nLab generalized homology

Idea

A generalized homology theory is a certain functor from suitable topological spaces to graded abelian groups which satisfies most, but not all, of the abstract properties of ordinary homology functors (e.g. singular homology).

By the Brown representability theorem, under certain conditions every spectrum $K$ is the coefficient object of a generalized cohomology theory and S-dually of a generalized homology theory. For $K = H R$ an Eilenberg-MacLane spectrum this reduces to ordinary homology.

See at generalized (Eilenberg-Steenrod) cohomology for more.

Definition

Reduced homology

Throughout, write Top${}_{CW}$ for the category of topological spaces homeomorphic to CW-complexes. Write $Top^{\ast/}_{CW}$ for the corresponding category of pointed topological spaces.

Recall that colimits in $Top^{\ast/}$ are computed as colimits in $Top$ after adjoining the base point and its inclusion maps to the given diagram

Example

The coproduct in pointed topological spaces is the wedge sum, denoted $\vee_{i \in I} X_i$.

Write

$$\Sigma \coloneqq S^1 \wedge (-) \;\colon\; Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW}$$

for the reduced suspension functor.

Write $Ab^{\mathbb{Z}}$ for the category of integer-graded abelian groups.

Definition

A reduced homology theory is a functor

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

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

$$\tilde E _\bullet \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E_\bullet(X) \stackrel{f_\ast}{\longrightarrow} \tilde E_\bullet(Y)) \,,$$

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

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

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

   $$\tilde E_\bullet(A) \overset{i_\ast}{\longrightarrow} \tilde E_\bullet(X) \overset{j_\ast}{\longrightarrow} \tilde E_\bullet(Cone(i)) \,.$$

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 morphism

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

  from the direct sum of the value on the summands to the value on the wedge sum, example , is an isomorphism.

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

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

Unreduced homology

In the following a pair $(X,A)$ refers to a subspace inclusion of topological spaces (CW-complexes) $A \hookrightarrow X$. Whenever only one space is mentioned, the subspace is assumed to be the empty set $(X, \emptyset)$. Write $Top_{CW}^{\hookrightarrow}$ for the category of such pairs (the full subcategory of the arrow category of $Top_{CW}$ on the inclusions). We identify $Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}$ by $X \mapsto (X,\emptyset)$.

Definition

A homology theory (unreduced, relative) is a functor

$$E_\bullet : (Top_{CW}^{\hookrightarrow}) \longrightarrow Ab^{\mathbb{Z}}$$

to the category of $\mathbb{Z}$-graded abelian groups, as well as a natural transformation of degree +1, to be called the connecting homomorphism, of the form

$$\delta_{(X,A)} \;\colon\; E_{\bullet + 1}(X, A) \longrightarrow E_\bullet(A, \emptyset) \,.$$

such that:

1. (homotopy invariance) For $f \colon (X_1,A_1) \to (X_2,A_2)$ a homotopy equivalence of pairs, then

   $$E_\bullet(f) \;\colon\; E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2)$$

   is an isomorphism;

2. (exactness) For $A \hookrightarrow X$ the induced sequence

   $$\cdots \to E_{n+1}(X, A) \stackrel{\delta}{\longrightarrow} E_n(A) \longrightarrow E_n(X) \longrightarrow E_n(X, A) \to \cdots$$

   is a long exact sequence of abelian groups.

3. (excision) For $U \hookrightarrow A \hookrightarrow X$ such that $\overline{U} \subset Int(A)$, then the natural inclusion of the pair $i \colon (X-U, A-U) \hookrightarrow (X, A)$ induces an isomorphism

   $$E_\bullet(i) \;\colon\; E_n(X-U, A-U) \overset{\simeq}{\longrightarrow} E_n(X, A)$$

We say $E_\bullet$ is additive if it takes coproducts to direct sums:

• (additivity) If $(X, A) = \coprod_i (X_i, A_i)$ is a coproduct, then the canonical comparison morphism

  $$\oplus_i E_n(X_i, A_i) \overset{\simeq}{\longrightarrow} E_n(X, A)$$

  is an isomorphismfrom the direct sum of the value on the summands, to the value on the total pair.

We say $E_\bullet$ is ordinary if its value on the point is concentrated in degree 0

• (Dimension): $E_{\bullet \neq 0}(\ast,\emptyset) = 0$.

A homomorphism of unreduced homology theories

$$\eta \;\colon\; E_\bullet \longrightarrow F_\bullet$$

is a natural transformation of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these squares commute:

$$\array{ E_{\bullet +1}(X,A) &\overset{\eta_{(X,A)}}{\longrightarrow}& F_{\bullet +1}(X,A) \\ {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}} \\ E_\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F_\bullet(A,\emptyset) } \,.$$

Lemma

The excision axiom in def. is equivalent to the following statement:

For all $A,B \hookrightarrow X$ with $X = A \cup B$, then the inclusion

$$i \colon (A, A \cap B) \longrightarrow (X,B)$$

induces an isomorphism,

$$i_\ast \;\colon\; E_\bullet(A, A \cap B) \overset{\simeq}{\longrightarrow} E_\bullet(X, B) \,.$$

(e.g Switzer 75, 7.2, 7.5)

Proof

First consider the statement under the condition that $X = Int(A) \cup Int(B)$.

