Homotopy

This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.

Topological Invariants

In general, when mathematical objects are given, we are interested in classifying them (up to isomorphism). For instance, sets are completely classified by their cardinality, and \(\mathbb{k}\)-vector spaces are completely classified by their dimension. However, in most cases such classification is not easy, and topological spaces are no exception.

The homologies of topological spaces defined in our previous posts are topological invariants by functoriality: if two topological spaces \(X\) and \(Y\) are homeomorphic, then they are also homologous. Yet the converse does not hold in general. Finding a topological invariant that completely determines a topological space would thus be desirable, but the following remarkable result stands in the way.

Theorem 1 (Markov 1958) There exists no finite algorithm that decides whether two topological manifolds of dimension at least 4 are homeomorphic.

Put naïvely, even if a topological invariant capable of distinguishing arbitrary topological spaces up to homeomorphism were to exist, there would generally be no effective procedure to compute it. From this perspective, topological invariants are useful not for proving that two spaces are homeomorphic, but only for showing that two spaces are not homeomorphic.

Homotopy Equivalence

The notion of homotopy equivalence introduced in this post is likewise useful for showing that two topological spaces are not homeomorphic, and it is finer than the equivalence relation defined by homology. That is, the implication chain

\[X,Y\text{ homeomorphic}\implies X,Y \text{ homotopically equivalent}\implies X,Y\text{ homologous}\tag{$\ast$}\]

holds, but neither converse does. Moreover, homotopy equivalence is somewhat more geometrically intuitive than homology.

Definition 2 Let \(f_0,f_1:X \rightarrow Y\) be continuous maps between two topological spaces \(X,Y\). We say that \(f_0\) and \(f_1\) are homotopic if there exists a continuous map \(F:X\times [0,1]\rightarrow Y\) such that the two equations

\[F(x,0)=f_0(x),\qquad F(x,1)=f_1(x)\tag{1}\]

hold for all \(x\in X\). In this case, \(F\) is called a homotopy between \(f_0\) and \(f_1\), and when a homotopy exists we write \(f_0\simeq f_1\).

Intuitively, this means that \(f_0\) can be continuously deformed into \(f_1\). In the definition above, specifying a continuous map \(F\) is equivalent to specifying a family \((F(-,t))_{t\in[0,1]}\) of continuous maps. From this viewpoint, we sometimes denote a homotopy \(F\) between \(f_0\) and \(f_1\) by \((f_t)_{t\in[0,1]}\).

Proposition 3 The relation \(\simeq\) is an equivalence relation on \(C(X,Y)\).

Proof

1. First, \(\simeq\) is reflexive: for any \(f\in C(X,Y)\), defining \(F(x,t)=f(x)\) yields a homotopy between \(f\) and itself.

2. Second, \(\simeq\) is symmetric. Assume \(f_0\simeq f_1\). Then there exists a homotopy \(F\) satisfying (1). Defining \(\tilde{F}(x,t)=F(x,1-t)\), we obtain a continuous map satisfying the two equations

   \[\tilde{F}(x,0)=f_1(x),\qquad\tilde{F}(x,1)=f_0(x)\]

   and therefore \(f_1\simeq f_0\).

3. Finally, \(\simeq\) is transitive. Let \(f_0,f_1,f_2\in C(X,Y)\) satisfy \(f_0\simeq f_1\) and \(f_1\simeq f_2\). Then there exist homotopies \(F_0(x,t)\) and \(F_1(x,t)\) with \(F_0(x,0) = f_0(x)\), \(F_0(x,1) = f_1(x)\), \(F_1(x,0) = f_1(x)\), and \(F_1(x,1) = f_2(x)\). Defining \(F(x,t)\) by

   \[F(x,t) = \begin{cases} F_0(x,2t) & \text{if } 0 \leq t \leq \frac{1}{2} \\ F_1(x,2t-1) & \text{if } \frac{1}{2} \leq t \leq 1 \end{cases}\]

   yields a homotopy between \(f_0\) and \(f_2\).

