4.1 Singular Homology

First, we briefly introduce homology groups, for a complete—or even axiomatic—treatment, see for example [1, 4, 8]. In particular, there is a multitude of homology theories, all with their relative merits. We highlight Eilenberg’s singular homology. The intuition goes back to Riemann and Poincaré and is as follows. Compact two dimensional surfaces can be characterized by their genus, that is, the number of holes (or handles). Extending this, one can consider classifying topological spaces based on how many k-dimensional “holes” they have and so forth. Here, the dimension of the hole should be understood as a the smallest dimension of a closed manifold enclosing the hole, e.g., \(\mathbb {S}^1\) is said to have a single  1-dimensional hole.

More formally, let \(\textsf{X}\) be a topological space and let \(\sigma :\Delta ^k\rightarrow \textsf{X}\) be any continuous map from the standard k -simplex \(\Delta ^k=\{p\in \mathbb {R}^{k+1}:\sum \nolimits ^k_{i=0}p_i=1,\,p_i\ge 0\) for \(i=0,1,\dots ,k\}\) into \(\textsf{X}\), called a singular k -simplex. Now recall the introductory remarks on groups from Sect. 1.3, yet, to work with these singular maps we need to introduce more concepts from algebra. Let \(\textsf{G}\) be an Abelian (commutative) group, then \(\mathscr {B}\subseteq \textsf{G}\) is a basis of \(\textsf{G}\) when \(\textsf{G}\) is the smallest subgroup of \(\textsf{G}\) that contains \(\mathscr {B}\) and is such that every \(g\in \textsf{G}\) can be expressed as a formal sum (meaning, for general “\(+\)”) \(g=\sum \nolimits _i \alpha _i b_i\) with \(\alpha _i\in \mathbb {Z}\), \(b_i\in \mathscr {B}\) and only finitely many \(\alpha _i\) being non-zero [5, Theorem I.2.8]. Then, while omitting the (motivating) details, given any set Y the so-called free Abelian group generated by Y is given by \(\{\sum \nolimits _i \alpha _i y_i: y_i\in Y,\, \alpha _i\in \mathbb {Z}\) and finitely many \(\alpha _i\) are non-zero \(\}\) [5, Chap. II], which is an additive group.

We also recall the notion of a group homomorphism, that is, given two groups \((\textsf{G},\cdot ^{(\textsf{G})})\), \((\textsf{H},\cdot ^{(\textsf{H})})\), a map \(z:\textsf{G}\rightarrow \textsf{H}\) such that \(z(g_1\cdot ^{(\textsf{G})} g_2)=z(g_1)\cdot ^{(\textsf{H})} z(g_2)\) for all \(g_1,g_2\in \textsf{G}\). The set of group homomorphisms between \(\textsf{G}\) and \(\textsf{H}\) is denoted by \(\textrm{Hom}(\textsf{G},\textsf{H})\). In case \(\textsf{H}\) is Abelian, \((\textrm{Hom}(\textsf{G},\textsf{H}),\cdot )\) is itself an Abelian group with the group operation being defined pointwise by \((z_1\cdot z_2)(g)=z_1(g)\cdot ^{(\textsf{H})}z_2(g)\) for all \(z_1,z_2\in \textrm{Hom}(\textsf{G},\textsf{H})\) and all \(g\in \textsf{G}\).

Now let \(C_k(\textsf{X})\) denote the free Abelian group generated by all singular k-simplices, called the singular chain group, containing elements, called k-chains, of the form \(\sum _i\alpha _i\sigma _i\) for \(\sigma _i\) a singular k-simplex. For instance, for \(C_1(X)\), one can think of \(\sigma _i:[0,1]\rightarrow \textsf{X}\) as giving rise to a path (or point) in \(\textsf{X}\). Recall that paths enclosing on itself (loops), and in general maps from n-spheres, contain a lot of topological information. However, this homotopy approach does not necessarily detect all “holes” we are after in an intuitive manner, e.g., Hopf found a non-trivial map from the 3-sphere to the 2-sphere. This is one reason to use the singular chain groups instead. In particular, one can define a boundary operator \(\partial _k\) that acts on \(C_k(\textsf{X})\) as

