CW Approximation

A weak homotopy equivalence is a map f:X\to Y that induces isomorphisms \pi_n(X,x_0)\to\pi_n(Y,f(x_o)) for all n\geq 0 and all choices of basepoint x_0.

In other words, Whitehead’s theorem says that a weak homotopy equivalence between CW complexes is a homotopy equivalence. Just to recap, a map f:X\to Y is said to be a homotopy equivalence if there exists a map g:Y\to X such that fg\cong id_Y and gf\cong id_X. The spaces X and Y are called homotopy equivalent.

It turns out that for any space X there exists a CW complex Z and a weak homotopy equivalence f:Z\to X. This map f:Z\to X is called a CW approximation to X.

Excision for Homotopy Groups

According to Hatcher (Chapter 4.2), the main difficulty of computing homotopy groups (versus homology groups) is the failure of the excision property. However, under certain conditions, excision does hold for homotopy groups:

Theorem (4.23): Let X be a CW complex decomposed as the union of subcomplexes A and B with nonempty connected intersection C=A\cap B. If (A,C) is m-connected and (B,C) is n-connected, m,n\geq 0, then the map \pi_i(A,C)\to\pi_i(X,B) induced by inclusion is an isomorphism for i<m+n and a surjection for i=m+n.

Miscellaneous Definitions

Suspension: Let X be a space. The suspension SX is the quotient of X\times I obtained by collapsing X\times\{0\} to one point and X\times\{1\} to another point.

The definition of suspension is similar to that of the cone in the following way. The cone CX is the union of all line segments joining points of X to one external vertex. The suspension SX is the union of all line segments joining points of X to two external vertices.

The classical example is X=S^n, when SX=S^{n+1} with the two “suspension points” at the north and south poles of S^{n+1}, the points (0,\dots,0,\pm 1).

Here are some graphical sketches of the case where X is the 0-sphere and the 1 sphere respectively.

s0

s1