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MATH31052 Topology

6 The Fundamental Group

  • 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 \(X\), \(\sigma _0\sim \sigma _1\Leftrightarrow [\sigma _0] = [\sigma _1]\).

The algebra of homotopy classes of paths
  • 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 \(I^2\).  □

The algebra of homotopy classes of loops
  • 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 PoincarĂ© 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"  □ >

Dependence on the base point
  • 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.

Functorial properties of the fundamental group
  • 6.22 Theorem.   A continuous map of topological spaces \(f\colon X\to Y\) induces a homomorphism

    \[f_*\colon \pi _1(X,x_0) \to \pi _1\bigl (Y,f(x_0)\bigr )\]

    by \(f_*([\sigma ]) = [f\circ \sigma ]\) for any choice of base point \(x_0\in X\). This has the following properties.

    • (a) The identity map \(I_X\colon X\to X\) induces the identity map

      \[(I_X)_* = I_{\pi _1(X,x_0)}\colon \pi _1(X,x_0)\to \pi _1(X,x_0).\]

    • (b) Given continuous maps \(f\colon X\to Y\) and \(g\colon Y\to Z\) then

      \[ (g\circ f)_* = g_*\circ f_*\colon \pi _1(X,x_0) \to \pi _1\bigl (Z,gf(x_0)\bigr ).\]

  • 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 \(I^n\to X\).

Exercises

  • 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.