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Caractérisation des opérations d’algèbres sur les modules différentiels. (Characterization of actions of algebras on differential modules). (French) Zbl 0652.55012
Let \(\Lambda\) be an associative algebra with unity and \(M_*\) a \(\Lambda\)-differential graded module. To \(M_*\) we associate a class \[ \theta (M_*)\in Ext_{\Lambda}^{2.-1}(H_*(M_*),\quad H_*(M_*)) \] which characterizes the “2-quasi isomorphism class” of \(M_*\). This construction, applied to the singular complex of a G-space X, where G is a discrete group, gives a class \(\theta\) (X) which determines the second differential of the spectral sequence of the equivariant homology of X. In the special case of the \({\mathbb{N}}\)-structure determined by X equipped with an endomorphism f such that \(H_*(f)=1\), the class \(\theta\) (f) is in \(Ext^{1.-1}(H_*(X),H_*(X))\) and defines an obstruction for f to be an nth power. An action of \({\mathbb{Z}}_ p\) on a surface X yields \(\theta\) (X)\(\in H^ 2({\mathbb{Z}}_ p,Hom(H_ 1(X),H_ 2(X)))\) which classifies the action up to quasi-isomorphism.

MSC:
55N91 Equivariant homology and cohomology in algebraic topology
55T10 Serre spectral sequences
57S20 Noncompact Lie groups of transformations
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
54H15 Transformation groups and semigroups (topological aspects)
55U10 Simplicial sets and complexes in algebraic topology
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