MATH31052 Topology
6.1 Definition. Suppose that \(X\) is a topological space and \(x_0\), \(x_1\in X\). Write \(I=[0,1]\) with the usual topology. Two paths \(\sigma _0\) and \(\sigma _1\colon I\to X\) from \(x_0\) to \(x_1\) are said to be homotopic, written \(\sigma _0\sim \sigma _1\), when there exists a continuous map \(H\colon I^2\to X\) such that
\begin{eqnarray*} H(s,0) & = & \sigma _0(s),\\ H(s,1) & = & \sigma _1(s),\\ H(0,t) & = & x_0,\\ H(1,t) & = & x_1 \end{eqnarray*}
for \(s\), \(t\in I\). We say that \(H\) is a homotopy between \(\sigma _0\) and \(\sigma _1\) and indicate this by writing \(H\colon \sigma _0\sim \sigma _1\).
6.2 Remark. In the above situation, if we define \(\sigma _t\colon I\to X\), for \(t\in I\), by \(\sigma _t(s) = H(s,t)\), then \(\sigma _t\) is a path from \(x_0\) to \(x_1\) in \(X\) and \(\{\,\sigma _t\mid t\in I\,\}\) provides a `continuous family' of such maps betweem \(\sigma _0\) and \(\sigma _1\).
6.3 Example.
(a) Given \(\x _0\), \(\x _1\in D^2\), all paths in \(D^2\) from \(\x _0\) to \(\x _1\) are homotopic.
Proof. Define \(H\colon I^2\to \R ^2\) by
\[H(s,t) = (1-t)\sigma _0(s) + t\sigma _1(s).\]
This is a continuous map since \(\sigma _0\) and \(\sigma _1\) are continuous and \(H(I^2)\subset D^2\) since, for \(s\), \(t\in I\),
\[|H(s,t)| = |(1-t)\sigma _0(s)+t\sigma _1(s)| \leq (1-t)|\sigma _0(s)| + t|\sigma _1(s)| \leq (1-t) + t = 1.\]
Hence, \(H\colon I^2\to D^2\) and \(H\colon \sigma _0\sim \sigma _1\). □
(b) The same argument works for paths in any convex subset of \(\R ^n\).
(c) In \(S^1\subset \C \) with the usual topology, the paths \(\sigma _0(s) = \exp (i\pi s)\) and \(\sigma _1(s) = \exp (-i\pi s)\) are both from 1 to \(-1\) but are not homotopic.
The proof of this will be given in Section 7.
6.4 Proposition. Homotopy of paths between two points \(x_0\) and \(x_1\) in a topological space \(X\) is an equivalence relation.
Proof. We check the conditions for an equivalence relation (Definition 0.15).
The reflexive property. For any path \(\sigma \colon I\to X\) from \(x_0\) to \(x_1\) then a homotopy \(H\colon \sigma \sim \sigma \) is given by \(H(s,t)=\sigma (s)\) (the constant homotopy).
The symmetric property. Given a homotopy \(H\colon \sigma _0\sim \sigma _1\) between two paths in \(X\) from \(x_0\) to \(x_1\) then a homotopy \(\overline {H}\colon \sigma _1\sim \sigma _0\) is given by \(\overline {H}(s,t) =
H(s,1-t)\) (the reverse homotopy).
The transitive property. Given homotopies \(H\colon \sigma _0\sim \sigma _1\) and \(K\colon \sigma _1\sim \sigma _2\) where the \(\sigma _i\) are paths in \(X\) from \(x_0\) to \(x_1\) then a homotopy \(L\colon \sigma _0\sim \sigma
_2\) is given by
\[L(s,t) = \left \{\begin {array}{ll}H(s,2t) & \mbox {for $s\in I$ and $0\leq t\leq 1/2$,}\\ K(s,2t-1)&\mbox {for $s\in I$ and $1/2\leq t\leq 1$}.\end {array}\right .\]
This is well defined since, for \(t=1/2\), \(H(s,1) = \sigma _1(s) = K(s,0)\). In addition, \(L\) is continuous by the Gluing Lemma since \(I\times [0,1/2]\) and \(I\times [1/2,1]\) are closed subsets of \(I^2\). □
6.5 Definition. We write \([\sigma ]\) for the homotopy class of a path \(\sigma \) in a topological space \(X\). Thus, given two paths \(\sigma _0\) and \(\sigma _1\) from \(x_0\) to \(x_1\) in a topological space \(X\), \(\sigma _0\sim \sigma _1\Leftrightarrow [\sigma _0] = [\sigma _1]\).
