These notes are a record of what will be covered in the first half (roughly speaking) of Michaelmas term 2020. These notes are not exhaustive. In particular, proofs will be given in lectures and made available as hand-written notes. Proofs of most statements can also be found in Hatcher’s book Algebraic Topology - available for free online. Examples will also be given in lectures and made available as hand-written notes.
A set of typed notes covering the second half (roughly speaking) of Michaelmas 2020 will be made available after a few weeks.
In this section we give the definition of the singular homology of a topological space. This definition is good since it just relies on the topological space (and not on some decomposition, description, or presentation of the space). The definition is problematic since it will be hard to make any computations using it until we have developed some algebraic machinery.
A chain complex is a sequence of abelian groups \(\{C_i\}_{i \in \mathbb Z}\) and homomorphisms \(\partial_i \colon C_i \to C_{i-1}\) such that \(0= \partial_i \circ \partial_{i+1} \colon C_{i+1} \to C_{i-1}\). The maps \(\partial_i\) are called boundary maps or differentials.
The reason for the word ‘boundary’ will be become apparent in a short while, the reason for the word ‘differential’ is that there exist formulations of homological-style invariants that use so-called differential forms - although we shall not come across these in the current course.
The homology \(H_*(C)\) of a chain complex \((C_*,\partial_*)\) is \[H_i(C) := \frac{\ker(\partial_i \colon C_i \to C_{i-1})}{\mathop{\mathrm{im}}(\partial_{i+1} \colon C_{i+1} \to C_i)}.\]
The standard \(n\)-simplex \(\Delta^n\) is \[\{\underline{x} \in \mathbb R^{n+1} \mid x_0 + x_1 + \cdots + x_n =1 , x_i \geq 0 \,\, {\rm for} \,\, 0 \leq i \leq n\}.\] A singular \(n\)-simplex of a topological space \(X\) is a continuous map \(\sigma \colon \Delta^n \to X\).
For \(n\in \mathbb N_0\), the singular \(n\)-chains of \(X\), \(C_n(X)\), is the free abelian group generated by the singular \(n\)-simplices. \[C_n(X) := \{n_1 \sigma_1 + \cdots n_k\sigma_k \mid k \geq 1, n_1,\dots,n_k \in \mathbb Z, \sigma_1,\dots,\sigma_k \colon \Delta^n \to X \text{ sing.\ simplices}\}\] Note that \(C_n(X)=0\) for \(n <0\).
Let \(v_0,\dots,v_n\) be the standard basis of \(\mathbb R^{n+1}\). Note that each \(v_i\) corresponds to a vertex of \(\Delta^n\). There are \(n+1\) inclusion maps \(\iota_j \colon \Delta^{n-1} \to \Delta^n\), \(j=0,\dots,n\), defined by restricting the linear maps \(\mathbb R^{n} \to \mathbb R^{n+1}\) that send \[\begin{cases} v_i \mapsto v_i & i<j \\ v_i \mapsto v_{i+1} & i \geq j. \end{cases}\] We also write this map as \[[v_0,\dots,\widehat{v_j},\dots,v_n].\] Here, as is customary, the hat denotes the missing vertex / coordinate. Now we can define \[\partial(\sigma) := \sum_{j=0}^n (-1)^j \sigma \circ \iota_j.\] Then we extend the boundary map \(\partial\) by linearity to define it on formal sums of singular simplices.
\(\partial^2 \colon C_n(X) \to C_{n-2}(X)\) is the zero map, so we see that \((C_*(X),\partial)\) is a chain complex.
The \(n\)th singular homology of \(X\) is \[H_n(X) := H_n(C_*(X)) = \frac{\ker(\partial_i \colon C_i(X) \to C_{i-1}(X))}{\mathop{\mathrm{im}}(\partial_{i+1} \colon C_{i+1}(X) \to C_i(X))}.\]
We also write \(Z_n(X) := \ker(\partial_i \colon C_i(X) \to C_{i-1}(X))\), the \(n\)-cycles, and \(B_n(X) := \mathop{\mathrm{im}}(\partial_{i+1} \colon C_{i+1}(X) \to C_i(X))\), the \(n\)-boundaries, so that
\[H_n(X) = \frac{Z_n(X)}{B_n(X)} {\rm .}\]
When \(X\) is a point, \(H_0(X) \cong \mathbb Z\) and \(H_i(X) = 0\) for \(i \neq 0\).
