# Composite of consecutive maps of cellular chain complex is zero

## Contents

## Statement

### Statement independent of the language of topological spaces

Suppose are topological spaces, each having the subspace topology from the next. Suppose is a (nonnegative, though it doesn't matter) integer. Consider the following three cases of the long exact sequence of homology of a pair (we write different parts of the long exact sequences because we need different parts for the later steps):

- The pair :

- The pair :

- The pair :

Define:

Then .

### Statement in terms of cellular homology

This is a special case of the previous statement, where are the successive skeleta of a cellular filtration of a topological space . In this case, the map defined above is the boundary map and the map defined above is the boundary map in the Cellular chain complex (?). The statement is thus that , i.e., the cellular chain complex *is* indeed a chain complex.

## Proof

The proof is a one-liner: . The intermediate composite is zero since it is a composite of successive maps in a long *exact* sequence of homology. Hence, the overall composite is the zero map.