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A Möbius inversion formula for generalized Lefschetz numbers

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  • Mathematics

Abstract

Ferrario, D.L. Osaka J. Math. 40 (2003), 345–363 A M ¨OBIUS INVERSION FORMULA FOR GENERALIZED LEFSCHETZ NUMBERS DAVIDE L. FERRARIO (Received November 12, 2001) 1. Introduction Let be a finite group and : → an equivariant map. A common way of studying the properties of is looking at the restrictions : → to the spaces fixed by the subgroups of , as non-equivariant maps. For example, if and are -CW complexes, then : → is a -equivariant homotopy equivalence if and only if for every the map is a homotopy equivalence; a similar result re- lated to a -retraction is due to Jaworowski: a locally compact, separable metric and finite-dimensional -space is a -ENR if and only if for every the fixed point set is an ENR [11]. This paper is addressed to studying fixed points (up to com- pactly fixed -homotopy) of a -equivariant self-map : ⊂ → , where is a -ENR or a smooth -manifold. If there is a compactly fixed -homotopy , ∈ , such that 0 = and 1 is fixed point free, then for every subgroup there is a compactly fixed homotopy such that 0 = and 1 is fixed point free, and this means that every restriction can be deformed to a fixed point free map. To investigate under which conditions the converse of this statement is true, it is neces- sary to exhibit the algebraic obstructions of the existence of the equivariant deforma- tion , and then relate them to the corresponding obstructions of the non-equivariant restrictions . Under some dimensional assumptions, Nielsen theory is exactly what describes these invariants; if is a manifold of dimension different from 2 then the generalized Lefschetz number L( ) or equivalently the Nielsen number ( ) van- ish if and only if can be deformed to be fixed point free (it is the Converse of the Lefschetz Property). So the problem can be stated algebraically as: under which conditions does the knowledge of the generalized Lefschetz numbers L( ) allow to compute the obstructions to an equivariant deformation? Again, it is necessary to relate the latter obstruction to

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