Computation of cohomology operations of 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 the (co)homology groups of the complex [Mun84], or the “incremental algorithm” for computing Betti numbers [DE93]. Nevertheless, little is known about general methods for computing cohomology operations.

For any finite simplicial complex $K$, we give a procedure including the computation of some primary and secondary cohomology operations. This method is based on the transcription of the reduction algorithm mentioned above, in terms of a special type of algebraic homotopy equivalences, called contractions [McL75], of the (co)chain complex of $K$ to a “minimal” (co)chain complex $M(K)$. More concretely, 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 [GR99] and [Gon00], these operations at cohomology level can be computed. Finally, a method for calculating Adem secondary cohomology operations $\Phi_q:\; Ker(Sq^2H^q(K)) \to H^{q+3}(K)/Sq^2H^q(K)$ is showed.

Keywords

cohomology operations, finite simplicial complexes

2010 Mathematics Subject Classification

55S05, 55U10

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Published 1 January 2003