The equivalence relation of being homotopic is, as above, defined for maps. By extending it, we can define what it means for two topological spaces to be homotopically equivalent.

Definition 4 Two topological spaces \(X,Y\) are homotopically equivalent if there exist continuous maps \(f:X\rightarrow Y\) and \(g:Y\rightarrow X\) such that \(f\circ g\simeq \id_Y\) and \(g\circ f\simeq\id_X\).

In this case, the two maps \(f\) and \(g\) satisfying the above condition are each called a homotopy equivalence.

Example 5 For any natural number \(n\), Euclidean space \(\mathbb{R}^n\) is homotopically equivalent to the one-point space \(\{\ast\}\). The homotopy equivalences are given by

\[f:\mathbb{R}^n \rightarrow \{\ast\};\quad x\mapsto \ast,\qquad g:\{\ast\}\rightarrow \mathbb{R}^n;\quad \ast\mapsto 0\]

Then \(f\circ g=\id_{\{\ast\}}\) is obvious, and for \(g\circ f\simeq \id_{\mathbb{R}^n}\) we define the continuous map \(t\cdot\id_{\mathbb{R}^n}\) in the variable \(t\in[0,1]\) by

\[t\cdot\id_{\mathbb{R}^n}:\mathbb{R}^n\rightarrow \mathbb{R}^n;\qquad \mathbf{x}\mapsto t\mathbf{x}\]

For completeness we must establish the implication (\(\ast\)). This is obtained from the following more general proposition.

Proposition 6 For continuous maps \(f_0,f_1:X\rightarrow Y\), if \(f_0\) and \(f_1\) are homotopic then \(C_\bullet(f_0)\) and \(C_\bullet(f_1)\) are chain homotopic. (Homological Algebra, §Homology, ¶Definition 5)

Proof

By definition, we must construct \(h_n:C_n(X) \rightarrow C_{n+1}(Y)\) satisfying

\[C_n(f_1)-C_n(f_0)=\partial_{n+1}^Y h_n+h_{n-1}\partial_n^X\tag{1}\]

The information at our disposal is the continuous map

\[F:X\times I \rightarrow Y\]

and by definition elements of \(C_n\) are continuous maps from \(\Delta^n\) to \(X\), so the composition

\[F\circ(\sigma\times\id_I):\Delta^n\times I \rightarrow Y\]

is well-defined. We must first use this to produce an element of \(C_{n+1}(Y)\). Since the domain \(\Delta^n\times I\) of this continuous map is not an \((n+1)\)-simplex, the correspondence itself does not lie in \(C_{n+1}(Y)\). Instead, we decompose it into a sum of \((n+1)\)-simplices and define the chain homotopy thereby.

Let the vertices on the bottom face (\(t=0\)) of the domain \(\Delta^n\times I\) be \(v_0,\ldots, v_n\) and those on the top face (\(t=1\)) be \(w_0,\ldots,w_n\). The case \(n=2\) is illustrated below.

Then we can decompose this into \((n+1)\) copies of the \((n+1)\)-simplex

\[[v_0,\ldots, v_n,w_n],\quad [v_0,\ldots, v_{n-1}, w_{n-1}, w_n],\quad\ldots,\quad[v_0,w_0,\ldots, w_n]\]

and for \(n=2\) the picture is as follows.

Using this decomposition we define

\[h_n(\sigma)=\sum_i (-1)^iF\circ(\sigma\times\id_I)\vert_{[v_0,\ldots, v_i, w_i,\ldots, w_n]}\]

and verifying that this satisfies (1) is a straightforward computation.

Hence homotopic continuous maps induce the same map on homology. (Homological Algebra, §Homology, ¶Proposition 6) In particular, if two spaces \(X\) and \(Y\) are homotopically equivalent and \(f:X \rightarrow Y\) and \(g:Y\rightarrow X\) are given as in Definition 4, then the homologies \(H_\bullet(X)\) and \(H_\bullet(Y)\) coincide.