$$\begin{aligned} \cdots \overset{\partial _{k+1}}{\rightarrow } C_k(\textsf{X})\overset{\partial _k}{\rightarrow }C_{k-1}(\textsf{X})\overset{\partial _{k-1}}{\rightarrow }\cdots 0 \end{aligned}$$

and satisfies \(\partial _k\circ \partial _{k+1}=0\) (boundaries have no boundaries). In the context of singular homology this map can be made explicit. To that end, define the face embedding \(F_{i,k}:\Delta ^{k-1}\hookrightarrow \Delta ^k\) as follows. Let \(\{e_0,e_1,\dots ,e_{\ell }\}\) be the set of vertices of \(\Delta ^{\ell }\). Then, \(F_{i,k}\) is such that it maps \(\Delta ^{k-1}\) to the face opposite to the vertex \(e_i\in \Delta ^k\). Note that \(\Delta ^{k-1}\) itself is a face of \(\Delta ^k\). Now for \(\sigma \in C_k(\textsf{X})\) the boundary operator can be defined as \(\partial _k \sigma = \sum \nolimits ^k_{i=0}(-1)^i \sigma \circ F_{i,k}\), where \(\partial _k\circ \partial _{k+1} = 0\) can be verified.

Then, a k-chain \(c\in C_k(\textsf{X})\) is called a k-cycle when \(\partial _k c=0\). Differently put, the set \(\textrm{ker}(\partial _k)\subseteq C_k(\textsf{X})\) contains all k-cycles. See also that when \(b\in C_{k+1}(\textsf{X})\), then, \(\partial _{k+1}b \in \textrm{ker}(\partial _k)\) since \(\partial _k (\partial _{k+1} b)=0\). Now, the \(k^{\textrm{th}}\) singular homology group of \(\textsf{X}\) is defined as \(H_k(\textsf{X};\mathbb {Z})=H_k(\textsf{X})=\textrm{ker}(\partial _k)/\textrm{im}(\partial _{k+1})\). As such, \(H_k(\textsf{X};\mathbb {Z})=0\) when all k-cycles are boundaries of \((k+1)\)-chains, that is, there are no k-dimensional holes.

Example 4.1

(The 0th singular homology group) Let \(\textsf{X}\) be path-connected.Footnote 1 we will follow [6] and show that \(H_0(\textsf{X};\mathbb {Z})=H_0(\textsf{X})=\textrm{ker}(\partial _0)/\textrm{im}(\partial _1)\simeq \mathbb {Z}\). First of all, by (4.1) see that \(\textrm{ker}(\partial _0)=C_0(\textsf{X})\). Now for any 0-chain \(c=\sum \nolimits _i\alpha _i\sigma _i\) construct the index map \(I:C_0(\textsf{X})\rightarrow \mathbb {Z}\) by \(I(c)=\sum \nolimits _i\alpha _i\). Evidently, this map is a surjective homomorphism. We will show that \(\textrm{ker}(I)=\textrm{im}(\partial _1)\), which implies that \(\mathbb {Z}\) is isomorphic to \(H_0(\textsf{X})\) by the first isomorphism theorem for groups.Footnote 2 Pick any singular 1-simplex \(\sigma \), then \(\partial \sigma = \sigma (1)-\sigma (0)\) and indeed \(I(\partial \sigma )=0\). This implies that \(\textrm{im}(\partial _1)\subseteq \textrm{ker}(I)\). For the other direction, fix a point in \(\textsf{X}\), say \(x'\), and let \(\psi (x)\) denote a continuous curve from \(x\in \textsf{X}\) to \(x'\), which always exists as \(\textsf{X}\) is path-connected. This means that for any 0-chain \(c=\sum \nolimits _i \alpha _i x_i\) (recall that \(\sigma _i=x_i\) in this case), we have \(\partial (\alpha _i\sum \nolimits _i \psi (x_i))=\sum \nolimits _i\alpha _i x_i - I(c)x'\). Therefore, if \(I(c)=0\), then c can be written as the boundary of a 1-chain and hence \(\textrm{ker}(I)\subseteq \textrm{im}(\partial _1)\). This concludes showing that \(H_0(\textsf{X})\simeq \mathbb {Z}\). When \(\textsf{X}\) consists out of p components, this argument is generalized to showing that \(H_0(\textsf{X})\simeq \mathbb {Z}^p\). See also [1, p. 172] for a similar explanation.

