Singular chains and singular homology

Overview

A singular chain $\sigma$ is a formal linear combination of continuous maps from generalized triangles to an arbitrary space $X$. Two chains $\sigma ,\sigma ’\in C_p(X)$ are homologous if their difference is a boundary, meaning there exists some formal linear combination $\alpha \in C_{p+1}(X)$ for which ${\partial }_{p+1}\alpha =\sigma -\sigma ’$. Homology is an equivalence relation, and the $p$th singular homology group is the group of $p$-cycles modulo the relation of being homologous.

Key terms and notation

• Standard $p$-simplex ${\Delta }^p$: the $p$-dimensional “triangle” whose vertices are the standard basis vectors of ${\mathbb{R}}^{p+1}$.
• Boundary $\partial {\Delta }^p$: the set obtained by inserting a $0$ at the $i$th coordinate for each dimension $0\le i\le p$ and each point in ${\Delta }^{p-1}$.
• Singular $p$-simplex $\sigma$: any continuous map ${\Delta }^p\to X$ for some topological space $X$.
• ${\text{Sin}}_p(X)$: the set of all singular $p$-simplices in $X$.
• Face of a $p$-simplex $d_i(\sigma )$: the $(p-1)$-simplex obtained by first inserting $0$ at the $i$th coordinate and then applying some singular $p$-simplex $\sigma$. Note that this procedure is a map ${\text{Sin}}_p(X)\to {\text{Sin}}_{p-1}(X)$.
• Singular $p$-chain: a formal linear combination of $p$-simplices.
• $C_p(X)$: the free abelian group $\mathbb{Z}{\text{Sin}}_p(X)$ (i.e., all formal linear combinations of $p$-simplices in $X$ with coefficients in $\mathbb{Z}$).
• Boundary operator ${\partial }_p$: the group homomorphism $C_p(X)\to C_{p-1}(X)$ given by adding together oriented (i.e., scaled by some power of $-1$) faces of $p$-simplices.
• $p$-cycle: an element in the kernel of the boundary operator $\partial$, which is denoted $\ker \partial =Z_p(X)\subset C_p(X)$.
• Boundary: an element in the image of $\partial$, which is denoted $\text{Im}\partial =B_{p-1}(X)\subset C_{p-1}(X)$.

Singular homology is the composition of functors $\mathsf{Top}\to \mathsf{Ch}\to \mathsf{Ab}$ defined by the mappings

$$X\mapsto (C_{\ast }(X),\partial )\mapsto H_p(X)f\mapsto f_{\ast }\mapsto H_p(f),$$

where $X$ is any space, $f:X\to Y$ is a continuous function, $f_{\ast }:C_p(X)\to C_p(Y)$ is the homomorphism defined on generators by $\sigma \mapsto f\circ \sigma$, and $H_p(f):H_p(X)\to H_p(Y)$ is the homomorphism defined by $\left[\sum c_{\sigma }\sigma \right]\mapsto \left[\sum c_{\sigma }(f\circ \sigma )\right]$ for cycles $c=\sum c_{\sigma }\sigma$.

Relevant theorems:

• (Theorem) The first homology group is isomorphic to the abelianization of the fundamental group for path-connected spaces

Related notes:

• Homology of general chain complexes
• Simplicial chains and simplicial homology

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Standard simplices

Simplices are higher-dimensional analogues of triangles, and play a role in singular homology that is similar to the role of the unit interval $I=[0,1]$ for fundamental groups, where every representative in the fundamental group is a loop out of $I$. The basic standard simplices are spaces with $p$ vertices that are realized in $p+1$ Euclidean space.

Singular $p$-simplex

Given an integer $p\ge 0$, the standard $p$-simplex is the space ${\Delta }^p=\left\{(t_0,\dots ,t_p)\in {\mathbb{R}}^{p+1}:{\sum }_{i=0}^p{t}_i=1,t_i\ge 0\text{ for all }i\right\},$ which is equipped with the subspace topology induced by the usual Euclidean topology on ${\mathbb{R}}^{p+1}$.

