Modification of CW-structures
There is a technique, developed by Whitehead, for replacing a CW-complex with a homotopy-equivalent CW-complex which has a simpler CW-decomposition.
Consider, for example, an arbitrary CW-complex. Its 1-skeleton can be fairly complicated, being an arbitrary graph. Now consider a maximal forest F in this graph. Since it is a collection of trees, and trees are contractible, consider the space where the equivalence relation is generated by if they are contained in a common tree in the maximal forest F. The quotient map is a homotopy equivalence. Moreover, naturally inherits a CW-structure, with cells corresponding to the cells of which are not contained in F. In particular, the 1-skeleton of is a disjoint union of wedges of circles.
Another way of stating the above is that a connected CW-complex can be replaced by a homotopy-equivalent CW-complex whose 0-skeleton consists of a single point.
Consider climbing up the connectivity ladder—assume X is a simply-connected CW-complex whose 0-skeleton consists of a point. Can we, through suitable modifications, replace X by a homotopy-equivalent CW-complex where consists of a single point? The answer is yes. The first step is to observe that and the attaching maps to construct from form a group presentation. The Tietze theorem for group presentations states that there is a sequence of moves we can perform to reduce this group presentation to the trivial presentation of the trivial group. There are two Tietze moves:
- 1) Adding/removing a generator. Adding a generator, from the perspective of the CW-decomposition consists of adding a 1-cell and a 2-cell whose attaching map consists of the new 1-cell and the remainder of the attaching map is in . If we let be the corresponding CW-complex then there is a homotopy-equivalence given by sliding the new 2-cell into X.
- 2) Adding/removing a relation. The act of adding a relation is similar, only one is replacing X by where the new 3-cell has an attaching map that consists of the new 2-cell and remainder mapping into . A similar slide gives a homotopy-equivalence .
If a CW-complex X is n-connected one can find a homotopy-equivalent CW-complex whose n-skeleton consists of a single point. The argument for is similar to the case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for (using the presentation matrices coming from cellular homology. i.e.: one can similarly realize elementary matrix operations by a sequence of addition/removal of cells or suitable homotopies of the attaching maps.
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