nLab reduced cohomology

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

$$\tilde E^\bullet \;\colon\; \big( Top^{\ast/}_{CW} \big)^{op} \longrightarrow Ab^{\mathbb{Z}}$$

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

$$\tilde E \;\colon\; \big( X \stackrel{f}{\longrightarrow} Y \big) \;\mapsto\; \big( \tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X) \big) \,,$$

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

$$\sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-)$$

such that:

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

   $$f_1^\ast = f_2^\ast \,.$$

2. (exactness) For $i \colon A \hookrightarrow X$ an inclusion of pointed topological spaces, with $j \colon X \longrightarrow Cone(i)$ the induced mapping cone, then this gives an exact sequence of graded abelian groups

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

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

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

  $$\tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i)$$

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

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

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

A homomorphism of reduced cohomology theories

$$\eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet$$

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

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

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

Properties

Brown representability

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

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

Relation to unreduced cohomology

For an unreduced cohomology theory $E^\bullet$ the induced reduced cohomology is the kernel of operation of pullback to the base point.

$$\tilde E^k(X,x_0) \;\coloneqq\; E^k(X,\{x_0\}) \;=\; ker\big( H^k(X) \to H^k(\{x_0\}) \big)$$

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

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

Related entries

• reduced homology

• vanishing at infinity

References

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

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 12 of: Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

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