§ CW Complexes and HEP
If is a CW complex and is a closed subcomplex, then it has the HEP.
A closed subcomplex is a union of closed cells of such that is obtained
by adding more cells to .
If is a disk, then there is a continuous map from to
If is obtained from by attaching one -cell, then has HEP.
Given a homotopy and a new homotopy
such that , we want to complete
such that .
The only part I don't know where to define on is the new added portion.
So I need to construct on . Use the previous map to get to
. This is in the domain of or ,
and thus we are done.
§ CW Complexes have HEP
Induction on lemma. base case is empty set.
§ Connected 1D CW Complex
Theorem: any connected 1D CW complex is homotopic to wedge of circles.
- Find a contractible subcomplex of that passes through all cells.
- By HEP, . has only one zero-cell and other one cells. One cells are only attached to zero cells. Hence, is a wedge of circles.
- The idea to find a contractible subcomplex is to put a partial order on the set of all contractible cell complexes by inclusion.
- Pick a maximal element with respect to this partial order.
- Claim: maximal element must contain all zero cells. Suppose not. Then I can add the new zero cell into the maximal element (why does it remain contractible? Fishy!)