Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
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Section 5.1 Definition of \(\pi_1\)
Definition 5.1.1 .
Let \(X\) be a topological space and let \(x_0 \in X\text{.}\) A loop in \(X at x_0 \) is a continuous function \(f:I \to X\) with \(f(0) = f(1) = x_0\text{.}\)
Definition 5.1.2 .
Two paths \(f,g:I \to X\) in a space \(X\) are path homotopic, written \(f \simeq_p g\text{,}\) if there is a continuous function \(H: I \times I \to X\) such that \(H(s,0) = f(s)\) and \(H(s,1) = g(s)\) for all \(s \in I\text{,}\) and \(H(0,t) = f(0)\) and\(H(1,t) = f(1)\) for all \(t \in I\text{.}\) The map \(H\) is called a path homotopy.
Lemma 5.1.3 .
If \(X\) is a topological space, then \(\simeq_p\) is an equivalence relation on the set of paths in \(X\text{.}\)
Definition 5.1.4 .
Let \(f,g:I \to X\) be paths in a space \(X\text{.}\)
If \(f(1) = g(0)\text{,}\) then their product, written \(f∗g\text{,}\) is the path in \(X\) from \(f(0)\) to \(g(1)\) defined by \((f∗g)(s) := f(2s)\) for all \(s \in [0,1/2]\) and \((f∗g)(s) := g(2s-1)\) for all \(s \in [1/2,1]\text{.}\)
The constant path at \(x_0\text{,}\) written \(c_{x_0}\text{,}\) is the path (loop) in X defined by c_{x_0} := x_0 for all s \in I.
The reverse of the path f is the path frev:I \to X defined by frev(s) := f(1-s) for all s \in I.
Lemma 5.1.5 .
Let \(f,g,h,k:I \to X\) be paths in a space \(X\text{,}\) and let \(x_0 := f(0)\) and \(x1 := f(1)\text{.}\)
If \(f \simeq_p g\) and \(h \simeq_p k\text{,}\) then \(f∗h \simeq_p g∗k\text{.}\)
\(c_{x_0}∗f \simeq_p f \simeq_p f∗c_{x_1}\text{.}\)
\(f∗f_{rev} \simeq_p c_{x_0}\) and \(f_{rev∗} f \simeq p c_{x_1}\text{.}\)
Definition 5.1.6 .
Let \(X\) be a topological space and let \(x_0 \in X\text{.}\) The fundamental group of \(X\) at the basepoint \(x_0\text{,}\) written \(\pi_1(X,x_0)\text{,}\) is the set of path homotopy equivalence classes of loops in \(X\) at \(x_0\text{,}\) with group operation defined by \([f][g] := [f∗g]\) for all \([f],[g] \in \pi_1(X,x_0).\)
Theorem 5.1.7 .
\(\pi_1(X,x_0)\) is a group.
Example 5.1.8 .
Examples
Theorem 5.1.9 .
\(\pi_1(X,x_0) = 1\) for all \(x_0\) in \(X\) iff every continuous map \(S^1 \to X\) extends to a continuous map \(D^2 \to X\text{.}\)
