From: Boudewijn Moonen <Boudewijn.Moonen@ipb.uni-bonn.de>
Subject: Re: betti numbers
Date: Fri, 25 May 2001 14:32:15 +0200
Newsgroups: sci.math
Summary: Equivalent complexes to determine homology of a CW complex

Michael wrote:

> 
> I was wondering if there is a way to compute the betti numbers of a manifold
> without knowing the de rham cohomology of the manifold?

A supplement to what Robin wrote:

Often it is more convenient and economical to use a finite
CW-decomposition of a space X in place of a simplicial one.
Given such, define the *n-th cellular* chain group C_n(X)
as

        C_n(X) := free abelian group on the n-cells of X

and the boundary map

        del_n : C_n --> C_{n-1}(X)

as

        del_n(e) := sum_{(n-1) - cells e'} [e:e']e'

where the *incidence number* [e:e'] of e with e' is defined as follows:
Let X_n denote the union of all k-cells with k <= n. Let

        f : S^{n-1} --> X_{n-1}

be the attaching map for e. Then [e:e'] = deg(g), the mapping
degree of the map

   g :  S^{n-1} --> X_{n-1} -->  X_{n-1}/ X_{n-1}-e' = S^{n-1}

the left map being f and the second map collapsing everything in
X_{n-1} outside of e' to a point. Then the homology of the chain
complex

                    del_n              del_{n-1}
    ... --> C_n(X) ------> C_{n-1}(X) -----------> ...

computes the homology, and hence the Betti numbers, of X. This
method is quite effective in computing these for the complex and
real projective spaces.

This all is explained in Dold's Algebraic Topology book. An online
reference is Allen Hatcher's book; see

        http://math.cornell.edu/~hatcher

The main difficulty, compared to a simplicial decomposition, is to
figure out the incidence numbers. The gain is that cellular 
decompositions have usually a far lesser number of cells.

Regards,

-- 
    Boudewijn Moonen
    Institut fuer Photogrammetrie der Universitaet Bonn
    Nussallee 15
    
    D-53115 Bonn
    
    GERMANY

    e-mail: Boudewijn.Moonen@ipb.uni-bonn.de
    Tel.:   GERMANY +49-228-732910
    Fax.:   GERMANY +49-228-732712