Cellular Homology

Paris, Chennai

Cellular homology

Cellular homology provide an efficient way to compute homology groups of CW complexes, based on degree calculations.

Let be a CW complex; then:

  1. is zero for , and free abelian otherwise, with basis in one-to-one correspondence with the n-cells of . This follows from the observation that is a good pair, and is the wedge sum of -spheres, one for each -cell of .
  2. for .
  3. The inclusion induces the isomorphism if , since

for all , as in (b).

To prove (b), consider the l.e.s of the pair :

If , then the outer two groups are zero, and we get . If , .

Pairs fit into the l.e.s:

are relativizations of the boundary map , and the composition is zero. The horizontal row in the diagram is termed cellular chain complex in . Since is free with basis in one-to-one correspondence with the -cells, it can be thought of as linear combinations of -cells of . The homology groups of this cellular chain complex are called cellular homology groups.

. From the diagram, can be identified with . Further, since is injective, it maps isomorphically onto , and onto . Since is injective, . Thus, induces a isomorphism of the quotient onto .

  1. if is a CW complex with no -cells.
  2. If is a CW complex with -cells, is generated by atmost elements; is free abelian on generators, so must be generated by at most elements, hence also the quotient .
  3. If is a CW complex with no two cells in adjacent dimensions, then it is free abelian with basis in one-to-one correspondence with the -cells of . This is because the cellular boundary maps are zero.

has a CW-structure with each cell of each even dimension, where is the complex projective -space of complex lines through the origin in .

As another example, has a product CW-structure, with one -cell, two -cells, and one -cell.

To compute cellular boundary maps , note that is the same as the simplicial boundary map . If is connected, and has only one -cell, must be zero, otherwise would not be . When , can be computed in terms of degrees:

Cellular boundary formula: where is the degree of the map that is the composition of attaching map with the quotient map collapsing to a point. The cellular boundary formula can then be obtained from the following diagram: