Generalized orbifold Euler characteristics for general orbifolds and wreath products

Abstract

We introduce the -Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G , extending to orbifolds that are not global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as the Euler-Satake characteristic of the space of -sectors of the orbifold where is a finitely generated discrete group. We study the behavior of these Euler characteristics under product operations applied to the group as well as the orbifold and establish their relationships to existing Euler characteristics for orbifolds. As applications, we generalize formulas of Tamanoi, Wang and Zhou for the Euler characteristics and Hodge numbers of wreath symmetric products of global quotient orbifolds to the case of quotients by compact, connected Lie groups acting locally freely, in particular including all closed, effective orbifolds. 22A22, 55S15; 58E40, 55N91

Algebraic Structures Associated to Orbifold Wreath Products

“…For k < n, G.S k / is considered a subgroup of G.S n / corresponding to the first k factors, and G.S k / acts on M n D M k M n k with S k acting trivially on the M n k -factor. This notation follows that of [29]; see also [15] for more details.…”

“…Let D fm r .c j /g denote the partition function of ..g;s// for r D 1;:::;n and j D 1;:::;N . The conjugacy classes of G.S n / are parameterized by the partition functions ; see [15]. Also define Z , Z j , and Z j;r , the "cut-down" orders associated to an element of type D fm r .c j /g, r D 1;:::;n, j D 1;:::;N , in the following way…”

“…L c2G ; K.M hci =C G .c// denotes the natural identification of the fixed-point-set decomposition. That is, we decompose elements according to their product decomposition as products of elements of K.M hci =C G .c// for c 2 G ; (see [15,Proposition 2.11] and [29, Lemma 5 pg. 6]).…”

“…The authors study these and other sector decompositions of orbifolds in [13] and [14]. In [15], the authors consider these decompositions for wreath symmetric products of orbifolds presented by the quotient of a smooth, closed manifold by a compact, connected Lie group acting smoothly and almost freely. In particular, we extend to this case many of the results describing the structure of a wreath symmetric product required in [29].…”

“…Given the results of [15], these generalizations require only minor modifications to Wang's proofs. In Theorem 2.5, we show that the ring F G .M / is isomorphic as a .Z C ;Z 2 /-graded algebra to a supersymmetric algebra expressed in terms of the equivariant K-theory K G .M /˝C only.…”