Revision as of 21:24, 21 February 2017

Problem statement

Let $n \geq 1$. We show that the map $H_k(B^n, S^{n-1}) \to H_k(B^n, B^n \setminus \{ 0 \})$ by the inclusion $i: S^{n-1} \to B^n \setminus \{0\}$ is an isomorphism.

Tools

First we show that the map $H_k(S^{n-1}) \to H_k(B^n \setminus \{0\})$ induced by the inclusion $i: S^{n-1} \to B^n \setminus \{0\}$ is an isomorphism, for $k \geq 0$. Note that $S^{n-1}$ is a retract of $B^n \setminus \{0\}$.

• Define the map $r: B^n \setminus \{ 0 \} \to S^{n - 1}$ by $r: x \mapsto \frac{x}{\vert\vert x \vert\vert}$, where $\vert\vert x \vert\vert$ denotes the norm of $x$.
• Then $r \circ i \eq \mathrm{id}_{S^{n-1} }$.
• So $i_*: S^{n - 1} \to B^n \setminus \{ 0 \}$ is a monomorphism.
• Also, $i \circ r$ is homotopy equivalent to the identity map on $B^n \setminus \{ 0 \}$. [left as an exercise to the reader]

Final proof

Alec's formatting