Cellular Approximation for Pairs

(This is Example 4.11 in Hatcher’s book).

Cellular Approximation for Pairs: Every map f:(X,A)\to (Y,B) of CW pairs can be deformed through maps (X,A)\to (Y,B) to a cellular map g:(X,A)\to (Y,B).

What “map of CW pairs” mean, is that f is a map from X to Y, and the image of A\subseteq X under f is contained in BCW pair (X,A) means that X is a cell complex, and A is a subcomplex.

First, we use the ordinary Cellular Approximation Theorem to deform the restriction f:A\to B to be cellular. We then use the Homotopy Extension Property to extend this to a homotopy of f on all of X. Then, use Cellular Approximation Theorem again to deform the resulting map to be cellular staying stationary on A.

We use this to prove a corollary: A CW pair (X,A) is n-connected if all the cells in X-A have dimension greater than n. In particular the pair (X,X^n) is n-connected, hence the inclusion X^n\hookrightarrow X induces isomorphisms on \pi_i for i<n and a surjection on \pi_n.

First we note that being n-connected means that the space is non-empty, path-connected, and the first n homotopy groups are trivial, i.e. \pi_i(X)\cong 0 for 1\leq i\leq n.

Proof: First, we apply cellular approximation to maps (D^i,\partial D^i)\to (X,A) with i\leq n, thus the map is homotopic to a cellular map of pairs g. Since all the cells in X-A have dimension greater than n, the n-skeleton of X must be inside A. Therefore g is homotopic to a map whose image is in A, and thus it is 0 in the relative homotopy group \pi_i(X,A). This proves that the CW pair (X,A) is n-connected. Note that 0-connected means path-connected.

Consider the long exact sequence of the pair (X,X^n):

\dots\to\pi_n(X^n,x_0)\xrightarrow{i_*}\pi_n(X,x_0)\xrightarrow{j_*}\pi_n(X,X^n,x_0)\xrightarrow{\partial}\pi_{n-1}(X^n,x_0)\to\dots\to\pi_0(X,x_0)

Since it is an exact sequence, the image of any map equals the kernel of the next. Thus, \text{Im}(i_*)=\ker j_*=\pi_n(X,x_0) (since \pi_n(X,X^n,x_0)=0). Thus i_* is surjective. Since \pi_n(X,X^n,x_0)=0, the later terms in the long exact sequence are also 0, thus, the inclusion X^n\hookrightarrow X induces isomorphisms on \pi_i for i<n, since the first n homotopy groups all vanish.

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