A Lefschetz fixed point formula in the relative elliptic theory. (English) Zbl 1089.58500
Let \((E,d)\) be an elliptic pseudodifferential complex on a smooth closed manifold \(M\); the Atiyah-Bott Lefschetz formula [M. F. Atiyah and R. Bott, Ann. Math. (2) 86, 374–407 (1967; Zbl 0161.43201)], expresses the Lefschetz number of any geometric endomorphism of \((E,d)\) in terms of topological invariants of the fixed point submanifold of the underlying smooth map \(f: M \to M\). The aim of the paper under review is to derive a similar fixed point formula for the mapping-cone of a chain mapping of two such elliptic complexes.
Let \(S\) be a second closed manifold and let \((E_M,d_M)\) and \((E_S,d_S)\) be two elliptic pseudodifferential complexes over \(M\) and \(S\), respectively. Given a cochain map \(F\) between \((E_M,d_M)\) and \((E_S,d_S)\), the authors consider the elliptic mapping-cone complex \(C^*(F)\) associated with \(F\) and prove, under some shape assumptions, that the Lefschetz number \(L(T,F)\) associated with a given mapping \(T\) of \(C^*(F)\) is given by the formula \[ L(T,F) = L(T_M;E_M,d_M) - L(T_S; E_S,d_S). \] Here \(L(T_M;E_M,d_M)\) and \(L(T_S; E_S,d_S)\) are the corresponding Lefschetz numbers on \(M\) and \(S\), respectively. For geometric endomorphisms induced by smooth maps \(f_M\) and \(f_S\) of \(M\) and \(S\), the authors then deduce a fixed point formula in the case of simple fixed points, by using the Atiyah-Bott corresponding formulae on \(M\) and \(S\), respectively.
A particularly important example corresponds to the imbedding of a submanifold \(S\) of \(M\). Then the mapping-cone complex associated with de Rham complexes computes the relative cohomology of the pair \((M,S)\). The authors obtain in this way a relative Lefschetz formula.
Finally the authors obtain a generalization to more general pseudodifferential complexes by using a method due to B. V. Fedosov [Partial differential equations VIII. Encycl. Math. Sci. 65, 155–251 (1996); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 65, 165–268 (1991; Zbl 0884.58087)].
It is worth pointing out that such an approach should give corresponding generalizations of the Atiyah-Segal formula when the fixed point submanifold has positive dimension and is transverse to a fixed closed submanifold.
Let \(S\) be a second closed manifold and let \((E_M,d_M)\) and \((E_S,d_S)\) be two elliptic pseudodifferential complexes over \(M\) and \(S\), respectively. Given a cochain map \(F\) between \((E_M,d_M)\) and \((E_S,d_S)\), the authors consider the elliptic mapping-cone complex \(C^*(F)\) associated with \(F\) and prove, under some shape assumptions, that the Lefschetz number \(L(T,F)\) associated with a given mapping \(T\) of \(C^*(F)\) is given by the formula \[ L(T,F) = L(T_M;E_M,d_M) - L(T_S; E_S,d_S). \] Here \(L(T_M;E_M,d_M)\) and \(L(T_S; E_S,d_S)\) are the corresponding Lefschetz numbers on \(M\) and \(S\), respectively. For geometric endomorphisms induced by smooth maps \(f_M\) and \(f_S\) of \(M\) and \(S\), the authors then deduce a fixed point formula in the case of simple fixed points, by using the Atiyah-Bott corresponding formulae on \(M\) and \(S\), respectively.
A particularly important example corresponds to the imbedding of a submanifold \(S\) of \(M\). Then the mapping-cone complex associated with de Rham complexes computes the relative cohomology of the pair \((M,S)\). The authors obtain in this way a relative Lefschetz formula.
Finally the authors obtain a generalization to more general pseudodifferential complexes by using a method due to B. V. Fedosov [Partial differential equations VIII. Encycl. Math. Sci. 65, 155–251 (1996); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 65, 165–268 (1991; Zbl 0884.58087)].
It is worth pointing out that such an approach should give corresponding generalizations of the Atiyah-Segal formula when the fixed point submanifold has positive dimension and is transverse to a fixed closed submanifold.
Reviewer: Moulay-Tahar Benameur (MR1855985)
MSC:
| 58J20 | Index theory and related fixed-point theorems on manifolds |
| 55M20 | Fixed points and coincidences in algebraic topology |
