A weak homotopy equivalence is a map that induces isomorphisms for all and all choices of basepoint .

In other words, Whitehead’s theorem says that a weak homotopy equivalence between CW complexes is a homotopy equivalence. Just to recap, a map is said to be a homotopy equivalence if there exists a map such that and . The spaces and are called homotopy equivalent.

It turns out that for any space there exists a CW complex and a weak homotopy equivalence . This map is called a **CW approximation** to .

## 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 be a CW complex decomposed as the union of subcomplexes and with nonempty connected intersection . If is m-connected and is n-connected, , then the map induced by inclusion is an isomorphism for and a surjection for .

## Miscellaneous Definitions

**Suspension: **Let be a space. The suspension is the quotient of obtained by collapsing to one point and to another point.

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

The classical example is , when with the two “suspension points” at the north and south poles of , the points .

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