Hatcher 0.16

Hatcher, Algebraic Topology, Chapter 0

16. Show that S^\infty is contractible.

Proof. Dr. Hatcher himself gives the proof of this in Example 1B.3 on Page 88, a proof the ideas of which are replicated here.

The goal is to show that X=S^\infty is contractible, i.e. that the identity map 1_X is homotopic to a constant map. It will be shown that 1_X\simeq (1,0,0,\ldots) where (1,0,0,\ldots) is the map \mathbf{x}\mapsto(1,0,0,\ldots)\in S^\infty for all \mathbf{x}\in S^\infty. First, note that S^\infty\subset\mathbb{R}^\infty is the collection of all points \mathbf{x} for which \|\mathbf{x}\|=1: This fact will be used to build the desired homotopy, the construction of which will take place in two parts.

Let F_t:\mathbb{R}^\infty\to\mathbb{R}^\infty be the homotopy

F_t:(x_1,x_2,\ldots)\mapsto (1-t)(x_1,x_2,\ldots)+t(0,x_1,x_2,\ldots)

for all t\in I. This is a straight-line homotopy between the point \mathbf{x}=(x_1,\ldots) and the point (0,x_1,\ldots) from which we define the restriction f_t=F_t/|F_t|:S^\infty\to \mathbb{R}^\infty for all t\in I. Clearly, then, f_t is a homotopy between points y=(y_1,y_2,\ldots)\in S^\infty and (0,y_1,\ldots)\in\mathbb{R}^\infty. Next, let G_t:\mathbb{R}^\infty\to S^\infty be given by

G_t:(0,x_1,x_2,\ldots)\mapsto (1-t)(0,x_1,x_2,\ldots)+t(1,0,0,\ldots).

As with F_t, G_t is a straight-line homotopy, this time between the points (0,x_1,x_2,\ldots)\in\mathbb{R}^\infty and (1,0,0,\ldots)\in S^\infty. As above, defining the restriction g_t=G_t/|G_t| for all t\in I gives a chain of homotopies

\left\{1_X\right\} \mapsto \left\{\mathbf{x}\mapsto (0,x_1,x_2,\ldots)\right\} \mapsto \left\{(0,x_1,x_2,\ldots)\mapsto(1,0,0,\ldots)\right\}.

In particular, then, the composition h_t is the desired homotopy between 1_X and the constant map \mathbf{x}\mapsto(1,0,0,\ldots), where

h_t = \left\{       \begin{array}{rl}         f_{1-2t}, & 0\leq t\leq 1/2\\         g_{2t-1}, & 1/2\leq t\leq 1       \end{array}     \right..   \square

2 thoughts on “Hatcher 0.16

  1. I’m interested why you need to apply the straight line homotopy f first, instead of just using g.

    Best,
    Lee

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