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Topology

Section 5.2 Group homomorphisms

Subsection 5.2.1 Change of basepoint and PC spaces

Definition 5.2.1.

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 \([f] \in \pi_1(X,x_0)\text{.}\)

Definition 5.2.4.

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 \(\pi_1(X) = 1\text{.}\)

Subsection 5.2.2 Homomorphisms induced by continuous maps

Definition 5.2.5.

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 \(h:(X,x_0) \to (Y,y_0)\text{.}\)

Definition 5.2.6.

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 \([f] \in \pi_1(X,x_0)\text{.}\)

Remark 5.2.7.

The function \(h∗\) is also sometimes written \(\pi_1(h)\text{.}\)

Subsection 5.2.3

Subsection 5.2.4 Interactions with constructions

Remark 5.2.15.

  1. If \(A\) is a subspace of \(X\text{,}\) then \(\pi_1(A)\) might not be a subgroup of \(\pi_1(X)\text{.}\)
  2. For equivalence relation \(\sim \)on a space \(X, \pi_1(X/\sim )\) might not be a quotient (group) of \(\pi_1(X)\text{.}\)