nLab
reduced cohomology

Contents

Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Given any notion of cohomology defined on pointed objects, the corresponding reduced cohomology is that part of the cohomology which vanishes on the basepoint.

Specifically for Whitehead-generalized cohomology theories the reduced cohomology is the cohomology relative to the base point, hence is the kernel of the operation of pullback to the base point See below and see at generalized cohomology – Relation between reduced and unreduced for more.

Definition

Definition

A reduced cohomology theory is a functor

E˜ :(Top CW */) opAb \tilde E^\bullet \;\colon\; \big( Top^{\ast/}_{CW} \big)^{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\; \big( X \stackrel{f}{\longrightarrow} Y \big) \;\mapsto\; \big( \tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X) \big) \,,

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 , 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 the kernel of operation of pullback to the base point.

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\big( H^k(X) \to H^k(\{x_0\}) \big)

(e.g. AGP 02, theorem 12.1.12).

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

References

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

Last revised on January 5, 2021 at 04:05:47. See the history of this page for a list of all contributions to it.