Definition (homotopy)
maps. A homotopy from to is a map s.t. and . If it exists, is homotopic to , or .
. The homotopy is relative to if additionally for all .
Proposition
The relation “homotopic relative to set ” on the set of all maps from to is an equivalence relation (any ).
Proof
Standard, use Gluing lemma for transitivity
Definition (homotopy-equivalence)
A map is a homotopy-equivalence if there is a map s.t. and .
is homotopy equivalent to if such exists.
Example
,
Take with .
Take to be the identity (ie )
One direction is easy.
For the other one, set .
Hence .
Contractible Space
Lemma
Suppose maps with and maps with
Then
Proof
Note
Also
Done.
Proposition
The relation of homotopy equivalence between spaces is an equivalence relation.
Retraction map
Deformation retraction map