Let \(A\) be the set of path components of \(X\). Then \(H_0(X) \cong \bigoplus_A \mathbb Z\). If \(X\) is path connected, then \(H_0(X)=\mathbb Z\).
Higher homology groups cannot be directly computed from the definition. We need to develop some tools and theory with which to compute. This means learning about exact sequences, which we will do in the next chapter. However, it might help to have to some intuition for what these homology groups will eventually turn out to be – so here are the homologies of some familiar spaces.
For \(X= \mathbb R^n\), the homology is the same as that of a point. \[H_i(\mathbb R^n) \cong \begin{cases} \mathbb Z, & \mbox{if } i=0 \\ 0, & \mbox{otherwise}. \end{cases}\]
The \(n\) dimensional sphere is defined as the subspace of \(\mathbb R^{n+1}\) \[S^n := \{\underline{x} \in \mathbb R^{n+1} \mid \|\underline{x}\| = 1\}.\] The homology is \[H_i(S^n) = \begin{cases} \mathbb Z, & \mbox{if } i=0,n \\ 0, & \mbox{otherwise}. \end{cases}\]
Products of spheres, \(S^n \times S^m\). First, if \(n=m\) we have: \[H_i(S^n \times S^n) = \begin{cases} \mathbb Z, & \mbox{if } i=0,2n \\ \mathbb Z\oplus \mathbb Z, & \mbox{if } i=n \\ 0, & \mbox{otherwise}. \end{cases}\] The torus \(S^1 \times S^1\) is a good first case to think about. On the other hand if \(n \neq m\) we have \[H_i(S^n \times S^m) = \begin{cases} \mathbb Z, & \mbox{if } i=0,n,m,n+m \\ 0, & \mbox{otherwise}. \end{cases}\]
What are the advantages of defining homology by considering the (usually) infinite rank abelian groups generated by all possible continuous maps of an \(n\)-simplex into \(X\)? One advantage is that it is easy to prove that homology behaves well with respect to maps between spaces i.e. that it is functorial.
A chain map \(F \colon C_* \to D_*\) between chain complexes \(C_*\) and \(D_*\) is a collection of homomorphisms \(F_n \colon C_n \to D_n\) such that \[\partial_{n+1}^D \circ F_{n+1} = F_n \circ \partial_n^C \colon C_{n+1} \to D_n\] for every \(n \in \mathbb Z\).
In other words, the diagram \[\xymatrix{\cdots \ar[r] & C_{n+1} \ar[r]^{\partial_{n+1}^C} \ar[d]^{F_{n+1}} & C_n \ar[r]^{\partial_{n}^C} \ar[d]^{F_{n}} & C_{n-1} \ar[r] \ar[d]^{F_{n-1}} & \cdots \\ \cdots \ar[r] & D_{n+1} \ar[r]_{\partial_{n+1}^D} & D_n \ar[r]_{\partial_{n}^D} & D_{n-1} \ar[r] & \cdots }\] commutes.
Commutative diagrams will be very important to us, so it’s important to understand what they mean.
A chain map \(F \colon C_* \to D_*\) induces a map on homology \[\begin{array}{rcl} F_* \colon H_n(C_*) &\to& H_n(D_*) \\ {[c]} &\mapsto & {[f(c)]} \end{array}\] for every \(n \in \mathbb N_0\).
In other words, this map is both defined, i.e. sends cycles to cycles, and well-defined, meaning it sends boundaries to boundaries.
Let \(f \colon X \to Y\) be a homeomorphism of topological spaces. Then the induced map \(f_* \colon H_n(X) \xrightarrow{\cong} H_n(Y)\) is an isomorphism for every \(n \geq 0\).
A sequence of abelian groups and homomorphisms \[A \xrightarrow{f} B \xrightarrow{g} C\] is exact at \(B\) if \(\mathop{\mathrm{im}}(f) = \ker(g)\). A sequence \(\cdots \to A_{i+1} \to A_i \to A_{i-1} \to \cdots\) is exact if \(A_{i+1} \to A_i \to A_{i-1}\) is exact at \(A_i\) for every \(i\).