On the other hand, from the computation in §Homology, ¶Example 8 we saw that for any space \(Y\), the singular \(k\)-complex (\(k>0\)) corresponding to a point \(y\in Y\) is always zero in \(H_k(Y)\). Therefore, if a continuous map \(f:X \rightarrow Y\) is homotopic to a constant map, then \(H_k(f)\) is the zero map for every \(k>0\). For this reason, a continuous map homotopic to a constant map is called null-homotopic. In particular, if the identity map \(\id_X:X \rightarrow X\) is null-homotopic, we say that \(X\) is contractible. Then by §Homology, ¶Proposition 11 and Proposition 6 above, the \(k\)-th homology of a contractible space is zero for every \(k>0\).

In the remainder of this post we examine homotopy equivalence and the fundamental group.

Deformation Retract

In many cases, two homotopically equivalent spaces arise from a deformation called a deformation retract, and moreover they bear a more direct geometric relationship than the somewhat abruptly posed Definition 2 might suggest. To define this we first need the notion of a retraction. (Set Theory, §Retraction and Section, ¶Definition 2)

Definition 7 Let \(X\) be a topological space and \(A\) its subspace. For the canonical inclusion \(\iota:A\rightarrow X\), if there exists a continuous map \(r:X\rightarrow A\) satisfying \(r\circ\iota=\id_A\), then \(r\) is called a (continuous) retraction onto the subspace \(A\), and in this case \(A\) is called a retract of \(X\).

From a set-theoretic standpoint, a map \(r\) satisfying the above condition always exists; the essential point is that \(r\) be continuous.

Example 8 For example, for the closed disk \(D^2\) in the plane and its boundary \(S^1\), no retraction from \(D^2\) onto \(S^1\) exists. If a retraction \(r:D^2\rightarrow S^1\) did exist, then by the functoriality of \(H_n\) we would have

\[H_n(r)\circ H_n(\iota)=H_n(r\circ\iota)=H_n(\id_{S^1})=\id_{H_n(S^1)}\]

for every \(n\). In particular, \(H_n(\iota):H_n(S^1)\rightarrow H_n(D^2)\) would have to be injective. Yet in §Homology, ¶Example 8 we showed that \(H_1(D^2)\cong 0\), and following the computation for \(D^2\setminus \left\{(0,0)\right\}\) we know that \(H_1(S^1)\neq 0\), so an injective homomorphism \(H_1(\iota)\) cannot exist.

Comparing this example with the homology computation for \(D^2\setminus \left\{(0,0)\right\}\) in §Homology, ¶Example 8, we see that the nontrivial homology appearing in \(D^2\setminus \left\{(0,0)\right\}\) already appears identically in its subset \(S^1\). From Proposition 6 we know that homotopic continuous maps induce homotopic chain maps and hence the same map on homology, so the direction in which to generalize is clear.

Definition 9 Let \(X\) be a topological space and \(A\) its subspace, and let \(r:X\rightarrow A\) be a retraction. If there exists a homotopy \(F\) from \(\id_X\) to \(r\), then it is called a deformation retraction onto \(A\), and \(A\) is called a deformation retract of \(X\).

Then for the retraction \(r\) in Example 8 above, defining

\[t\frac{\mathbf{x}}{\lvert\mathbf{x}\rvert}+(1-t)\mathbf{x}\]

yields a homotopy from \(\id_X\) to \(r\). That is, \(S^1\) is a deformation retract of \(D^2\setminus \left\{(0,0)\right\}\).

Fundamental Group

In §Homology, ¶Example 8 we observed that if we regard the \(1\)-simplex \(\Delta^1\) as \(I=[0,1]\), then by definition the generators of \(C_1(X)\) can be thought of as paths in \(X\), and in obtaining the homology \(H_1(X)\) we consider closed paths. This is essentially the same as looking at maps from \(S^1\) to \(X\). Let us examine this more carefully.

