# Homology theory

of topological spaces

A part of algebraic topology which realizes a connection between topological and algebraic concepts. By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings. Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. Homology of a polyhedron), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [3] (cf. Algebraic topology; Homology group; Steenrod–Eilenberg axioms).

According to this construction a homology theory $\{ H, \partial \}$ consists of three functions: 1) relative $r$-dimensional homology groups $H _ {r} ( X, A)$ of a pair of topological spaces $( X, A)$, $A \subset X$, which assign to each pair $( X, A)$ and each integer $r$ an Abelian group $H _ {r} ( X, A)$; 2) the homomorphism

$$H _ {r} ( f ) = \ f _ {*} : H _ {r} ( X, A) \rightarrow H _ {r} ( Y, B),$$

which is assigned to a continuous mapping $f: ( X, A) \rightarrow ( Y, B)$ and a number $r$, and which is called the homomorphism induced by $f$; and 3) the boundary operator $\partial$, which assigns to each pair $( X, A)$ and each $r$ a homomorphism $\partial$ of $H _ {r} ( X, A)$ into $H _ {r- 1} ( A)$ (the so-called absolute group of the space $A$, which is the group of the pair $( A, \emptyset )$). These functions must satisfy the following axioms.

1) If $f$ is the identity mapping, so is $f _ {*}$.

2) $( gf ) _ {*} = g _ {*} f _ {*}$.

3) $\partial f _ {*} = ( f \mid _ {A} ) _ {*} \partial$.

4) The exactness axiom. If

$$i: A \rightarrow X \ \textrm{ and } \ j: X \rightarrow ( X, A)$$

are the natural inclusions, then the sequence

$$\dots \rightarrow H _ {r} ( A) \rightarrow ^ { {i _ *} } \ H _ {r} ( X) \rightarrow ^ { {j _ *} } H _ {r} ( X, A) \mathop \rightarrow \limits ^ \partial H _ {r - 1 } ( A) \rightarrow \dots ,$$

the so-called homology sequence of the pair $( X, A)$, is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.

5) The homotopy axiom. If the mappings

$$f, g: ( X, A) \rightarrow ( Y, B)$$

are homotopic, then $f _ {*} = g _ {*}$.

6) The excision axiom. If $U$ is an open subset of the space $X$ and its closure is contained in the interior of the subspace $A$, then the inclusion

$$e: ( X \setminus U, A \setminus U) \rightarrow ( X, A)$$

induces an isomorphism $e _ {*}$.

7) The dimension axiom. If $X$ is a one-point space, then $H _ {r} ( X) = 0$ for all $r \neq 0$.

Instead of taking the category of all pairs of spaces as the domain of definition of $H _ {r}$, it is also possible to take an arbitrary category of pairs of spaces, e.g. the category of pairs of compact spaces or the category of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with $( X, A)$ also the pairs $( \emptyset , \emptyset )$, $( X, \emptyset ) = X$, $( A, \emptyset ) = A$, $( A, A)$, $( X, X)$, the cylinder $( X, A) \times I$, where $I = [ 0, 1]$, and some one-point space $P _ {0}$, with all their inclusions. Such a category must also include all pairs and mappings which are encountered in the axioms or in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of $H _ {r}$, it is also possible to use other categories, e.g. the category of topological, in particular that of compact, groups with continuous homomorphisms, or the category of modules over some ring with linear homomorphisms.

Axioms 1 and 2 mean that $H _ {r}$ is a covariant functor from some category of pairs of spaces into the category of groups. Axiom 3 means that the boundary operator $\partial$ is a natural transformation of the functor $H _ {r} ( X , - )$ to the functor $H _ {r- 1} ( - )$. Axiom 4, which connects the functors of all dimensions $r$, is sometimes replaced by the weaker requirement: The sequence should only be partially exact, i.e. the image should be included in the kernel (cf. Exact sequence); an important example of a partially exact homology theory is the Aleksandrov–Čech homology theory. Axiom 5 may be written in an equivalent form: If

$$f _ {0} , f _ {1} : ( X, A) \rightarrow ( X, A) \times I$$

are mappings defined by the formulas $f _ {0} ( x) = ( x, 0)$, $f _ {1} ( x) = ( x, 1)$, then $f _ {0*} = f _ {1*}$. Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory. Axiom 7, which ensures the geometric meaning of the dimensionality index $r$, is often omitted in modern studies. One then obtains so-called generalized homology theories, an important example of which is bordism theory.