6.6 Proposition. Given two homotopic paths \(\sigma _0\sim \sigma _1\) from \(x_0\) to \(x_1\) and two homotopic paths \(\tau _o\sim \tau _1\) from \(x_1\) to \(x_2\) in a topological space \(X\), then
\[\sigma _0\ast \tau _0 \sim \sigma _1\ast \tau _1.\]
Proof. Suppose that \(H\colon \sigma _0\sim \sigma _1\) and \(K\colon \tau _0\sim \tau _1\). Then we may define a homotopy \(L\colon \sigma _0\ast \tau _0\sim \sigma _1\ast \tau _1\) by
\[L(s,t) = \left \{\begin {array}{ll}H(2s,t) & \mbox {for $0\leq s\leq 1/2$ and $t\in I$,}\\ K(2s-1,t)&\mbox {for $1/2\leq s\leq 1$ and $t\in I$}.\end {array}\right .\]
This is well defined since, for \(s=1/2\), \(H(1,t) = x_1 = K(0,t)\). In addition, \(L\) is continuous by the Gluing Lemma since \([0,1/2]\times I\) and \([1/2,1]\times I\) are closed subsets of \(I^2\). □
6.7 Definition. Given a homotopy class \([\sigma ]\) of paths from \(x_0\) to \(x_1\) and \([\tau ]\) a homotopy class of paths from \(x_1\) to \(x_2\) in a topological space \(X\) then we define their product \([\sigma ][\tau ]\), a homotopy class of paths from \(x_0\) to \(x_2\), by
\[ [\sigma ][\tau ] = [\sigma \ast \tau ].\]
This is well-defined by Proposition 6.6.
6.8 Proposition. Given a path \(\sigma \) from \(x_0\) to \(x_1\) in a topological space \(X\), then
\[\e _{x_0}\ast \sigma \sim \sigma \sim \sigma \ast \e _{x_1}\]
or, equivalently,
\[ [\e _{x_0}][\sigma ] = [\sigma ] = [\sigma ][\e _{x_1}].\]
Proof. A homotopy \(H\colon \e _{x_0}\ast \sigma \sim \sigma \) is given by
\[ H(s,t) = \left \{\begin {array}{ll}x_0 & \mbox {for $0\leq s \leq (1-t)/2$,}\\\sigma \Bigl (\frac {s - (1-t)/2}{1 - (1-t)/2}\Bigr ) & \mbox {for $(1-t)/2 \leq s \leq 1$.}\end {array}\right .\]
This is well-defined since, for \(s=(1-t)/2\), \(x_0 = \sigma (0)\). It is continuous by the Gluing Lemma since \(\{(s,t)\in I^2\mid 0\leq s\leq (1-t)/2\}\) and \(\{(s,t)\in I^2\mid (1-t)/2\leq s\leq 1\}\) are closed subsets of \(I^2\).
There is a similar homotopy \(\sigma \sim \sigma \ast \e _{x_1}\) (Exercise). □
6.9 Proposition. Given a path \(\sigma \) from \(x_0\) to \(x_1\) in a topological space \(X\), then
\[\sigma \ast \overline {\sigma } \sim \e _{x_0}\quad \mbox { and }\quad \overline {\sigma }\ast \sigma \sim \e _{x_1}\]
or, equivalently,
\[ [\sigma ][\overline {\sigma }] = [\e _{x_0}]\quad \mbox { and }\quad [\overline {\sigma }][\sigma ] = [\e _{x_1}].\]
Proof. A homotopy \(H\colon \sigma \ast \overline {\sigma } \sim \e _{x_0}\) is given by
\[ H(s,t) = \left \{\begin {array}{ll}\sigma \bigl (2(1-t)s\bigr )&\mbox {for $0\leq s\leq 1/2$,}\\ \sigma \bigl (2(1-t)(1-s)\bigr )&\mbox {for $1/2\leq s\leq 1$.}\end {array}\right .\]
This is well-defined since, for \(s=1/2\), both formulae give \(\sigma (1-t)\). It is continuous by the Gluing Lemma since \([0,1/2]\times I\) and \([1/2,1]\times I\) are closed in \(I^2\).
In a similar way we may write down a homotopy \(\overline {\sigma }\ast \sigma \sim \e _{x_1}\) [Exercise].