First, for two homotopic continuous maps \(f,g:X \rightarrow Y\) and a homotopy \(F\) between them, if for a subset \(A\subseteq X\) the equation

\[F(x,t)=f(x)\qquad\text{for all $t\in[0,1]$}\]

holds, then \(F\) is called a homotopy relative to \(A\). If in Definition 9 the homotopy \(F\) is a homotopy relative to \(A\), then we say that \(A\) is a strong deformation retract of \(X\).

Now for any two paths \(\alpha_0,\alpha_1:I\rightarrow X\), a path homotopy between them means a homotopy relative to \(\{0,1\}\). That is, two paths \(\alpha_0,\alpha_1\) share their endpoints (so \(\alpha_0(0)=\alpha_1(0)\) and \(\alpha_0(1)=\alpha_1(1)\)) and the homotopy \((\alpha_t)_{0\leq t\leq 1}\) preserves the endpoints:

\[\alpha_0(0)=\alpha_t(0)=\alpha_1(0),\qquad \alpha_0(1)=\alpha_t(1)=\alpha_1(1)\qquad\text{for all $0\leq t \leq 1$}\]

Definition 10 If a path homotopy exists between two paths \(\alpha_0,\alpha_1:I\rightarrow X\), we say that they are path homotopic and write \(\alpha_0\sim \alpha_1\).

Then, adapting the proof of Proposition 3 slightly, we see that path homotopy gives an equivalence relation on the set of paths with endpoints \(p\) and \(q\). Moreover, it is clear that this equivalence relation preserves reparametrization: for any path \(\alpha:I \rightarrow X\) and any reparametrization \(\varphi:I\rightarrow I\) (that is, a homeomorphism fixing \(0\) and \(1\)), defining

\[\alpha_t(s)=\alpha(t\varphi(s)+(1-t)s)\]

yields a path homotopy between \(\alpha_0=\alpha\) and \(\alpha_1=\alpha\circ\varphi\). Using this, we define the product of two paths by

\[(\alpha\ast \beta)(s)=\begin{cases}\alpha(2s)&0\leq s \leq 1/2\\ \beta(2s-1)&1/2\leq s \leq 1\end{cases}\tag{$\ast\ast$}\]

For this to be a continuous path, of course \(\ast\) requires \(\alpha(1)=\beta(0)\). The following properties then hold.

1. If \(\alpha_0\sim \alpha_1\) and \(\beta_0\sim \beta_1\) and \(\alpha_0\ast \beta_0\) is well-defined, then \(\alpha_1\ast \beta_1\) is also well-defined and \(\alpha_0\ast \beta_0\sim \alpha_1\ast \beta_1\). This is clear by considering the homotopy \(\alpha_t\ast \beta_t\).
2. Hence, imposing the equivalence relation of path homotopy on \(C(I,X)\), the formula \([\alpha]\ast[\beta]=[\alpha\ast \beta]\) gives a well-defined operation on \([\alpha]\) and \([\beta]\) for each pair \(\alpha,\beta\) satisfying \(\alpha(1)=\beta(0)\).
3. Then for the constant paths \(c_{\alpha(0)}\) and \(c_{\alpha(1)}\) staying at \(\alpha(0)\) and \(\alpha(1)\) respectively, we have \([c_{\alpha(0)}]\ast [\alpha]=[\alpha]=[\alpha]\ast[c_{\alpha(1)}]\). Roughly speaking, path homotopy preserves reparametrization, so in (\(\ast\ast\)) we may replace \(1/2\) by any number between \(0\) and \(1\), and letting this number approach \(0\) or \(1\) gives the desired homotopy. By essentially the same argument, \(\ast\) is associative.
4. Inverses also exist: for any path \(\alpha\), defining \(\bar{\alpha}(t)=\alpha(1-t)\) gives \([\alpha]\ast[\bar{\alpha}]=[c_{\alpha(0)}]\) and \([\bar{\alpha}]\ast[\alpha]=[c_{\alpha(1)}]\).

Summarizing these results, we obtain the following.