There exists a cohomology theory dual to a homology theory (cf. Duality in topology). It is given by relative $r$-dimensional cohomology groups $H ^ {r} ( X, A)$, which are contravariant functors from the category of pairs of topological spaces into the category of Abelian groups, with induced homomorphisms

$$H ^ {r} ( f ) = f ^ { * } : H ^ {r} ( Y, B) \rightarrow H ^ {r} ( X, A)$$

and coboundary operators

$$\delta : H ^ {r} ( A) \rightarrow H ^ {r + 1 } ( X, A).$$

The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms. For instance, the exactness axiom requires the existence of an exact cohomology sequence

$$\dots \rightarrow H ^ {r - 1 } ( A) \mathop \rightarrow \limits ^ \delta \ H ^ {r} ( X, A) \rightarrow ^ { {j ^ {*}} } H ^ {r} ( X) \rightarrow ^ { {i ^ {*}} } H ^ {r} ( A) \rightarrow \dots .$$

And, analogously, there are also generalized cohomology theories, important examples of which are $K$-theory and cobordism theory. The facts given below concerning homology theory have cohomology analogues.

The coefficient group of a homology or a cohomology theory is the group $H _ {0} ( P _ {0} )$ or, respectively, $H ^ {0} ( P _ {0} )$. The so-called reduced groups $\widetilde{H} _ {r} ( X, A)$ are often conveniently substituted for the groups $H _ {r} ( X, A)$: the reduced zero-dimensional homology group $\widetilde{H} _ {0} ( X)$ is the kernel of the homomorphism

$$l _ {*} : H _ {0} ( X) \rightarrow H _ {0} ( P _ {0} )$$

induced by the mapping $l: X \rightarrow P _ {0}$, while the reduced zero-dimensional cohomology group $\widetilde{H} {} ^ {0} ( X)$ is the quotient group of $H ^ {0} ( X)$ by the image $l ^ {*} ( H ^ {0} ( P _ {0} ))$; the reduced groups of other dimensions are identical with the initial ones: $\widetilde{H} _ {r} ( X) = H _ {r} ( X)$, $r \neq 0$. Thus, $H _ {0} ( X) \sim \widetilde{H} _ {0} ( X) \oplus G$. If $A \neq \emptyset$, then $\widetilde{H} _ {r} ( X, A) = H _ {r} ( X, A)$ for all $r$. The replacement of the ordinary by the reduced groups permits one to convert the homology sequence into a reduced homology sequence.

The axioms of homology theory are not independent. E.g., axiom 1 is a consequence of the axioms 2, 3 and 4. The system of axioms is compatible, as is seen from the example of the trivial theory $H _ {r} ( X, A) = 0$; non-trivial examples include the Aleksandrov–Čech cohomology theory, singular homology, etc. Regarding uniqueness, the following holds: A homomorphism of a homology theory $\{ H, \delta \}$ into a homology theory $\{ H ^ \prime , \partial ^ \prime \}$ is a system of homomorphisms

$$h ( X, A; r): H _ {r} ( X, A) \rightarrow H _ {r} ^ \prime ( X, A)$$

such that

$$H ^ \prime ( f ) \circ h ( X, A; r) = h ( Y, B; r) \circ H ( f )$$

and

$$\partial ^ \prime \circ h ( X, A; r) = h ( A; r - 1) \circ \partial .$$

If all $h( X, A; r)$ are isomorphisms, then the homology theories $\{ H, \partial \}$ and $\{ H ^ \prime , \partial ^ \prime \}$ are called isomorphic. A homology theory on finite polyhedra is uniquely determined by its coefficient group. More precisely, if $h _ {0} : G \rightarrow G ^ \prime$ is an arbitrary homomorphism of the coefficient group $G$ of $\{ H, \partial \}$ into the coefficient group $G ^ \prime$ of $\{ H ^ \prime , \partial ^ \prime \}$, then for each polyhedral pair $( X, A)$ there exists a unique homomorphism

$$h ( X, A; r): H _ {r} ( X, A) \rightarrow H _ {r} ^ \prime ( X, A)$$

with the property $h( P _ {0} ; 0) = h _ {0}$. If $h _ {0}$ is an isomorphism, so are all $h( X, A; r)$. Since the homology groups of negative dimension of a triangulable pair are trivial, the equality $H _ {r} ( X, A) = 0$, $r < 0$, is valid for any homology theory $\{ H, \partial \}$ as well. The uniqueness theorem is also valid for wider categories of spaces if the homology theory satisfies appropriate additional axioms.

