generalized homology




Special and general types

Special notions


Extra structure





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 KK is the coefficient object of a generalized cohomology theory and S-dually of a generalized homology theory. For K=HRK = H R an Eilenberg-MacLane spectrum this reduces to ordinary homology.

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


Reduced homology

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

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


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


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

for the reduced suspension functor.

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


A reduced homology theory is a functor

E˜ :(Top CW */)Ab \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

E˜ :(XfY)(E˜ (X)f *E˜ (Y)), \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

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

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

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 morphism

    iIE˜ (X i)E˜ ( iIX i) \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 1, is an isomorphism.

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) } \,.

Unreduced homology

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


A homology theory (unreduced, relative) is a functor

E :(Top CW )Ab 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

δ (X,A):E +1(X,A)E (A,). \delta_{(X,A)} \;\colon\; E_{\bullet + 1}(X, A) \longrightarrow E^\bullet(A, \emptyset) \,.

such that:

  1. (homotopy invariance) For f:(X 1,A 1)(X 2,A 2)f \colon (X_1,A_1) \to (X_2,A_2) a homotopy equivalence of pairs, then

    E (f):E (X 1,A 1)E (X 2,A 2) 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 AXA \hookrightarrow X the induced sequence

    E n+1(X,A)δE n(A)E n(X)E n(X,A) \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 UAXU \hookrightarrow A \hookrightarrow X such that U¯Int(A)\overline{U} \subset Int(A), then the natural inclusion of the pair i:(XU,AU)(X,A)i \colon (X-U, A-U) \hookrightarrow (X, A) induces an isomorphism

    E (i):E n(XU,AU)E n(X,A) E_\bullet(i) \;\colon\; E_n(X-U, A-U) \overset{\simeq}{\longrightarrow} E_n(X, A)

We say E E^\bullet is additive if it takes coproducts to direct sums:

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

    iE n(X i,A i)E n(X,A) \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 E_\bullet is ordinary if its value on the point is concentrated in degree 0

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

A homomorphism of unreduced homology theories

η:E F \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:

E +1(X,A) η (X,A) F +1(X,A) δ E δ F E (A,) η (A,) F (A,). \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) } \,.

The excision axiom in def. 2 is equivalent to the following statement:

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

i:(A,AB)(X,B) i \colon (A, A \cap B) \longrightarrow (X,B)

induces an isomorphism,

i *:E (A,AB)E (X,B). i_\ast \;\colon\; E_\bullet(A, A \cap B) \overset{\simeq}{\longrightarrow} E_\bullet(X, B) \,.

(e.g Switzer 75, 7.2, 7.5)


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

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

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

and that

(XU,BU)=(A,AB). (X-U, B-U) = (A, A \cap B) \,.

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

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

Int(XU)Int(A) =(XU¯)Int(A) (XInt(A))Int(A) =X \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

(XU,(XU)A)=(XU,AU). (X-U, (X-U) \cap A) = (X-U, A - U) \,.


E (XU,AU)E (XU,(XU)A)E (X,A). 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)Int(U)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).


(exact sequence of a triple)

For E E_\bullet an unreduced generalized cohomology theory, def. 2, then every inclusion of two consecutive subspaces

ZYX Z \hookrightarrow Y \hookrightarrow X

induces a long exact sequence of homology groups of the form

E q(Y,Z)E q(X,Z)E q(X,Y)δ¯E q1(Y,Z) \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


δ¯:E q(X,Y)δE q1(Y)E q1(Y,Z). \bar \delta \;\colon \; E_{q}(X,Y) \stackrel{\delta}{\longrightarrow} E_{q-1}(Y) \longrightarrow E_{q-1}(Y,Z) \,.

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

(graphics from this Maths.SE comment)

See here for details.


Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence

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

Whitehead theorem


Let ϕ:EF\phi \colon E \longrightarrow F be a morphism of reduced generalized (co-)homology functors, def. 1 (a natural transformation) such that its component

ϕ(S 0):E(S 0)F(S 0) \phi(S^0) \colon E(S^0) \longrightarrow F(S^0)

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

For EE and FF 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 ϕ\phi induces an isomorphism between the second pages of the AHSSs for EE and FF). A complete proof of the general result is also given as (Switzer 75, theorem 7.55, theorem 7.67)



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

Original articles include

Textbook accounts include

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)

Revised on June 6, 2016 14:36:56 by Urs Schreiber (