Note that for chain complexes, \(\partial_n \circ \partial_{n+1} =0\) means that \(\mathop{\mathrm{im}}\partial_{n+1} \subseteq \ker \partial_n\). A chain complex \(C_*\) is an exact sequence if and only if \(H_n(C_*)=0\) for every \(n\).
A short exact sequence is five-term exact sequence \(0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0\). So \(f\) is injective, \(g\) is surjective, and \(\mathop{\mathrm{im}}f = \ker g\). Note that \(C \cong B/\ker g \cong B/\mathop{\mathrm{im}}f \cong B/f(A)\).
A short exact sequence of chain complexes is a sequence of chain maps \[0 \to C_* \to D_* \to E_* \to 0\] with \(0 \to C_n \to D_n \to E_n \to 0\) being a short exact sequence of abelian groups for every \(n\).
A short exact sequence of chain complexes \[0 \to C_* \xrightarrow{f} D_* \xrightarrow{g} E_* \to 0\] determines a long exact sequence in homology groups \[\begin{aligned} \cdots \to H_{n+1}(E) \xrightarrow{\delta} H_n(C) \xrightarrow{f_*} H_n(D) \xrightarrow{g_*} H_n(E) \xrightarrow{\delta} H_{n-1}(C) \to \cdots\end{aligned}\]
Here is the definition of the map \(\delta\). Let \([e] \in H_{n+1}(E)\), so \(e \in E_{n+1}\) and \(\partial(e)=0\). Since \(g_{n+1}\) is surjective, there exists \(d \in D_{n+1}\) with \(g_{n+1}(d)=e\). Then \[g_n\circ \partial^D(d) = \partial^E \circ g_{n+1}(d) = \partial^E(e)=0.\] Therefore \(\partial^D(d) \in \ker(g_n) \subseteq \mathop{\mathrm{im}}f_n\). Since \(f_n\) is injective, there is a unique \(c \in C_n\) with \(f_n(c) = \partial^D(d)\). We define \[\delta([e]) = [c].\] We need to see that this defines a homology class, that is \(\partial^C(c)=0\), and that this is well defined: we chose a representative \(e\) and we chose \(d\). There are also six things to show to check exactness. Details can be found in Hatcher pages 116–7.
Let \(f,g \colon X \to Y\) be continuous maps between topological spaces. A homotopy from \(f\) to \(g\) is a continuous map \(h \colon X \times I \to Y\) with \(h|_{X \times \{0\}} = f \colon X \times \{0\} = X \to Y\) and \(h|_{X \times \{1\}} = g \colon X \times \{0\} = X \to Y\). We write \(f \sim_h g\) or just \(f \sim g\).
If \(f,f' \colon X \to Y\) are homotopic and \(g,g' \colon Y \to Z\) are homotopic then \(g\circ f \sim g' \circ f' \colon X \to Z\) are also homotopic.
Homotopy gives rise to an equivalence relation on maps \(X \to Y\).
A map \(f\colon X \to Y\) is a homotopy equivalence if there exists a map \(g \colon Y \to X\) such that \(f \circ g \sim \mathop{\mathrm{Id}}_Y\) and \(g \circ f \sim \mathop{\mathrm{Id}}_X\). The map \(g\) is called the homotopy inverse. We say that the spaces \(X\) and \(Y\) are homotopy equivalent, and write \(X \simeq Y\).
Homotopy equivalence is an equivalence relation on spaces. Please do not say or write that two spaces \(X\) and \(Y\) are homotopic. There is no meaning attached to such a phrase.
Any two homotopy inverses \(g_1,g_2 \colon Y \to X\) for a homotopy equivalence \(f \colon X \to Y\) are homotopic.
We have a sequence of homotopies \[g_1 = \mathop{\mathrm{Id}}_Y \circ g_1 \sim g_2 \circ f \circ g_1 \sim g_2 \circ \mathop{\mathrm{Id}}_X = g_2.\qedhere\]
A space \(X\) is contractible if \(X \simeq \{\mathop{\mathrm{pt}}\}\).
Here is the most important result for us about homotopy equivalences, which will be proven in the next section..
Let \(f \colon X \to Y\) be a homotopy equivalence. Then \(f_* \colon H_n(X) \to H_n(Y)\) is an isomorphism for every \(n \geq 0\).
