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Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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Role of determinant of the matrix corresponding to $i$-th Homology group.

Now suppose for any finite-dimensional simplex $K$ I have the $i$-th Homology group with rational coefficients, $H_i(K,\mathbb{Q})$ and $f:K\rightarrow K$. … Since $f_*$ is a map from a finite-dimensional vetor space to itself, we can talk about its matrix and the alternating sum of the trace of the matrices of $i$-th Homology groups give the Lefschetz's Number …
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