In one direction, suppose that $E^\bullet$ satisfies the original excision axiom. Given $A,B$ with $X = \Int(A) \cup Int(B)$, set $U \coloneqq X-A$ and observe that

$$\begin{aligned} \overline{U} & = \overline{X-A} \\ & = X- Int(A) \\ & \subset Int(B) \end{aligned}$$

and that

$$(X-U, B-U) = (A, A \cap B) \,.$$

Hence the excision axiom implies $E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)$.

Conversely, suppose $E^\bullet$ satisfies the alternative condition. Given $U \hookrightarrow A \hookrightarrow X$ with $\overline{U} \subset Int(A)$, observe that we have a cover

$$\begin{aligned} Int(X-U) \cup Int(A) & = (X - \overline{U}) \cap \Int(A) \\ & \supset (X - Int(A)) \cap Int(A) \\ & = X \end{aligned}$$

and that

$$(X-U, (X-U) \cap A) = (X-U, A - U) \,.$$

Hence

$$E^\bullet(X-U,A-U) \simeq E^\bullet(X-U, (X-U)\cap A) \simeq E^\bullet(X,A) \,.$$

This shows the statement for the special case that $X = Int(A)\cup Int(U)$. The general statement reduces to this by finding a suitable homotopy equivalence to a slightly larger covering pair (e.g Switzer 75, 7.5).

Proposition

(exact sequence of a triple)

For $E_\bullet$ an unreduced generalized cohomology theory, def. , then every inclusion of two consecutive subspaces

$$Z \hookrightarrow Y \hookrightarrow X$$

induces a long exact sequence of homology groups of the form

$$\cdots \to E_q(Y,Z) \stackrel{}{\longrightarrow} E_q(X,Z) \stackrel{}{\longrightarrow} E_q(X,Y) \stackrel{\bar \delta}{\longrightarrow} E_{q-1}(Y,Z) \to \cdots$$

where

$$\bar \delta \;\colon \; E_{q}(X,Y) \stackrel{\delta}{\longrightarrow} E_{q-1}(Y) \longrightarrow E_{q-1}(Y,Z) \,.$$

Proof

Apply the braid lemma to the interlocking long exact sequences of the three pairs $(X,Y)$, $(X,Z)$, $(Y,Z)$:

(graphics from this Maths.SE comment)

See here for details.

Properties

Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence

The Atiyah-Hirzebruch spectral sequence serves to express generalized homology $E_\bullet$ in terms of ordinary homology with coefficients in $E_\bullet(\ast)$.

Whitehead theorem

Proposition

Let $\phi \colon E \longrightarrow F$ be a morphism of reduced generalized (co-)homology functors, def. (a natural transformation) such that its component

$$\phi(S^0) \colon E(S^0) \longrightarrow F(S^0)$$

on the 0-sphere is an isomorphism. Then $\phi(X)\colon E(X)\to F(X)$ is an isomorphism for $X$ any CW-complex with a finite number of cells. If both $E$ and $F$ satisfy the wedge axiom, then $\phi(X)$ is an isomorphism for $X$ any CW-complex, not necessarily finite.

For $E$ and $F$ ordinary cohomology/ordinary homology functors a proof of this is in (Eilenberg-Steenrod 52, section III.10). From this the general statement follows (e.g. Kochman 96, theorem 3.4.3, corollary 4.2.8) via the naturality of the Atiyah-Hirzebruch spectral sequence (the classical result gives that $\phi$ induces an isomorphism between the second pages of the AHSSs for $E$ and $F$). A complete proof of the general result is also given as (Switzer 75, theorem 7.55, theorem 7.67)

Examples

• stable homotopy homology theory is the homology theory represented by the sphere spectrum

• ordinary homology is the homology theory represented by an Eilenberg-MacLane spectrum

• bordism homology theory is the homology theory represented by a Thom spectrum;

Related concepts

• generalized cohomology

• Kronecker pairing

• Atiyah-Hirzebruch spectral sequence

• Steenrod algebra

• phantom map

• bivariant cohomology

• Bousfield equivalence

• Hurewicz homomorphism, Boardman homomorphism

References

(For more see the references at generalized (Eilenberg-Steenrod) cohomology.)

Original articles include

• George Whitehead, Generalized homology theories (1961) (pdf)

Textbook accounts include

• Stanley Kochmann, section 3.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Robert Switzer, chapter 7 (and 8-12) of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

• Stefan Schwede, chapter II, section 6 of Symmetric spectra, 2012 (pdf)

See also

• Friedrich Bauer, Classifying spectra for generalized homology theories Annali di Maternatica pura ed applicata

  (IV), Vol. CLXIV (1993), pp. 365-399

• Friedrich Bauer, Remarks on universal coefficient theorems for generalized homology theories Quaestiones Mathematicae

  Volume 9, Issue 1 & 4, 1986, Pages 29 - 54

A general construction of homologies by “geometric cycles” similar to the Baum-Douglas geometric cycles for K-homology is discussed in

• S. Buoncristiano, C. P. Rourke and B. J. Sanderson, A geometric approach to homology theory, Cambridge Univ. Press, Cambridge, Mass. (1976)

Further generalization of this to bivariant cohomology theories is in

• Martin Jakob, Bivariant theories for smooth manifolds, Applied Categorical Structures 10 no. 3 (2002)