So spaces \(X\) and \(Y\) with different homology are not homotopy equivalent. In particular we see that \(H_k(D^n) \cong H_k(\mathbb R^n) \cong H_k(\{\mathop{\mathrm{pt}}\})\), and also \(H_k(\mathbb R^2 {\smallsetminus}\{\mathop{\mathrm{pt}}\}) \cong H_k(S^1)\) for every \(k\).
A similar result holds for fundamental groups, provided we are careful with base points.
Let \(f \colon (X,x) \to (Y,y)\) be a based homotopy equivalence with \(f(x)=y\). Then \(f_* \colon \pi_1(X,x) \to \pi_1(Y,y)\) is an isomorphism.
Here are some important homotopy equivalent spaces.
Let \(f \colon X \to Y\) be a continuous map. The mapping cylinder of \(f\), \(M_f\), is \[X \times I \coprod Y / (X \times \{1\} \sim f(x) \text{ for all }x \in X).\]
For every \(f \colon X \to Y\), we have that \(M_f \simeq Y\).
The cone on a map \(f\) is \[\operatorname{Cone}(f) = C_f := M_f / X \times \{0\}.\]
For any space \(X\), \(C_{\mathop{\mathrm{Id}}_X} \simeq \{\mathop{\mathrm{pt}}\}\). i.e. the cone on the identity map is contractible.
An important special case of homotopy equivalences is deformation retracts. Let \(i_A \colon A \subseteq X\) be a subspace.
A deformation retract of \(X\) onto \(A\) is a map \(f \colon X \to A\) with a homotopy \(H \colon X \times I \to X\) with \(\mathop{\mathrm{Id}}_X \sim_H f\) such that \(H|_{A\times \{t\}} \colon A \times \{t\} \to X\) coincides with \(i_A\) for every \(t \in I\).
A retraction is a continuous map \(r \colon X \to X\) with \(r(X)=A\) and \(r \circ i_A \colon A \to X\) equal to \(i_A\).
Every nonempty space \(X\) retracts to a point for all \(X\), but this is not true for deformation retracts.
A deformation retract determines a retract, but the converse does not hold. A deformation retract is a homotopy equivalence, but homotopy equivalence is a notion defined when neither space is a subspace of the other.
Chain homotopies are algebraic models of homotopies, and chain homotopy equivalences model homotopy equivalences.
Two chain maps \(f,g \colon C_* \to D_*\) are chain homotopic if there exists a homomorphism \(P_n \colon C_n \to D_{n+1}\) for every \(n\) with \[f_n - g_n = \partial \circ P_n + P_{n-1} \circ \partial \colon C_n \to D_n.\] We write \(f \sim g\).
If chain maps \(f \sim g \colon C_* \to D_*\) then \(f_* = g_* \colon H_n(C_*) \to H_n(D_*)\) for every \(n\).
Let \(c \in C_n\) be a cycle, so \(\partial c =0\). Then \(f_n(c) - g_n(c) = \partial(P(c)) + P(\partial(c)) = \partial(P(c))\), so \[[f_n(c)] = [g_n(c) + \partial(P(c))] = [g_n(c)].\qedhere\]
If maps of spaces \(f,g \colon C_*(X) \to C_*(Y)\) are homotopic, then \(f_* \sim g_* \colon C_*(X) \to C_*(Y)\) are chain homotopic.
The idea of the proof is to convert the product \(\Delta^n \times I\) into a sum of simplices, and use this to take the data of a homotopy and convert it into a chain homotopy.
A chain map \(f \colon C_* \to D_*\) is a chain homotopy equivalence if there is a chain map \[g \colon D_* \to C_*\] with \(g \circ f \sim \mathop{\mathrm{Id}}_C\) and \(f \circ g \sim \mathop{\mathrm{Id}}_D\). We write \(C_* \simeq D_*\) and we say that \(C_*\) and \(D_*\) are chain homotopy equivalent. If \(C_* \simeq 0\) then we say that \(C_*\) is chain contractible.
If \(X \simeq Y\) then \(C_*(X) \simeq C_*(Y)\).