Definition 11 By the above results, \(C(I,X)/{\sim}\) forms a groupoid, called the fundamental groupoid of \(X\) and denoted \(\Pi_1(X)\). (Category Theory, §Categories, ¶Definition 11)

That is, for any space \(X\), the fundamental groupoid \(\Pi_1(X)\) is the category whose objects are the points of \(X\) and whose morphisms between two points \(x\) and \(y\) are the homotopy classes of paths from \(x\) to \(y\). As a full subcategory of \(\Cat\), a morphism in \(\Grpd\) is simply a functor. Explicitly, given any continuous map \(f:X \rightarrow Y\), the map \(\Pi_1(f):\Pi_1(X)\rightarrow\Pi_1(Y)\) is defined on objects by \(x\mapsto f(x)\) and on morphisms by

\[\Pi_1(f)([\alpha])=[f\circ\alpha]\]

That this is well-defined is clear: for two paths \(\alpha_0\sim\alpha_1\), if \(\alpha_t\) is a path homotopy between them then \(f\circ \alpha_t\) is a path homotopy between \(f\circ\alpha_0\) and \(f\circ\alpha_1\). Moreover, if two continuous maps \(f_0,f_1:X \rightarrow Y\) are homotopic, then there exists a natural isomorphism between the two functors \(\Pi_1(f_0)\) and \(\Pi_1(f_1)\) they induce. (Category Theory, §Natural Transformations, ¶Definition 1) That is, for any path \(\alpha:I \rightarrow X\) with starting point \(x_0\) and endpoint \(x_1\), the following diagram

commutes (up to path homotopy). Here \(f_t(x_0)\) and \(f_t(x_1)\) are the paths arising from the homotopy \((f_t)_{0\leq t\leq 1}\), and commutativity follows from the path homotopy

\[F(s,t)=f_t(\alpha(s))\]

In particular, if we consider only paths with a fixed point \(x\in X\) as both starting point and endpoint (that is, loops based at \(x\)), this is the same as considering the endomorphism monoid at \(x\) in the category \(\Pi_1(X)\), and since \(\Pi_1(X)\) is a groupoid this is in fact an automorphism group. We denote it by

\[\pi_1(X,x)=\Aut_{\Pi_1(X)}(x)\]

and if \(X\) is path-connected, this group does not depend on the choice of \(x\). It is called the fundamental group of \(X\), and it is a skeleton of \(\Pi_1(X)\). (Category Theory, §Natural Transformations, ¶Definition 4) Therefore, \(\pi_1(X,x)\) is equivalent to \(\Pi_1(X)\) as a category. As we saw above, homotopic continuous maps induce natural isomorphisms between fundamental groupoids, so the fundamental groupoid and the fundamental group are homotopy invariants.

Example 12 For instance, the fundamental groupoid \(\Pi_1(\mathbb{R^n})\) of the space \(\mathbb{R}^n\) is the category described by the following data.

• The objects of \(\Pi_1(\mathbb{R}^n)\) are precisely the points of \(\mathbb{R}^n\).

• For any \(\mathrm{x}_1,\mathrm{x}_2\in \mathbb{R}^n\), there exists a unique morphism from \(\mathrm{x}_1\) to \(\mathrm{x}_2\). That is, any path \(\alpha_1:I \rightarrow \mathbb{R}^n\) starting at \(\mathrm{x}_1\) and ending at \(\mathrm{x}_2\) is always path homotopic to the path

  \[\alpha_0:t\mapsto (1-t)\mathrm{x}_1+t\mathrm{x}_2\]

  This is easily verified by setting \(\alpha_t=(1-t)\alpha_1+t\alpha_0\).

Therefore, for any \(\mathrm{x}\), the group \(\pi_1(\mathbb{R},\mathrm{x})\) is trivial.

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References

[Hat] A. Hatcher, Algebraic Topology. Cambridge University Press, 2022.
[Mun] James Munkres, Topology. Prentice Hall, 2000.

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