The homology groups are topological and also homotopy invariants: If $f$ is a homotopy equivalence, then $f _ {*}$ is an isomorphism. If $X$ is a contractible space — a cell, in particular — then $H _ {r} ( X) = 0$, $r \neq 0$, and $H _ {0} ( X) \sim G$. If $i : A \subset X$ is a homotopy equivalence, then $H _ {r} ( X, A) = 0$ and, for any $X$, $H _ {r} ( X, X) = 0$. If $A$ is a retract of the space $X$, then $i _ {*}$ is a monomorphism, $j _ {*}$ is an epimorphism, the operation $\partial$ is trivial, and

$$H _ {r} ( X) \sim H _ {r} ( A) \oplus H _ {r} ( X, A).$$

If $X$ is deformable into $A$, then $i _ {*}$ is an epimorphism, $j _ {*}$ is trivial, $\partial$ is a monomorphism, and

$$H _ {r} ( A) \sim H _ {r} ( X) \oplus H _ {r + 1 } ( X, A).$$

Let $S( X)$ denote the suspension over $X$. The following isomorphism is then valid:

$$\widetilde{H} _ {r} ( X) \sim \widetilde{H} _ {r + 1 } ( S ( X)).$$

This makes it possible to compute the homology groups of the spheres $S ^ {n}$: $\widetilde{H} _ {r} ( S ^ {n} ) = 0$ if $r \neq n$, and $\widetilde{H} _ {n} ( S ^ {n} ) \sim G$; consequently, $H _ {r} ( S ^ {n} ) = 0$ if $n \neq r \neq 0$; $H _ {r} ( S ^ {n} ) \sim G$ if $n \neq r = 0$ or $n = r \neq 0$; and $H _ {0} ( S ^ {0} ) \sim G \oplus G$.

An important role in homology theory is played by homology sequences of triples and triads. In the case of a triple $( X, A, B)$, $X \supset A \supset B$, of spaces, the boundary operator $\partial ^ \prime = k _ {*} ^ \prime \partial$ is defined as the composition $k _ {*} ^ \prime \circ \partial$, where $k ^ \prime : A \rightarrow ( A, B)$ is the inclusion. There results the so-called homology sequence of the triple $( X, A, B)$ (which, for $B = \emptyset$, reduces to the homology sequence of the pair $( X, A)$):

$$\dots \rightarrow H _ {r} ( A, B) \rightarrow ^ { {i _ *} ^ \prime } \ H _ {r} ( X, B) \ \rightarrow ^ { {j _ *} ^ \prime } \ H _ {r} ( X, A) \ \rightarrow ^ { {\partial ^ \prime } }$$

$$\rightarrow ^ { {\partial ^ \prime } } H _ {r - 1 } ( A, B) \rightarrow \dots ,$$

where $i ^ \prime : ( A, B) \rightarrow ( X, B)$ and $j ^ \prime : ( X, B) \rightarrow ( X, A)$ are the inclusions. This sequence is exact. If the groups $H _ {r} ( X, A)$, respectively $H _ {r} ( X, B)$, $H _ {r} ( A, B)$, are trivial for all $r$, then $i _ {*} ^ \prime$, respectively $\partial ^ \prime$, $j _ {*} ^ \prime$, are isomorphisms, and vice versa. If $X$ is the union of non-intersecting closed sets $X _ {i}$, $i = 1 \dots n$, and $A = A _ {1} \cup \dots \cup A _ {n}$, where $A _ {i} \subset X _ {i}$, then $H _ {r} ( X, A)$ is isomorphic to the direct sum of the groups $H _ {r} ( X _ {i} , A _ {i} )$, $i = 1 \dots n$. A triad $( X; A, B)$ is a space $X$ with an ordered pair of subspaces $A, B$ (cf. also Triads). It is a proper triad if the inclusions

$$k: ( A, A \cap B) \rightarrow ( A \cup B, B),\ \ l: ( B, A \cap B) \rightarrow ( A \cup B, A)$$

induce isomorphisms, or if the decomposition

$$H _ {r} ( A \cup B, A \cap B) \sim \ H _ {r} ( A, A \cap B) \oplus H _ {r} ( B, A \cap B)$$

is valid. Further, the boundary operator

$$\overline \partial : H _ {r} ( X, A \cup B) \rightarrow \ H _ {r - 1 } ( A, A \cap B)$$

is defined as $k _ {*} ^ {- 1} \circ m _ {*} \circ \partial$, where $m : A \cup B \subset ( A \cup B, B)$. This generates the exact homology sequence of the triad:

