## Algebraic & Geometric Topology

### Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K–theory

Hirotaka Tamanoi

#### Abstract

We introduce the notion of generalized orbifold Euler characteristic associated to an arbitrary group, and study its properties. We then calculate generating functions of higher order ($p$–primary) orbifold Euler characteristic of symmetric products of a $G$–manifold $M$. As a corollary, we obtain a formula for the number of conjugacy classes of $d$–tuples of mutually commuting elements (of order powers of $p$) in the wreath product $G≀Sn$ in terms of corresponding numbers of $G$. As a topological application, we present generating functions of Euler characteristic of equivariant Morava K–theories of symmetric products of a $G$–manifold $M$.

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 115-141.

Dates
Revised: 16 February 2001
Accepted: 16 February 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882586

Digital Object Identifier
doi:10.2140/agt.2001.1.115

Mathematical Reviews number (MathSciNet)
MR1805937

Zentralblatt MATH identifier
0965.57033

#### Citation

Tamanoi, Hirotaka. Generalized orbifold Euler characteristic of symmetric products and equivariant Morava K–theory. Algebr. Geom. Topol. 1 (2001), no. 1, 115--141. doi:10.2140/agt.2001.1.115. https://projecteuclid.org/euclid.agt/1513882586

#### References

• J. Bryan and J. Fulman, Orbifold Euler characteristics and the number of commuting $m$-tuples in the symmetric groups , Annals of Combinatorics, 2 , 1–6 (1998) \ref
• R. Dijkgraaf, Fields, strings, matrices, and symmetric products \jour\nl arxiv:hep-th/9912104 \ref
• R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde, Elliptic genera of symmetric products and second quantized strings , Comm. Math. Phys., 185 (1997), 197–209 \ref
• L. Dixon, J. Harvey, C. Vafa and E. Witten, Strings on orbifolds , Nuclear Physics, B 261 (1985), 678–686 \ref
• F. Hirzebruch and H. Höfer, On the Euler number of an orbifold , Math. Annalen, 286 , 255–260 (1990) \ref
• M. J. Hopkins, The Poincaré series of the $E_n$ Dyer-Lashof algebra , preprint \ref
• M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel, Generalized group characters and complex oriented cohomology theories , J. Amer. Math. Soc., 13 (2000), 553–594 \ref
• N. J. Kuhn, Character rings in algebraic topology , London Math. Soc. Lecture Notes, 139 (1989), 111–126 \ref
• I. G. Macdonald, Poincaré polynomials of symmetric products , Proc. Camb. Phil. Soc., 58 (1962), 123–175 \ref
• I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press (1995, second edition) \ref
• P. Shanahan, The Atiyah-Singer Index Theorem, Lecture Notes in Math., 638 (1978, Springer-Verlag, New York) \ref
• L. Solomon, Relations between combinatorics and other parts of mathematics , Proc. Symp. Pure Math., 34 , Amer. Math. Soc., 309–330 (1979) \ref
• R. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge (1999) \ref
• W. Wang, Equivariant K-theory, wreath products, and Heisenberg algebra , Duke Math. J., 103 (2000), 1–23