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reduced cohomology

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Cohomology

cohomology

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Definition

Definition

A reduced cohomology theory is a functor

E˜ :(Top CW */) opAb \tilde E^\bullet \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}}

from the opposite of pointed topological spaces (CW-complexes) to \mathbb{Z}-graded abelian groups (“cohomology groups”), in components

E˜:(XfY)(E˜ (Y)f *E˜ (X)), \tilde E \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X)) \,,

and equipped with a natural isomorphism of degree +1, to be called the suspension isomorphism, of the form

σ:E˜ +1(Σ)E˜ () \sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)

such that:

  1. (homotopy invariance) If f 1,f 2:XYf_1,f_2 \colon X \longrightarrow Y are two morphisms of pointed topological spaces such that there is a (base point preserving) homotopy f 1f 2f_1 \simeq f_2 between them, then the induced homomorphisms of abelian groups are equal

    f 1 *=f 2 *. f_1^\ast = f_2^\ast \,.
  2. (exactness) For i:AXi \colon A \hookrightarrow X an inclusion of pointed topological spaces, with j:XCone(i)j \colon X \longrightarrow Cone(i) the induced mapping cone, then this gives an exact sequence of graded abelian groups

    E˜ (Cone(i))j *E˜ (X)i *E˜ (A). \tilde E^\bullet(Cone(i)) \overset{j^\ast}{\longrightarrow} \tilde E^\bullet(X) \overset{i^\ast}{\longrightarrow} \tilde E^\bullet(A) \,.

We say E˜ \tilde E^\bullet is additive if in addition

  • (wedge axiom) For {X i} iI\{X_i\}_{i \in I} any set of pointed CW-complexes, then the canonical comparison morphism

    E˜ ( iIX i) iIE˜ (X i) \tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i)

    is an isomorphism, from the functor applied to their wedge sum, example \ref{WedgeSumAsCoproduct}, to the product of its values on the wedge summands, .

We say E˜ \tilde E^\bullet is ordinary if its value on the 0-sphere S 0S^0 is concentrated in degree 0:

  • (Dimension) E˜ 0(𝕊 0)0\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0.

A homomorphism of reduced cohomology theories

η:E˜ F˜ \eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet

is a natural transformation between the underlying functors which is compatible with the suspension isomorphisms in that all the following squares commute

E˜ (X) η X F˜ (X) σ E σ F E˜ +1(ΣX) η ΣX F˜ +1(ΣX). \array{ \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F^\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E^{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F^{\bullet + 1}(\Sigma X) } \,.

(e.g. AGP 02, def. 12.1.4)

Properties

Brown representability

The Brown representability theorem says that for any reduced cohomology theory E˜ \tilde E^\bullet there is an Omega-spectrum EE which represents E˜ \tilde E^\bullet on pointed connected CW-complex XX, in that

E˜ n(X)[X,E n] *. \tilde E^n(X) \simeq [X,E_n]_\ast \,.

Relation to unreduced cohomology

For an unreduced cohomology theory E E^\bullet the induced reduced cohomology is

E˜ k(X,x 0)E k(X,{x 0})=ker(H k(X)H k({x 0})) \tilde E^k(X,x_0) \coloneqq E^k(X,\{x_0\}) = ker(H^k(X)\to H^k(\{x_0\}))

e.g. AGP 02, theorem 12.1.12

For more see at generalized cohomology – Relation btween reduced and unreduced.

References

See the references at generalized (Eilenberg-Steenrod) cohomology.

Revised on April 21, 2016 05:47:54 by Urs Schreiber (131.220.184.222)