ACTA issues

Quasicomplexes and Lefschetz numbers

Jörg Eschmeier

Acta Sci. Math. (Szeged) 79:3-4(2013), 611-621

Abstract. In a recent paper of Tarkhanov and Wallenta [TW] a definition of Lefschetz numbers for morphisms $a = (a^\bullet )$ of Fredholm quasicomplexes $E^\bullet = (E^\bullet, d^\bullet )$ with trace class curvature is proposed. In the present note we show that there always exist trace class perturbations of $a$ and $E^\bullet $ to a cochain mapping $A = (A^\bullet )$ of a Fredholm complex $(E^\bullet,D^\bullet )$, and we clarify the relation between the Lefschetz number of $A$ relative to the perturbed complex $(E^\bullet,D^\bullet )$ and the Lefschetz number of $a$ relative to the original quasicomplex $(E^\bullet,d^\bullet )$. Furthermore, we prove that the Lefschetz numbers relative to $E^\bullet $ satisfy a natural commutativity property.

AMS Subject Classification (1991): 47A53; 46M20, 47A13

Keyword(s): Fredholm complexes, quasicomplexes of Banach spaces, Lefschetz numbers

Received September 22, 2012, and in revised form June 15, 2013. (Registered under 78/2012.)