Reidemeister torsion

For a based finite acyclic chain complex, Reidemeister torsion is an alternating product of determinants comparing each given chain basis with bases formed from boundaries and lifts of boundaries. With suitable local coefficients this gives an invariant of a space finer than its homology. Analytic torsion constructs a corresponding invariant from spectra of Laplacian operators, while Turaev torsion refines combinatorial sign and normalization choices.