To provide another important example, \(H_k(\mathbb {S}^{n\ge 1};\mathbb {Z})\simeq \mathbb {Z}\) for \(k\in \{0,n\}\) and 0 otherwise. Similarly, given a subspace \(\textsf{A}\) of \(\textsf{X}\), one can consider the homology group of \(\textsf{X}\)modulo \(\textsf{A}\)” as follows. Define the relative chain group \(C_k(\textsf{X},\textsf{A})=C_k(\textsf{X})/C_k(\textsf{A})\) and analogously the (relative) boundary operator \(\partial _{\textsf{A},k}:C_k(\textsf{X},\textsf{A})\rightarrow C_{k-1}(\textsf{X},\textsf{A})\). Then, the k-th singular homology group of \(\textsf{X}\) relative to \(\textsf{A}\) is defined as \(H_k(\textsf{X},\textsf{A};\mathbb {Z})=H_k(\textsf{X},\textsf{A})=\textrm{ker}(\partial _{\textsf{A},k})/\textrm{im}(\partial _{\textsf{A},k+1})\) [4, p. 115]. When \(\textsf{A}\ne \emptyset \) is closed and a neighbourhood deformation retract of \(\textsf{X}\), then \(H_k(\textsf{X},\textsf{A};\mathbb {Z})=\widetilde{H}_k(\textsf{X}/\textsf{A};\mathbb {Z})\) [4, Proposition 2.22], for \(\widetilde{H}_k\) the reduced homology, i.e., \(H_k\simeq \widetilde{H}_k\) for \(k>0\) and \(H_0\simeq \widetilde{H}_0\oplus \mathbb {Z}\) [4, p. 110]. Whenever k is irrelevant, we write \(H_{(\cdot )}\), where \(H_{\bullet }\) is also common notation. Omitting details, when \(\textsf{X}\) is a compact manifold, then \(H_k(\textsf{X})\) is a finitely-generated Abelian group such that the rank of \(H_{(\cdot )}\) is simply the number of \(\mathbb {Z}\) summands used to describe the group [4, Sect. 2.2]. However, \(H_k(\textsf{X})\) is by no means finitely-generated in general, consider a plane with uncountably many holes.

Now, dual to the homology groups, one can define via \(C^k(\textsf{X})=\textrm{Hom}(C_k(\textsf{X}),\mathbb {Z})\) the k-th singular cohomology group of \(\textsf{X}\) via the so-called induced coboundary operator \(\delta ^k:C^k(\textsf{X})\rightarrow C^{k+1}(\textsf{X})\) as \(H^k(\textsf{X};\mathbb {Z})=H^k(\textsf{X})=\textrm{ker}(\delta ^k)/\textrm{im}(\delta ^{k+1})\) [4, 6]. Then, Poincaré duality allows for linking homology and cohomology groups of \(\textsf{X}\) [4, Sect. 3.3], e.g., \(H_k(\textsf{M}^m;\mathbb {Z})\simeq H^{m-k}(\textsf{M}^m;\mathbb {Z})\), for appropriate \(\textsf{M}^m\) (see below).

