Betti Numbers of Finite Abstract Simplicial Complexes

1 Betti Numbers of Finite Abstract Simplicial Complexes 

In the first section of this chapter we would like to introduce the reader to the basic invariants as quickly as possible. We therefore restrict our attention to finite abstract simplicial complexes for now. 

Before we proceed, we need the following combinatorial leimna. 

Lemma 3.1. Let S be a finite set, and let S be a family of subsets of S that contains the empty set and that is closed under the operation . Then we have 

Remark 3.2. Lemma 3.1 is an easy consequence of linear algebra over the finite field Z2. Indeed, the elements of the set S can be identified with the coordinates of which is an 5 -dimensional vector field over Z2. Under this identification, the subsets of S are vectors in that vector space, and the condition that the family E is closed under the operation translates to the fact that U is a linear subspace of Z2. Say this subspace has dimension d. Then by linear algebra over Z2, we see that L = 2.  

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