{n_1}(X,A) \otimes E_2^{n_2}(Y,B) \longrightarrow E_3^{n_1 + n_2}(X\times Y \;,\; A\times Y \cup X \times B)$$

such that this is compatible with the connecting homomorphisms $\delta_i$ of $E_i$, in that the following are commuting squares

$$\array{ E_1^{n_1}(A) \otimes E_2^{n_2}(Y,B) &\overset{\delta_1 \otimes id_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1+1, n_2}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , X \times B)} {E_3^{n_1 + n_2}(A \times Y, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) }$$

and

$$\array{ E_1^{n_1}(X,A) \otimes E_2^{n_2}(B) &\overset{(-1)^{n_1} id_1 \otimes \delta_2}{\longrightarrow}& E_1^{n_1+1}(X,A) \otimes E_2^{n_2}(Y,B) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1, n_2 + 1}}} \\ \underoverset {E_3^{n_1 + n_2}(A \times Y \cup X \times B , A \times Y)} {E_3^{n_1 + n_2}(X \times B, A \times B)} {\simeq} &\overset{\delta_3}{\longrightarrow}& E_3^{n_1 + n_2+ 1}(X \times Y, A \times B) } \,,$$

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

Definition

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

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

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

such that

1. (associativity) $\mu \circ (id \otimes \mu) = \mu \circ (\mu \otimes id)$;

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

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

• (graded commutativity)

  $$\array{ E^{n_1}(X,A) \otimes E^{n_2}(Y,B) &\overset{(u \otimes v) \mapsto (-1)^{n_1 n_2} (v \otimes u) }{\longrightarrow}& E^{n_2}(Y,B) \otimes E^{n_1}(X,A) \\ {}^{\mathllap{\mu_{n_1,n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2}}} \\ E^{n_1 + n_2}( X \times Y , A \times Y \cup X \times B) &\underset{(switch_{(X,A), (Y,B)})^\ast}{\longrightarrow}& E^{n_1 + n_2}( Y \times X , B \times X \cup Y \times A) } \,.$$

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

$$(-) \cup (-) \;\colon\; E^{n_1}(X,A) \otimes E^{n_2}(X,A) \overset{\mu_{n_1,n_2}}{\longrightarrow} E^{n_1 + n_2}( X \times X, \; A \times X \cup X \times A) \overset{\Delta^\ast_{(X,A)}}{\longrightarrow} E^{n_1 + n_2}(X, \; A \cup B) \,.$$

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

Properties

Ring and module structure on cohomology groups

Proposition

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

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

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

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

Proof

Regarding the third point:

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

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

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

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

Multiplicative Atiyah-Hirzebruch spectral sequences

Proposition

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

A proof of prop. 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 $(A,\mu,1)$ (def. ), then for every Serre fibration $X \to B$ all the differentials in the corresponding Atiyah-Hirzebruch spectral sequence

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

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

Proof

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

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

consisting of the pullback homomorphisms and the connecting homomorphisms of $A$.

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

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, for more details see here).

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

Examples

• ordinary cohomology with coefficients in a ring, in particular integral cohomology

• topological K-theory, K-theory

• cobordism cohomology theory

• tmf

• etc….

Related entries

• Kronecker pairing

• universal coefficient theorem, module cohomology theories, …

See also

• complex oriented cohomology theory, chromatic homotopy theory

• Thom isomorphism

• orientation in generalized cohomology

• fiber integration

• K-theory of a bipermutative category

• multiplicative spectral sequence

References

• Pierre Conner, Edwin Floyd, p. 23 (30 of 120) in: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• John Michael Boardman, Section 3 (p. 10-11) of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford 1995 (doi:10.1016/B978-0-444-81779-2.X5000-7)

• Dai Tamaki, Akira Kono, chapter 2.6 of Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (pdf, ISBN: 978-0-8218-3514-2)

• Jacob Lurie, A Survey of Elliptic Cohomology - cohomology theories

See also

• Stanley Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996