$$\dots \rightarrow H _ {r} ( A, A \cap B) \ \rightarrow ^ { {p _ *} } \ H _ {r} ( X, B) \ \rightarrow ^ { {q _ *} } \ H _ {r} ( X, A \cup B) \ \mathop \rightarrow \limits ^ { {\overline \partial }}$$

$$\mathop \rightarrow \limits ^ { {\overline \partial }} H _ {r - 1 } ( A, A \cap B) \rightarrow \dots ,$$

where $p: ( A, A \cap B) \rightarrow ( X, B)$, $q: ( X, B) \rightarrow ( X, A \cup B)$ are the inclusions (if $B \subset A$, this sequence reduces to the homology sequence of the triple $( X, A, B)$).

Let $X = A \cup B$, $A \cap B = C$ and suppose the mappings $h, h _ {1} , h _ {2} : ( X, C) \rightarrow ( Y, D)$ satisfy the relations $h _ {1} | _ {A} = h | _ {A}$, $h _ {2} | _ {B} = h | _ {B}$, $h _ {1} ( B) \subset D$, $h _ {2} ( A) \subset D$. The following addition theorems are then valid.

1) $h _ {*} = h _ {1* } + h _ {2* }$.

2) If $D$ is contractible and if $f, f _ {1} , f _ {2} : X \rightarrow Y$ are defined, respectively, by means of $h, h _ {1}$ and $h _ {2}$, then the equality $f _ {*} = f _ {1* } + f _ {2* }$ holds for the induced homomorphisms of the reduced groups $f _ {*} , f _ {1* } , f _ {2* } : \widetilde{H} _ {r} ( X) \rightarrow \widetilde{H} _ {r} ( Y)$.

Define the homomorphism

$$s : H _ {r} ( C) \rightarrow H _ {r} ( A) \oplus H _ {r} ( B)$$

by $s ( c) = ( s _ {1* } ( c), - s _ {2* } ( c))$, $c \in H _ {r} ( C)$, where $s _ {1} : C \rightarrow A$, $s _ {2} : C \rightarrow B$ are the inclusions. Similarly, define

$$t: H _ {r} ( A) \oplus H _ {r} ( B) \rightarrow H _ {r} ( X)$$

by $t ( a, b) = t _ {1* } ( a) + t _ {2* } ( b)$, $( a, b) \in H _ {r} ( A) \oplus H _ {r} ( B)$, where $t _ {1} : A \rightarrow X$, $t _ {2} : B \rightarrow X$ are the inclusions. Finally, define

$$\Delta : H _ {r} ( X) \rightarrow H _ {r} ( C),$$

by $\Delta = \partial u _ {*} ^ {- 1} v _ {*}$, where

$$v: X \rightarrow ( X, B),\ \ u: ( A, C) \rightarrow ( X, B)$$

are the inclusions. One then obtains the so-called Mayer–Vietoris sequence of the proper triad:

$$\dots \rightarrow H _ {r} ( C) \ \mathop \rightarrow \limits ^ { {s }} H _ {r} ( A) \oplus H _ {r} ( B) \ \mathop \rightarrow \limits ^ { {t }} H _ {r} ( X) \ \mathop \rightarrow \limits ^ \Delta \ H _ {r - 1 } ( C) \rightarrow \dots ,$$

which is exact and which relates the homology groups of the spaces $A$ and $B$ with the homology groups of their union and intersection. Then, if $C \neq \emptyset$, one may pass to a similar sequence for the reduced groups, which implies that:

1) if $A \cap B$ is contractible, then

$$\widetilde{H} _ {r} ( A \cup B) \sim \ \widetilde{H} _ {r} ( A) \oplus \widetilde{H} _ {r} ( B);$$

2) if $A \cup B$ is contractible, then

$$\widetilde{H} _ {r} ( A \cap B) \sim \ \widetilde{H} _ {r} ( A) \oplus \widetilde{H} _ {r} ( B);$$

