Cellular homology

In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules.


If   is a CW-complex with n-skeleton  , the cellular-homology modules are defined as the homology groups of the cellular chain complex


where   is taken to be the empty set.

The group


is free abelian, with generators that can be identified with the  -cells of  . Let   be an  -cell of  , and let   be the attaching map. Then consider the composition


where the first map identifies   with   via the characteristic map   of  , the object   is an  -cell of X, the third map   is the quotient map that collapses   to a point (thus wrapping   into a sphere  ), and the last map identifies   with   via the characteristic map   of  .

The boundary map


is then given by the formula


where   is the degree of   and the sum is taken over all  -cells of  , considered as generators of  .


The n-dimensional sphere Sn admits a CW structure with two cells, one 0-cell and one n-cell. Here the n-cell is attached by the constant mapping from   to 0-cell. Since the generators of the cellular homology groups   can be identified with the k-cells of Sn, we have that   for   and is otherwise trivial.

Hence for  , the resulting chain complex is


but then as all the boundary maps are either to or from trivial groups, they must all be zero, meaning that the cellular homology groups are equal to


When  , it is not very difficult to verify that the boundary map   is zero, meaning the above formula holds for all positive  .

As this example shows, computations done with cellular homology are often more efficient than those calculated by using singular homology alone.

Other propertiesEdit

One sees from the cellular-chain complex that the  -skeleton determines all lower-dimensional homology modules:


for  .

An important consequence of this cellular perspective is that if a CW-complex has no cells in consecutive dimensions, then all of its homology modules are free. For example, the complex projective space   has a cell structure with one cell in each even dimension; it follows that for  ,





The Atiyah-Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

Euler characteristicEdit

For a cellular complex  , let   be its  -th skeleton, and   be the number of  -cells, i.e., the rank of the free module  . The Euler characteristic of   is then defined by


The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of  ,


This can be justified as follows. Consider the long exact sequence of relative homology for the triple  :


Chasing exactness through the sequence gives


The same calculation applies to the triples  ,  , etc. By induction,



  • A. Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1.
  • A. Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1. A free electronic version is available on the author's homepage.