The power of singular homology does not necessarily lie in computation, but in the ability to prove relationships between several homology groups. To that end, given a continuous map \(G:\textsf{X}\rightarrow \textsf{Y}\) define the homomorphism \(G_{\#}:C_k(\textsf{X})\rightarrow C_k(\textsf{Y})\) by \(G_{\#}(\sigma )=G\circ \sigma \) for any singular k-simplex \(\sigma \in C_k(\textsf{X})\). The explicit formula for the boundary operator reveals that \(G_{\#}(\partial _k \sigma )=\partial _k(G_{\#}(\sigma ))\). Note that at the LHS of this equality the operator \(\partial _k\) acts on \(C_k(\textsf{X})\) whereas on the RHS \(\partial _k\) acts on \(C_k(\textsf{Y})\). This means that \(G_{\#}\) maps cycles to cycles and so forth. As such, \(G_{\#}\) induces a homomorphism \(G_{\star }:H_k(\textsf{X};\mathbb {Z})\rightarrow H_k(\textsf{Y};\mathbb {Z})\). It readily follows that for two continuous maps \(G_1:\textsf{X}\rightarrow \textsf{Y}\) and \(G_2:\textsf{Y}\rightarrow \textsf{Z}\) we have \((G_2\circ G_1)_{\star }={G_2}_{\star }\circ {G_1}_{\star }\). We are now equipped to state the following lemma.

Lemma 4.1

(Homology homotopy invariance [4, Corollary 2.11]) Let \(G:\textsf{X}\rightarrow \textsf{Y}\) be a homotopy equivalence, then, the induced homomorphism \(G_{\star }:H_k(\textsf{X};\mathbb {Z})\rightarrow H_k(\textsf{Y};\mathbb {Z})\) on singular homology is an isomorphism for any \(k\ge 0\).

Lemma 4.1 is commonly proved by first proving that for homotopic maps their induced homomorphims are equivalent [4, Theorem 2.10]. Then, one uses that if \(G:\textsf{X}\rightarrow \textsf{Y}\) is a homotopy equivalence, there must exist a map \(G':\textsf{Y}\rightarrow \textsf{X}\) such that \(G'\circ G{\simeq }_h\textrm{id}_{\textsf{X}}\) and similarly, \(G\circ G'{\simeq }_h\textrm{id}_{\textsf{Y}}\). However, this implies that \(G'_{\star }\circ G_{\star }=(\textrm{id}_{\textsf{X}})_{\star }\) and \(G_{\star }\circ G'_{\star }=(\textrm{id}_{\textsf{Y}})_{\star }\). Therefore, \(G_{\star }\) must be an isomorphism.

An elementary implication of Lemma 4.1—which will be of use in Chap. 6—is that by homotopy equivalence between \(\mathbb {R}^n\setminus \{0\}\) and \(\mathbb {S}^{n-1}\) for \(n\ge 2\), the singular homology groups of \(\mathbb {R}^n\setminus \{0\}\), for \(n\ge 2\), become

$$\begin{aligned} H_k(\mathbb {R}^n\setminus \{0\};\mathbb {Z}) \simeq {\left\{ \begin{array}{ll} \mathbb {Z} \quad \text {if }k\in \{0,n-1\}\\ 0 \quad \text {otherwise} \end{array}\right. }. \end{aligned}$$

Now, let \(G:\mathbb {S}^{n}\rightarrow \mathbb {S}^{n}\) be a continuous map, then, the degree of G as described in Sect. 3.4 can be equivalently defined by means of the induced homomorphism \(G_{\star }\), that is, as the integer \(\textrm{deg}(G)\) such that \(G_{\star }(H_n(\mathbb {S}^{n};\mathbb {Z}))=\textrm{deg}(G)H_n(\mathbb {S}^{n};\mathbb {Z})\), e.g., see [1, Chap. IV]. This definition of the degree also immediately works for maps \(G:\textsf{M}^m\rightarrow \textsf{N}^n\) over oriented, connected, compact manifolds as \(H_n(\textsf{M}^n;\mathbb {Z})\simeq \mathbb {Z}\) (by the universal coefficient theorem [1, Chap. V]) [7, Sect. III.2]. We return to this viewpoint below and in Chap. 6.