Alternatively, we may apply the first part of the result to the path \(\overline {\sigma }\) since \(\sigma \) is the reverse of the path \(\overline {\sigma }\). □
6.10 Proposition. Given two homotopic paths \(\sigma _0\sim \sigma _1\) from \(x_0\) to \(x_1\) in a topological space \(X\), then \(\overline {\sigma }_0\sim \overline {\sigma }_1\).
Proof. Exercise. □ class="theoremendmark" >
6.11 Definition. Given a homotopy class \([\sigma ]\) of paths from \(x_0\) to \(x_1\), we define its inverse \([\sigma ]^{-1}\), a homotopy class of paths from \(x_1\) to \(x_0\), by \([\sigma ]^{-1} = [\overline {\sigma }]\). This is well-defined by Proposition 6.10.
6.12 Remark. With this notation we may write the result of Proposition 6.9 as
\[ [\sigma ][\sigma ]^{-1} = [\e _{x_0}]\quad \mbox { and } [\sigma ]^{-1}[\sigma ] = [\e _{x_1}]. \]
6.13 Proposition. Given paths \(\sigma \) from \(x_0\) to \(x_1\), \(\tau \) from \(x_1\) to \(x_2\) and \(\rho \) from \(x_2\) to \(x_3\) in a topological space \(X\) then
\[(\sigma \ast \tau )\ast \rho \sim \sigma \ast (\tau \ast \rho )\]
or, equivalently,
\[([\sigma ][\tau ])[\rho ] = [\sigma ]([\tau ][\rho ])\quad \mbox {so that we may write $[\sigma ][\tau ][\rho ]$ without ambiguity.}\]
Proof. A homotopy \(H\colon (\sigma \ast \tau )\ast \rho \sim \sigma \ast (\tau \ast \rho )\) is given by
\[ H(s,t) = \left \{\begin {array}{ll}\sigma \bigl (4s/(1+t)\bigr ) & \mbox {for $0\leq s \leq (1+t)/4$,}\\ \tau \bigl (4(s - (1+t)/4)\bigr ) & \mbox {for $(1+t)/4 \leq s \leq (2+t)/4$,}\\ \rho \Bigl (\frac {s - (2+t)/4}{1 - (2+t)/4}\Bigr ) & \mbox {for $(2+t)/4 \leq s \leq 1$.}\end {array}\right .\]
This is well-defined since, when \(s=(1+t)/4\), \(\sigma (1) = x_1 = \tau (0)\) and when \(s=(2+t)/4\), \(\tau (1) = x_2 = \rho (0)\). It is continuous by the Gluing Lemma since \(\{(s,t)\in I^2\mid 0 \leq s \leq (1+t)/4\}\), \(\{(s,t)\in I^2\mid (1+t)/4\leq s \leq (2+t)/4\}\) and \(\{(s,t)\in I^2\mid (2+t)/4 \leq s \leq 1\}\) are closed in \(I^2\). □
6.14 Definition. Let \(X\) be a topological space and \(x_0\in X\). The a loop or closed path in \(X\) based at \(x_0\) is a path \(\sigma \colon I\to X\) from \(x_0\) to \(x_0\), i.e. such that \(\sigma (0) = \sigma (1) = x_0\).
6.15 Theorem. The set of homotopy classes of loops in a topological space \(X\) based at a point \(x_0\in X\) forms a group under the product \([\sigma ][\tau ] = [\sigma \ast \tau ]\). The identity is given by \(e = [\e _{x_0}]\) and the inverse \([\sigma ]^{-1} = [\overline {\sigma }]\).
Proof. This follows from the results of Proposition 6.13 (associative product), Proposition 6.8 (\(e\) is an identity element) and Proposition 6.9 (\([\sigma ]^{-1}\) is the inverse of \([\sigma ]\)). □
6.16 Definition. This group is called the fundamental group of \(X\) with base point \(x_0\) and is denoted \(\pi _1(X,x_0)\). [Other names are the first homotopy group or the Poincaré group. Sometimes the notation \(\pi (X,x_0)\) is used.]
6.17 Remark. The fundamental group is not necessarily an abelian group.
6.18 Example. If \(X\) is a convex subset of \(\R ^n\) with the usual topology and \(x_0\in X\), then \(\pi _1(X,x_0)\cong \{e\}=I\), the trivial group.