3) if both $A$ and $B$ are contractible, then $\Delta$ defines an isomorphism

$$\widetilde{H} _ {r} ( A \cup B) \sim \widetilde{H} _ {r - 1 } ( A \cap B).$$

These results make it possible to compute the homology groups of various spaces. For instance, if $X$ is a closed orientable surface of genus $n$, then $H _ {r} ( X)$ is isomorphic to the coefficient group $G$ if $r = 0, 2$; to the direct sum $G ^ {2n}$ of $2n$ copies of $G$ if $r = 1$; and to 0 in the remaining cases. If $X$ is a closed non-orientable surface of genus $n$, then $H _ {r} ( X)$ is isomorphic to $G$ if $r = 0$; to $G ^ {n- 1} \oplus G _ {2}$, where $G _ {2}$ is the quotient group $G/2G$, $2G = \{ {2g } : {g \in G } \}$, if $r = 1$; to the subgroup $T _ {2} ( G)$ of $G$ consisting of all elements $g \in G$ with $2g = 0$ if $r = 2$; and to 0 in the remaining cases. Thus, homology theory gives a topological classification of closed surfaces. For an $n$-dimensional real projective space $P ^ {n}$ the group $H _ {r} ( P ^ {n} )$ is isomorphic to $G$ if $r = 0$ or $r = n$ and odd; to $G _ {2}$ if $r$ is odd and $0 < r < n$; to $T _ {2} ( G)$ if $r$ is even and $0 < r \leq n$; and to 0 in the remaining cases. The homology group $H _ {r} ( \mathbf C P ^ {n} )$ of the complex projective space $\mathbf C P ^ {n}$ of dimension $2n$ is isomorphic to $G$ if $r$ is even and $0 \leq r \leq 2n$; and to 0 in the remaining cases. The homology group $H _ {r} ( L _ {p,q} )$ of the lens space $L _ {p,q}$ is isomorphic to $G$ if $r = 0, 3$; to $G _ {p} = G/pG$, where $pG = \{ {pg } : {g \in G } \}$, if $r = 1$; to $T _ {p} ( G)$, where $T _ {p} ( G) = \{ {g \in G } : {pg = 0 } \}$, if $r = 2$; and to 0 in the remaining cases.

From the various applications of the results described above certain fundamental theorems are stated here. First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension. Furthermore, the equality $f _ {*} i _ {*} = g _ {*}$, where $f: X \rightarrow Y$ is an extension of a given mapping $g: A \rightarrow Y$, $A \subset X$, yields various criteria of extendability and retractibility of mappings; for example, a mapping of a sphere $S ^ {n- 1}$, $n > 1$, of non-zero degree into itself is not extendable to the $n$-dimensional ball $E ^ {n}$ with boundary $S ^ {n- 1}$, and $S ^ {n- 1}$ is not a retract of $E ^ {n}$ for any natural $n$. This, in turn, yields Brouwer's fixed-point theorem: Any continuous mapping $E ^ {n} \rightarrow E ^ {n}$ has a fixed point. Finally, it may be proved that a unit tangent vector field exists on $S ^ {n}$ if and only if $n$ is odd, while the theory of triads yields several theorems on the degrees of mappings; in particular, it is possibly to give a new proof of the fundamental theorem of algebra.

#### References

 [1] P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian) MR0076324 Zbl 0441.55002 Zbl 0097.15903 Zbl 0024.08404 Zbl 66.0947.03 [2] S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1942) MR0007093 Zbl 0061.39302 [3] S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952) MR0050886 Zbl 0047.41402 [4] E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) MR0210112 MR1325242 Zbl 0145.43303 [5] S.-T. Hu, "Homology theory" , Holden-Day (1966) MR0217786 Zbl 0145.19705 [6] S.-T. Hu, "Cohomology theory" , Markham , Chicago (1968) MR0234448 Zbl 0165.26101 [7] A. Dold, "Lectures on algebraic topology" , Springer (1980) MR0606196 Zbl 0434.55001

A sequence of modules $K _ {n}$, $n \geq 0$, over a ring $A$ together with homomorphisms $\partial _ {n} : K _ {n} \rightarrow K _ {n- 1}$ such that $\partial _ {n} \partial _ {n+ 1} = 0$ for all $n \geq 0$ is often called a chain complex. Thus, in case of a partially exact homology theory $( H _ {n} , \partial )$ the sequence of a pair $( X , A )$ is a chain complex, or complex, instead of a long exact sequence. Dually one has cochain complexes $( K ^ {n} , \partial ^ {n} : K ^ {n} \rightarrow K ^ {n+ 1} )$. Many homology and cohomology theories are constructed via chain and cochain complexes: first to a pair $( X , A )$ there is associated a chain complex $( C _ {n} ( X , A ) , \partial _ {n} )$ and then $H _ {n} ( X , A ) = \mathop{\rm Ker} ( \partial _ {n} ) / \mathop{\rm Im} ( \partial _ {n+ 1} )$, and analogously for cohomology.