4.2 The Euler Characteristic

To link the previous chapter with the geometric definition of the Euler characteristic as set forth in Sect. 2.3, find Fig. 4.1. A single triangle is constructed using 3 vertices, 3 edges and 1 face, as such this adds up to an Euler characteristic of 1. Similarly, see that we can construct a vector field with 3 sources, 3 saddles and 1 sink, adding up the vector field indices agrees with the Euler characteristic.

Fig. 4.1
figure 1

Vector field on a triangulation

This can be formalized using homology theory, which will be briefly outlined below. First, one generalizes the 2-dimensional Euler characteristic formula by appealing to Whitehead’s CW complexes [4, 6]. Informally put, a CW complex is a space constructed by glueing together k-cells, that is, topological k-dimensional balls. These cells can be of different dimension and the glue is applied to the boundary of the cells. More formally, let \(X_0\) be some discrete space and construct \(X_1\) by attaching some collection \(\{C_{j}\}_{j\in J}\) of open 1-cells to \(X_0\) via a collection of continuous maps \(\varphi _j:\partial C_j \rightarrow X_0\), that is, \(X_1 = X_0 \cup \varphi (\sqcup _j \partial C_j)\). One can continue this procedure and construct \(X_2\subseteq X_3\subseteq \cdots \subseteq X_n\) by attaching open k-cells of appropriate dimension. If a topological space \(\textsf{X}\) can be written as \(X_n\) for some \(n\ge 0\), as defined above, then \(\textsf{X}\) has a cell decomposition \(\mathscr {C}\), where the 0-cells are given by \(X_0\), the 1-cells by \(X_1\setminus X_0\) and so forth. The pair \((\textsf{X},\mathscr {C})\) is called a cell complex. This cell complex is a CW complex when it satisfies the following two properties

  1. 1.

    The closure of each cell is contained in a union of finitely many cells;

  2. 2.

    The topology of \(\textsf{X}\) is coherent with \(\{\textrm{cl}\, C:C\in \mathscr {C}\}\).

See [6, p. 135] for examples that fail to meet both conditions.

Definition 4.1

(The (combinatorial) Euler characteristic [6, Chap. 6]) Let \(\textsf{X}^n\) be a finite dimensional CW complex, with \(n_k\) the number of k-cells of \(\textsf{X}^n\). Then, the Euler characteristic of \(\textsf{X}^n\) is defined as

$$\begin{aligned} \chi (\textsf{X}^n) = \textstyle \sum \limits ^n_{k=0}(-1)^k n_k. \end{aligned}$$

Although all equivalent under compactness assumptions, e.g., see Theorem 4.2 below, we call (4.3) the combinatorial definition of the Euler characteristic. This definition shows in particular that for non-compact spaces homotopy invariance of \(\chi (\cdot )\) does not necessarily hold, e.g., compare \(\chi \) for an open and a closed interval.

Example 4.2

(The CW structure is not unique: the 2-sphere) Recall that the sphere \(\mathbb {S}^2\) can be constructed from a 2-cell (a disk) and 0-cell (a point), hence \(\chi (\mathbb {S}^2)=2\). Similarly, one could fix two poles and construct \(\mathbb {S}^2\) from two 0-cells, two 1-cells (two intervals) and two 2-cells, adding up to \(\chi (\mathbb {S}^2)=2\).

See also [9] for a less straightforward matrix manifold example. The combinatorial formula (4.3) is particularly appealing due to the following result.

Proposition 4.1

(CW equivalences [4, Corollary A.12]) Any compact topological manifold is homotopy equivalent to a finite CW complex.

