Editing CW complex (section)

== Modification of CW structures ==

There is a technique, developed by Whitehead, for replacing a CW complex with a homotopy-equivalent CW complex that has a ''simpler'' CW decomposition.

Consider, for example, an arbitrary CW complex.  Its 1-skeleton can be fairly complicated, being an arbitrary [[Graph (discrete mathematics)|graph]]. Now consider a maximal [[Tree (graph theory)|forest]] ''F'' in this graph.  Since it is a collection of trees, and trees are contractible, consider the space <math>X/{\sim}</math> where the equivalence relation is generated by <math>x \sim y</math> if they are contained in a common tree in the maximal forest ''F''.  The quotient map <math>X \to X/{\sim}</math> is a homotopy equivalence.  Moreover, <math>X/{\sim}</math> naturally inherits a CW structure, with cells corresponding to the cells of <math>X</math> that are not contained in ''F''. In particular, the 1-skeleton of <math>X/{\sim}</math> 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 <math>X^1</math> consists of a single point?  The answer is yes.  The first step is to observe that <math>X^1</math> and the attaching maps to construct <math>X^2</math> from <math>X^1</math> form a [[Presentation of a group|group presentation]]. The [[Tietze transformations|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 <math>X^1</math>.  If we let <math>\tilde X</math> be the corresponding CW complex <math>\tilde X = X \cup e^1 \cup e^2</math> then there is a homotopy equivalence <math>\tilde X \to X</math> 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 <math>\tilde X = X \cup e^2 \cup e^3</math> where the new ''3''-cell has an attaching map that consists of the new 2-cell and remainder mapping into <math>X^2</math>.  A similar slide gives a homotopy-equivalence <math>\tilde X \to X</math>.

If a CW complex ''X'' is [[N-connected space|''n''-connected]] one can find a homotopy-equivalent CW complex <math>\tilde X</math> whose ''n''-skeleton <math>X^n</math> consists of a single point.  The argument for <math>n \geq 2</math> is similar to the <math>n=1</math> case, only one replaces Tietze moves for the [[fundamental group]] presentation by [[elementary matrix]] operations for the presentation matrices for <math>H_n(X;\mathbb Z)</math> (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.