Let \(f \colon X \to Y\) and \(g \colon Y \to X\) be maps witnessing the homotopy equivalence. Then \(f \circ g \sim \mathop{\mathrm{Id}}\) and \(g \circ f \sim \mathop{\mathrm{Id}}\) imply that \(f_* \circ g_* \sim \mathop{\mathrm{Id}}_* \colon C_*(Y) \to C_*(Y)\) and \(g_* \circ f_* \sim \mathop{\mathrm{Id}}_* \colon C_*(X) \to C_*(X)\) by Theorem [thm:chain-homotopy]. It follows that \(C_*(X) \simeq C_*(Y)\) as desired.
If \(C_* \simeq D_*\) then \(H_n(C_*) \cong H_n(D_*)\) for every \(n\).
By Proposition [prop:chain-map] we have that \(g_*\circ f_* = \mathop{\mathrm{Id}}_* \colon H_n(C_*) \to H_n(C_*)\) and \(f_*\circ g_* = \mathop{\mathrm{Id}}_* \colon H_n(D_*) \to H_n(D_*)\) for every \(n\).
By combining the above facts we obtain the following conclusion.
[cor:homotopy-invariance] If two spaces \(X\) and \(Y\) are homotopy equivalent via a homotopy equivalence \(f \colon X\to Y\), then \(f_* \colon H_n(X) \to H_n(Y)\) is an isomorphism for every \(n \in \mathbb N_0\).
We are about half-way through the term and as yet have not quite developed the algebraic technology to enable us to compute the homology groups of, for example, the circle \(S^1\). That changes in this chapter.
The next stage in computing homology groups is to develop the Mayer-Vietoris long exact sequence. We know that a short exact sequence of chain complexes gives rise to a long exact sequence in homology. So we need an auspicious choice of short exact sequence.
Given a subset \(U \subset X\) let \(\mathring{U}\) denote its interior. Let \(\mathcal{U} = \{U_i\}_{i \in \mathcal{I}}\) be a collection of subsets \(U_i \subseteq X\) with \(\{\mathring{U}_i\}\) an open cover of \(X\). Let \[\begin{aligned} C_n^{\mathcal{U}}(X) := &\text{ free abelian group on singular } n-\text{simplices} \sum_j n_j \sigma_j \\ &\text{ where for every } j,\, \sigma_j(\Delta^n) \subseteq U_{i_j} \text{ for some } i_j \in \mathcal{I}. \end{aligned}\]
The inclusion \(C_*^{\mathcal{U}}(X) \to C_*(X)\) is a chain homotopy equivalence.
The proof uses iterated subdivision of simplices by adding in the barycentre as an extra vertex, to construct a chain homotopy inverse. See Proposition 2.21 of Hatcher. We use this in the next theorem, giving the Mayer-Vietoris sequence.
[Mayer-Vietoris long exact sequence] Let \(X = \mathring{U} \cup \mathring{V}\) where \(U,V\) are subsets. Let \(\mathcal{U} := \{U,V\}\). There is a short exact sequence of chain complexes \[0 \to C_*(U \cap V) \xrightarrow{(\iota_U,-\iota_V)} C_*(U) \oplus C_*(V) \xrightarrow{j_U + j_V} C_*^{\mathcal{U}}(X) \to 0\] inducing a long exact sequence of homology groups \[\begin{aligned} \cdots \xrightarrow{} & H_{n+1}(X) \xrightarrow{\delta} \\ \to H_n(U \cap V) \xrightarrow{((\iota_U)_*,-(\iota_V)_*)} H_n(U) \oplus H_n(V) \xrightarrow{(j_U)_* + (j_V)_*} & H_n(X) \xrightarrow{\delta} \cdots\end{aligned}\]
We use Theorem [theorem:long-exact-from-short-exact] to convert the short exact sequence of chain complexes into the long exact sequence in homology. The key step in the proof is to apply Theorem [thm:open-cover] to equate \(H_n(C_*^{\mathcal{U}}(X))\) with \(H_n(X)\).
Using the Mayer-Vietoris sequence we can compute the homology groups of the spheres \(S^m\). This lets us deduce the following likely-sounding but nontrivial theorem.
The spaces \(\mathbb R^n\) and \(\mathbb R^m\) are homeomorphic if and only if \(n = m\).
The next theorem is rather more surprising.
Let \(n\geq 0\) and let \(f \colon D^n \to D^n\) be a continuous map. Then \(f\) has a fixed point, i.e. there is a point \(x \in D^n\) with \(f(x)=x\).
piglet
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