Math and stuff

Problem Statement For a path-connected space $X$, show that $\pi_1(X)$ is abelian iff all basepoint-change homomorphisms $\beta_h$ depend only on the endpoints of the path $h$. Solution First, suppose that $\pi_1(X)$ is abelian. Let $x_0, x_1 \in X$ be given. Let $h_1, h_2$ be paths from $x_0$ to $x_1$. We...

Proposition If $\phi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphism \(\phi_{*}: \pi_1(X, x_0) \rightarrow \pi_1(Y, \phi(x_0))\) is an isomorphism for all $x_0 \in x$. Solution Since $\phi: X \rightarrow Y$ is a homotopy equivalence, the following functions must exist: $\psi: Y \rightarrow X$, and $\psi$ is...

Proposition If $X_0$ is the path-component of a space $X$ containing the basepoint $x_0$, show that the inclusion $i: X_0 \rightarrow X$ induces an isomorphism $\pi_1(X_0, x_0) \rightarrow \pi_1(X, x_0)$. Solution Since the inclusion $i$ is a continuous function, \(i_*\) is a homomorphism. We will show that it is bijective....

Proposition The map $\beta_h: \pi_1(X, x_1) \rightarrow \pi_1(X, x_0)$ is an isomorphism. Solution Let $h$ be a path from $x_0$ to $x_1$. Define $\beta_h: \pi_1(X, x_1) \rightarrow \pi_1(X, x_0)$ such that $\beta_h([f]) = [h \cdot f \cdot \overline{h}]$. $\beta_h$ is well-defined because $h \cdot f \cdot \overline{h}$ is a loop...

Proposition If a space $X$ retracts onto a subspace $A$, then the homomorphism \(i_*: \pi_1(A, x_0) \rightarrow \pi_1(X, x_0)\) induced by the inclusion $i: A \rightarrow X$ is injecive. If $A$ is a deformation retract of $X$, then \(i_*\) is an isomorphism. Solution Let $i: A \rightarrow X$ be the...