If $X$ is any topological space, a singular $p$-simplex in $X$ is a continuous map $\sigma :{\Delta }^p\to X$. The set of all singular $p$-simplices in a space $X$ is denoted $\text{mathup}{Sin}_p(X)$.

Remark

If $e_0,\dots ,e_p\in {\mathbb{R}}^{p+1}$ is the standard basis, e.g., $e_0=(1,0,\dots ,0$), $e_1=(0,1,\dots ,0)$, then the point $t\in {\Delta }^p$ may be written as the linear combination $t=(t_0,\dots ,t_p)=t_0{e}_0+\cdots +t_p{e}_p.$

Exm

${\Delta }^0$ is the one-point space, ${\Delta }^1$ is homeomorphic to an interval (interpreted geometrically as a line between two points $(0,1)$ and $(1,0)$ in ${\mathbb{R}}^2$) , ${\Delta }^2$ is a solid triangle, and ${\Delta }^3$ is a tetrahedron.

Face, boundary of simplices

For $0\le i\le p$, the face map ${\delta }^i:{\Delta }^{p-1}\to {\Delta }^p$ is the map defined by

$${\delta }^i(t_0,\dots ,t_{p-1})=(t_0,\dots ,t_{i-1},0,t_i,\dots ,t_{p-1}).$$

The image ${\delta }^i({\Delta }^{p-1})\subset {\Delta }^p$ is called the $i$th face of ${\Delta }^p$, and the boundary of ${\Delta }^p$ is the union of all faces $\partial {\Delta }^p={\bigcup }_{i=0}^p{\delta }^i({\Delta }^{p-1})\subset {\Delta }^p.$

For a general singular $p$-simplex $\sigma :{\Delta }^p\to X$ in a space $X$, the $i$th face of $\sigma$ is a $(p-1)$-simplex given by the image of the map $d_i:{\text{Sin}}_p(X)\to {\text{Sin}}_{p-1}(X)$, the canonical linear homeomorphism that preserves the ordering of vertices defined by $d_i(\sigma )=\sigma \circ {\delta }^i.$

Exm

${\Delta }^1$ has two faces, which are endpoints of the interval; ${\Delta }^2$ has three faces, which are the three edges of a triangle; ${\Delta }^3$ has four faces, which are the triangles of in the boundary of a solid tetrahedron.

Exm

A $0$-simplex in a space $X$ is a point in $X$. By the homeomorphism ${\Delta }^1\cong I$, a $1$-simplex in $X$ is a path in $X$, and the $0$-simplex faces $d_0(\sigma )$ and $d_1(\sigma )$ are just the two endpoints of the path.

Lemma

For $0\le i<j\le p$, we have ${\delta }^j{\delta }^i={\delta }^i{\delta }^{j-1}:{\Delta }^{p-2}\to {\Delta }^p$.

Proof from Algebraic Topology. Both maps send a point $(t_0,\dots ,t_p-2)\in {\Delta }^p$ to

$$(t_0,\dots ,t_{i-1},0,t_i,\dots ,t_{j-2},0,t_{j-1},\dots ,t_{p-2}).\square$$

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Singular chains

Singular $p$-chains, $p$th singular chain group

For an integer $p\ge 0$ and a topological space $X$, a singular $p$-chain in $X$ is a formal linear combination of singular $p$-simplices in $X$, which are continuous maps $\sigma :{\Delta }^p\to X$.

