Computation of Cohomology Operations on Finite Simplicial Complexes

Abstract

We propose a method for calculating cohomology operations for finite simplicial complexes. Of course, there exist well--known methods for computing (co)homology groups, for example, the reduction algorithm consisting in reducing the matrices corresponding to the differential in each dimension to the Smith normal form, from which one can read off (co)homology groups of the complex, or the incremental algorithm (due to Edelsbrunner et al.) for computing Betti numbers. However, there is a gap in the literature concerning general methods for computing cohomology operations. For a given finite simplicial complex K, we sketch a procedure including the computation of some primary and secondary cohomology operations and the $A_{\infty}$--algebra structure on the cohomology of K. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called a contraction, of the (co)chain complex of K to a "minimal" (co)chain complex M(K). For instance, whenever the ground ring is a field or the (co)homology of K is free, then M(K) is isomorphic to the (co)homology of K. Combining this contraction with the combinatorial formulae for Steenrod reduced $p$th powers at cochain level developed in \cite{GR99} and \cite{Gon00}, these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations $\Phi_q: Ker(Sq^2H^q(K))\ra H^{q+3}(K)/Sq^2H^q(K)$ is showed.