Proof. For all loops \(\sigma \) in \(X\) based at \(x_0\), \(\sigma \sim \e _{x_0}\) by the argument of Example 6.3(b). Hence \([\sigma ]=e\). class="theoremendmark" □ >
6.19 Theorem. Let \(X\) be a topological space and \(\rho \) be a path in \(X\) from \(x_0\) to \(x_1\). Then \(\rho \) induces an isomorphism
\[ u_{\rho }\colon \pi _1(X,x_0) \to \pi _1(X,x_1)\]
by
\[u_{\rho }(\alpha ) = [\rho ]^{-1}\alpha [\rho ]\quad \mbox {for $\alpha \in \pi _1(X,x_0)$.}\]
Proof. To see that \(u_{\rho }\) is a homomorphism observe that, for \(\alpha \), \(\beta \in \pi _1(X,x_0)\),
\begin{eqnarray*} u_{\rho }(\alpha )u_{\rho }(\beta ) & = & [\rho ]^{-1}\alpha [\rho ][\rho ]^{-1}\beta [\rho ]\\ & = & [\rho ]^{-1}\alpha [\e _{x_0}]\beta [\rho ]\quad \mbox {since $[\rho ][\rho ]^{-1}=[\e _{x_0}]$ (by Remark~6.12)}\\ & = & [\rho ]^{-1}\alpha \beta [\rho ]\quad \mbox {since $\alpha [\e _{x_0}] = \alpha $ (by Proposition~6.8)}\\ & = & u_{\rho }(\alpha \beta ). \end{eqnarray*}
To see that \(u_{\rho }\) is an isomorphism observe that \(u_{\overline {\rho }}\colon \pi _1(X,x_1)\to \pi _1(X,x_0)\) provides an inverse since, for \(\alpha \in \pi _1(X,x_0)\),
\begin{eqnarray*} u_{\overline {\rho }}u_{\rho }(\alpha ) & = & [\overline {\rho }]^{-1}[\rho ]^{-1}\alpha [\rho ][\overline {\rho }]\\ & = & [\rho ][\rho ]^{-1}\alpha [\rho ][\rho ]^{-1}\quad \mbox {since $[\rho ]^{-1}=[\overline {\rho }]$ and $[\overline {\rho }]^{-1}=[\rho ]$ (by Definition~2.11)}\\ & = & [\e _{x_0}]\alpha [\e _{x_0}]\quad \mbox {since $[\rho ][\overline {\rho }]=[\e _{x_0}]$ (by Remark~6.12)}\\ & = & \alpha \quad \mbox {since $[\e _{x_0}]\alpha = \alpha = \alpha [\e _{x_0}]$ (by Proposition~6.8)} \end{eqnarray*}
so that \(u_{\overline {\rho }}u_{\rho } = I_{\pi _1(X,x_0)}\colon \pi _1(X,x_0)\to \pi _1(X,x_0)\).
And similarly, \(u_{\rho }u_{\overline {\rho }}= I_{\pi _1(X,x_1)}\colon \pi _1(X,x_1)\to \pi _1(X,x_1)\). □
\(\Box \)
6.20 Remark. It is clear from the definition that homotopic paths induce the same isomorphism. However, non-homotopic paths may induce different isomorphisms and in that case there is no natural choice of isomorphism. However, this result nevertheless means that, if \(X\) is path-connected, then \(\pi _1(X,x_0)\cong \pi _1(X,x_1)\) for any two base points \(x_0\), \(x_1\in X\). In this case we can refer to the fundamental group of the space without reference to a base point and this is is sometimes denoted \(\pi _1(X)\).
6.21 Definition. A topological space \(X\) is said to be simply-connected when it is path-connected and \(\pi _1(X)\cong I\), the trivial group.
Proof. The function \(f_*\) is well-defined since, if \([\sigma _0]=[\sigma _1]\) then \(\sigma _0\sim \sigma _1\) and so there exists a homotopy \(H\colon \sigma _0\sim \sigma _1\). Then \(f\circ
H\colon I^2\to Y\) gives a homotopy \(f\circ \sigma _0\sim f\circ \sigma _1\) and so \([f\circ \sigma _0]=[f\circ \sigma _1]\).