The $p$th singular chain group is the set of all singular $p$-chains. It is defined as the free abelian group with basis ${\text{Sin}}_p(X)$, i.e., the set of finite linear combinations of elements of ${\text{Sin}}_p(X)$ with coefficients in $\mathbb{Z}$:

$$C_p(X)=\mathbb{Z}{\text{Sin}}_p(X)=\left\{\sum _{\sigma :{\Delta }^p\to X}{n}_{\sigma }\cdot \sigma :n_{\sigma }=0\in \mathbb{Z}\text{ for all but finitely many }\sigma \right\}.$$

Remark

Any $p$-simplex in a space $X$, i.e., a continuous map $\sigma :{\Delta }^p\to X$, is a $p$-chain. A general $p$-chain is a $\mathbb{Z}$-linear combination of $p$-simplices.

Boundary operator

The boundary operator is the group homomorphism $\partial :C_p(X)\to C_{p-1}(X)$ defined by

$$\partial \sigma =\sum _{i=0}^p(-1{)}^i{d}_i(\sigma )$$

for all standard $p$-simplices $\sigma \in {\text{Sin}}_p(X)$.

Using the fact that functions out of a base space can be extended to homomorphisms out of free abelian group, for a general chain $c={\sum }_{\sigma }{c}_{\sigma }\sigma$ we can define

$$\partial c=\sum _{\sigma }\sum _{i=0}^p(-1{)}^i{c}_{\sigma }{d}_i(\sigma ).$$

This is equivalent to defining the general boundary operator by specifying its values on basis elements.

Remark

The same notation $\partial$ is typically used for all $p$, though, e.g., Hatcher specifies ${\partial }_p:C_p(x)\to C_{p-1}(X)$.

• Note that signs are inserted to account for orientation and coherent oritnations

Lemma

The composition ${\partial }^2=\partial \circ \partial :C_p(X)\to C_{p-1}(X)\to C_{p-2}(X)$ is the trivial homomorphism.

Sketch of proof from Algebraic Topology. It suffices to show this on a basis, i.e., that ${\partial }^2\sigma =0$ for any ${\text{Sin}}_p(X)$. Explicitly write out the sum, then use the fact that face maps have the relation ${\delta }^j{\delta }^i={\delta }^i{\delta }^{j-1}$ to conclude that the terms in the sum cancel. $\square$

Corollary

The image of the homomorphism ${\partial }_{p+1}:C_{p+1}(X)\to C_p(X)$ is contained in the kernel of ${\partial }_p:C_p(X)\to C_{p-1}(X)$.

Proof from Algebraic Topology. An element $c\in \mathrm{Im}({\partial }_{p+1})\subseteq C_p(X)$ can be written as ${\partial }_{p+1}(c’)=c$ for some $c’\in C_{p+1}(X)$. Since ${\partial }_p{\partial }_{p+1}(c^{\prime })=0$, we can conclude that $c\in \ker ({\partial }_p)$ as well. $\square$

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Singular homology

Cycle, boundary

The kernel of the group homomorphism ${\partial }_p:C_p(X)\to C_{p-1}(X)$ is denoted $Z_p(X)=\ker {\partial }_p\subset C_p(X);$ elements of this kernel are called $p$-cycles in $X$.

The image of the group homomorphism ${\partial }_{p+1}:C_{p+1}(X)\to C_p(X)$ is denoted $B_p(X)=\text{Im}{\partial }_{p+1}\subset C_p(X);$ elements of this image are called boundaries.

Example.

$p$th singular homology group, homologous

The $p$th singular homology group of $X$ is the abelian group formed by the group of $p$-cycles modulo boundaries in $X$:

$$H_p(X)=Z_p(X)/B_p(X)=\frac{\ker ({\partial }_p:C_p(X)\to C_{p-1}(X))}{\text{Im}({\partial }_{p+1}:C_{p+1}(X)\to C_p(X))}.$$

Two chains $c,c’\in C_p(X)$ are homologous if their difference is a boundary, i.e., there exists $\alpha \in C_{p+1}(X)$ such that $\partial \alpha =c-c’\in B_p(X).$ Thus, $H_p(X)$ is the group of $p$-cycles modulo the relation of being homologous.

