Let \(p:I \to X\) be a path in a space \(X\) from \(x_0\) to \(x_1\text{.}\) The change of basepoint homomorphism induced by \(p\) is the function \(\beta_p: \pi_1(X,x_0) \to \pi_1(X,x1)\) defined by \(\beta_p([f]) := [p_{rev∗}f∗p]\) for all \([f] \in \pi_1(X,x_0)\text{.}\)
Theorem5.2.2.
If \(p\) is a path in \(X\) from \(x_0\) to \(x_1\text{,}\) then \(\beta_p: \pi_1(X,x_0) \to \pi_1(X,x1)\) is an isomorphism of groups.
Corollary5.2.3.
If \(X\) is a path-connected space, then \(\pi_1(X)\) is independent of basepoint, up to isomorphism.
Definition5.2.4.
A space \(X\) is \(0\)-connected if \(X\) is path-connected. A space \(X\) is \(1\)-connected, or simply connected, if \(X\) is path-connected and \(\pi_1(X) = 1\text{.}\)
Subsection5.2.2Homomorphisms induced by continuous maps
Definition5.2.5.
A pointed space is a pair \((X,x_0)\) in which \(X\) is a topological space and \(x_0 \in X\text{.}\) A function \(h:X \to Y\) with \(h(x_0) = y_0\) is a map of pointed spaces \(h:(X,x_0) \to (Y,y_0)\text{.}\)
Definition5.2.6.
Let \(h:(X,x_0) \to (Y,y_0)\) be a continuous map of pointed spaces. The induced homomomorphism associated to \(h\) is the function \(h∗: \pi_1(X,x_0) \to \pi_1(Y,y_0)\) defined by \(h∗([f]) := [h \circ f]\) for all \([f] \in \pi_1(X,x_0)\text{.}\)
Remark5.2.7.
The function \(h∗\) is also sometimes written \(\pi_1(h)\text{.}\)
Proposition5.2.8.
If \(h:(X,x_0) \to (Y,y_0)\) is a continuous map of pointed spaces, then \(h∗\) is a (well-defined) group homomorphism.
If \(h:(X,x_0) \to (Y,y_0)\) and \(k:(Y,y_0) \to (Z,z_0)\) are continuous maps of pointed spaces, then \((k \circ h)∗ = k∗ \circ h∗\text{.}\)
For any pointed space \((X,x_0), (1(X,x_0))∗ = 1\pi_1(X,x_0)\text{.}\)
Subsection5.2.3
Theorem5.2.9.
Let \(X\) be a topological space and let \(x_0,x_1 \in X\text{.}\) Let \(h:X \to Y\) be a continuous function, and let \(h_0:(X,x_0) \to (Y,h(x_0))\) and \(h_1:(X,x1) \to (Y,h(x1))\) be the corresponding maps of pointed spaces. If \(p:I \to X\) is a path from \(x_0\) to \(x_1\text{,}\) then \(\beta h \circ p \circ (h_0)∗ = (h_1)∗ \circ \beta p\text{.}\)
Theorem5.2.10.
If \(X\) and \(Y\) are path-connected spaces and \(X \simeq Y\text{,}\) then \(\pi_1(X) \cong \pi_1(Y)\text{.}\)
Corollary5.2.11.
The isomorphism class of the fundamental group is a homotopy type invariant for PC spaces.
Corollary5.2.12.
If \(X\) is a contractible space then \(\pi_1(X) = 1\text{.}\)
Corollary5.2.13.
If \(X\) and \(Y\) are homotopy equivalent path-connected spaces, then \(\pi_1(X)\) is abelian [respectively, finite] if and only if \(\pi_1(Y)\) is abelian [respectively, finite].
Subsection5.2.4Interactions with constructions
Theorem5.2.14.
If \(X\) and \(Y\) are PC topological spaces and \(X \times Y\) has the product topology, then \(\pi_1(X \times Y) \cong \pi_1(X) \times \pi_1(Y)\text{.}\)
Remark5.2.15.
If \(A\) is a subspace of \(X\text{,}\) then \(\pi_1(A)\) might not be a subgroup of \(\pi_1(X)\text{.}\)
For equivalence relation \(\sim \)on a space \(X, \pi_1(X/\sim )\) might not be a quotient (group) of \(\pi_1(X)\text{.}\)
Theorem5.2.16.
If \(r:X \to A\) is a retraction and \(i:A \to X\) is the inclusion, then \(r∗\) is onto and \(i∗\) is one-to-one.
Theorem5.2.17.
Let \(A\) be a PC subspace of a PC topological space \(X\text{,}\) and let \(a_0 \in A\text{.}\) Let \(i: A \to X\) be the inclusion map, and let \(i∗: \pi_1(A,a_0) \to \pi_1(A,a_0)\) be the induced homomorphism. Then \(i∗\) is onto if and only if every path in \(X\) with endpoints in \(A\) is path homotopic to a path in \(A\text{.}\)