Next, we provide the homological definition of \(\chi (\textsf{X}^n)\), here, \(b_k=\textrm{rank}\,H_k(\textsf{X}^n;\mathbb {Z})\) denotes the k-th Betti number of \(\textsf{X}^n\), i.e., when \(H_k(\textsf{X};\mathbb {Z})\) is finitely generated, we have \(H_k(\textsf{X}^n;\mathbb {Z})\simeq \mathbb {Z}^{b_k}\oplus T_k\), for \(T_k\) the torsion [1, p. 258, 282]. Compute for instance \(H_{(\cdot )}(\mathbb {R}\mathbb {P}^n;\mathbb {Z})\) to see \(T_{(\cdot )}\ne 0\). In that sense, we should speak of “holesandtwists”.

Theorem 4.1

(The (homological) Euler characteristic [4, Theorem 2.44]) Let \(\textsf{X}^n\) be a finite CW complex. Then, the Euler characteristic of \(\textsf{X}^n\) equals

$$\begin{aligned} \chi (\textsf{X}^n) = \textstyle \sum \limits ^n_{k=0}(-1)^k b_k. \end{aligned}$$

It follows from Poincaré duality that for closed.Footnote 3 oriented manifolds one has \(\textrm{rank}\,H_k(\textsf{M}^m;\mathbb {Z})=\textrm{rank}\,H^{m-k}(\textsf{M}^m;\mathbb {Z})\), and by the universal coefficient theorem that \(\textrm{rank}\,H_k(\textsf{M}^m;\mathbb {Z})=\textrm{rank}\,H^k(\textsf{M}^m;\mathbb {Z})\) e.g., see [4, Corollary 3.37]. However, then we have

$$\begin{aligned} \textrm{rank}\,H_{m-k}(\textsf{M}^m;\mathbb {Z})=\textrm{rank}\,H_k(\textsf{M}^m;\mathbb {Z}), \end{aligned}$$

see [4, Sect. 3.3] for the omitted details. A similar argument holds for non-orientable manifolds, such that by (4.4) the following result follows.

Corollary 4.1

(Odd-dimension Euler characteristic [4, Corollary 3.37]) A closed manifold \(\textsf{M}\) of odd-dimension has \(\chi (\textsf{M})=0\).


(Sketch) Combining (4.4) with (4.5): \(\chi (\textsf{M})=b_0-b_1+\cdots +b_1-b_0=0\).

Note that for oriented manifolds, Corollary 4.1 can also be shown using oriented intersection theory, cf. (3.5). When \(\textsf{M}\) has a boundary, Corollary 4.1 is not true, consider the interval [0, 1]. We will return to Corollary 4.1 frequently.

We end with one of the pillars of topology, linking the seemingly different definitions of the Euler characteristic. See Sect. 8.3 for comments on Morse theory.

Theorem 4.2

(The Euler characteristic [2, Theorem 8.6.6]) Let \(\textsf{M}\) be a closed, oriented, smooth manifold, then the corresponding combinatorial Euler characteristic (4.3), homological Euler characteristic (4.4) and the Euler characteristic from oriented intersection theory (3.6) all agree.


(Sketch) Recall the relation between Morse indices and cells in a CW structure, e.g., see Theorem 8.2 below. Then, following [2], construct a Morse function \(g:\textsf{M}^m\rightarrow \mathbb {R}\) with \(m_k\) critical points of index k, for \(k=1,\dots ,m\). Now, using Theorem 8.2 and Definition 4.1 it follows that \(\chi (\textsf{M}^m)=\sum \nolimits _k (-1)^k m_k\). However, note that the vector field index for an equilibrium point (critical point of g) with Morse index k equals \((-1)^k\), and we have \(m_k\) of them. As such, Theorem 3.6 (Poincaré–Hopf) tells us that \(\chi (\textsf{M}^m)=\sum \nolimits _k(-1)^km_k\), which is exactly what we had before. Then, the relation to homology follows from Theorem 4.1.

Observe that compactness is important as \(\chi (\mathbb {R}^n)=(-1)^n\) according to the combinatorial definition (4.3) while \(\chi (\mathbb {R}^n)=1\) according to singular homology (4.4).

We refer the reader to  [3] for an illustrated introduction to algebraic topology and to [1, 4, 8] for complete treatments.