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Examples

Singular homology groups of the one-point space

Let $X=\{\ast \}$ be the one-point space. We have:

• The set of singular simplices ${\text{Sin}}_p(X)$ is the one-point set of the constant map $\{{\sigma }_p=c:{\Delta }^0\to X\}$. In other words, only on singular simplex exists in each dimension $p$.
• The $p$th singular chain group is $C_p(X)=\mathbb{Z}{\text{Sin}}_p(X)\cong \mathbb{Z},$ the free abelian group generated by the unique singular $p$-simplex $c$. The isomorphism $\mathbb{Z}\to C_p(X)$ is given by the mapping $n\mapsto n\cdot c$ for all $n\in \mathbb{Z}$.
• Since there is only one $p$-simplex in each $C_p(X)$, the boundary operator $\partial :C_p(X)\to C_{p-1}(X)$ defined by $\partial {\sigma }_p={\sum }_{i=0}^p(-1{)}^i{\sigma }_i{d}_i(\sigma )$, where $d_i^p:{\Delta }^{p-1}\to {\Delta }^p$ is a face map, becomes the mapping $\partial {\sigma }_p=({\sum }_{i=0}^p(-1{)}^n){\sigma }_{p-1}$ defined on the basis ${\text{Sin}}_p(X)$.
• This extends to $n\cdot c\mapsto n\cdot (c-c+c-\cdots )$ on $C_p(X)$. When $p$ is odd, the $p+1$ terms of the sum cancel each other out and we have $\partial {\sigma }_p=0$. When $p$ is even, we have $n\cdot {\sigma }_{p-1}$.
• wip

eAch $C_i(X)\cong \mathbb{Z}$

Similar method for each discrete top. space

$C_3(X){\to }_0{C}_2(X){\to }_{\text{id}}{C}_1{\to }_{\partial =0}{C}_0(X)$

$$ $$ H_1(X) = \frac{\ker\partial_1}{\textup{Im} \ \partial _2} = \frac{C_1(X)}{C_1(X)} \cong 0 $$ $$ H_2(X) = \frac{\ker \partial_2}{\textup{Im} \ \partial_3} = \frac{0}{0} = 0. $$ #### $H_0(X)$ for a path-connected space is always infinite cyclic >[!abstract] Proposition: The zeroth singular homology group for a [[(Path-)connectedness|path-connected]] space is isomorphic to the integers > >Given any space $X$, we have $$ H _0 (X) \cong \mathbb Z \iff X \text{ path-connected}. $$ *Proof from [[MATH-GU4053|Algebraic Topology]].* ($\impliedby$) By definition, $$ H_0(X) = \frac{\ker(\partial_0 : C_0 (X) \to 0) }{\textup{Im}(\partial_1 : C_1(X) \to C_0(X)) } = \frac{C_0(X)}{\textup{Im} \ \partial_1}, $$where the final quotient is the **cokernel** of $\partial_1$. #wip something to do with the one-point space? Define a map $\varepsilon : H_0(X) \to \mathbb Z$ by adding up coefficients $$ \left [ \sum_{x \in X} n_x \cdot x \right ] \mapsto \sum_{x \in X}n_x. $$ Our goal is to show that this map $\varepsilon$ is a [[Group homomorphisms and isomorphisms|group isomorphism]]: - **Well-definedness:** - **Surjectivity:** - **Injectivity:** The claim is that for any two $x, x’ \in X$ such that $[x], [x’] \in H_0(X)$, we have $[x] = [x’]$. Using the fact that $I \cong \Delta^1$, by [[(Path-)connectedness|path-connectedness]] of $X$ we can choose $\sigma : \Delta^1 \to X$ such that $\sigma(0, 1) = x$ and $\sigma (1, 0) = x’$, so $$ \partial \sigma = d_0\sigma - d_1 \sigma = x - x' \implies [x] = [x']. $$ #### $H_0(X)$ for a general space is isomorphic to a free abelian group of path-components --- # Code snippets ``` \partial_p : C_p(X) \to C_{p-1}(X) ```