To see that \(f_*\) is a homomorphism suppose that \([\sigma ]\), \([\tau ]\in \pi _1(X,x_0)\). Then
\[f_*([\sigma ][\tau ]) = f_*([\sigma \ast \tau ]) = [f\circ (\sigma \ast \tau )]\]
and
\[f_*([\sigma ])f_*([\tau ]) = [f\circ \sigma ][f\circ \tau ] = [(f\circ \sigma )\ast (f\circ \tau )]\]
and by writing out the formulae we see that \(f\circ (\sigma \ast \tau ) = (f\circ \sigma )\ast (f\circ \tau )\colon I\to Y\). Hence, \(f_*([\sigma ][\tau ]) = f_*([\sigma ])f_*([\tau ])\).
(a) Property (a) is immediate from the definition since \(I_X\circ \sigma =\sigma \).
(b) Property (b) is immediate from the definition since \((g\circ f)\circ \sigma = g\circ (f\circ \sigma )\). □
6.23 Corollary. The fundamental group is a topological invariant: if \(f\colon X\to Y\) is a homeomorphism then \(f_*\colon \pi _1(X,x_0) \to \pi _1\bigl (Y,f(x_0)\bigr )\) is an isomorphism.
Proof. This follows immediately from the functorial properties. Suppose that \(f\colon X\to Y\) is a homeomorphism with inverse \(f^{-1}=g\colon Y\to X\). Then \(g_*\colon \pi _1\bigl (Y,f(x_0)\bigr )\to \pi _1(X,x_0)\) is the inverse of \(f_*\) proving that \(f_*\) is an isomorphism. To prove this observe that
\[g_*\circ f_* = (g\circ f)_*\mbox { (by 6.22(b))} = (I_X)_*\mbox { (since $g=f^{-1}$)} = I_{\pi _1(X,x_0)}\mbox { (by 6.22(a))}\]
and similarly \(f_*\circ g_* = I_{\pi _1\bigl (Y,f(x_0)\bigr )}\colon \pi _1\bigl (Y,f(x_0)\bigr )\to \pi _1\bigl (Y,f(x_0)\bigr )\). class="theoremendmark" >
6.24 Remark. Later we will proof that the fundamental group of the circle \(S^1=\{z \in \C \mid |z|=1\}\) is isomorphic to \(\Z \). The intuitive idea is to count how often a given loop winds around the circle. Here, the sign accounts for the orientation of the loop (counter clockkwise vs. clockwise).
6.25 Remark. The fundamental group is the first homotopy group since the definition can be generalized to a definition of an \(n\)th homotopy group \(\pi _n(X,x_0)\) for each natural number \(n\). Whereas the first homotopy group is defined using certain continuous maps \(I\to X\) the \(n\)th homotopy group is defined using certain continuous maps \(I^n\to X\).
1 . (a) Given a path \(\sigma \colon I \to X\) from \(x_0\) to \(x_1\) in a topological space \(X\), prove that
\[ \sigma \ast \e _{x_1} \sim \sigma .\]
[Proposition 6.8, second part]
(b) Given two homotopic paths \(\sigma _0\sim \sigma _1\) from \(x_0\) to \(x_1\) in a topological space \(X\), prove that \(\overline {\sigma }_0\sim \overline {\sigma }_1\). [Proposition 6.10]
2 . Suppose that \(X\) is a convex subset of \(\R ^n\) with the usual topology [see Problems 1, Question 7.] Prove that, all paths from \(x_0\) to \(x_1\in X\) are homotopic. Deduce that \(\pi _1(X)\cong I\), the trivial group.
3 . Suppose that \(X\) is a path-connected space and \(x_0\), \(x_1\in X\). Prove that all paths from \(x_0\) to \(x_1\) are homotopic if and only if \(X\) is simply-connected.
4 . Suppose that \(X\) is a path-connected topological space and \(x_0\), \(x_1\in X\). Prove that all paths \(\rho \) from \(x_0\) to \(x_1\) induce the same isomorphism \(u_{\rho }\colon \pi _1(X,x_0) \to \pi _1(X,x_1)\) if and only if the fundamental group \(\pi _1(X)\) is abelian.
5 . Recall from the proof of Proposition 1.17 that a continuous function \(f\colon X\to Y\) of topological spaces induces a function \(f_*\colon \pi _0(X)\to \pi _0(Y)\) by \(f_*([x])=[f(x)]\). Which of the following assertions are true in general? Give a proof or counterexample for each.
(a) If \(f\) is surjective then \(f_*\) is surjective.
(b) If \(f\) is injective then \(f_*\) is injective.
(c) If \(f\) is bijective then \(f_*\) is bijective.