How to categorify dynamical zeta functions

Alexander Fel'shtyn

Knots in Washington XXVII; 3rd Japan-USA Workshop in Knot Theory
January 9-11, 2009
George Washington University
Washington, DC, USA

Organizers
Yoshiyuki Ohyama (TWCU), Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Kouki Taniyama (Waseda and GWU), Tatsuya Tsukamoto (Osaka IT), Hao Wu (GWU), Akira Yasuhara (Tokyo GU)

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How to categorify dynamical zeta functions
by
Alexander Fel'shtyn
University of Szczecin and Boise State University

A programm of a categorification a la Khovanov of Weil type dynamical zeta functions is proposed.

(Categorification of the Weil zeta function) Let φ: Σ→ Σ be a symplectomorphism of a compact surface. Then

L_φ(z) : = exp æ
è ∞
å
n=1
L(φⁿ)
n
zⁿ ö
ø = ∞
å
n=0
L(S^d(φ)) z^d = ∞
å
n=0
χ(φ, d) z^d,

where L(φⁿ) is Lefschetz number, S^d(φ): S^d(Σ) → S^d(Σ) is induced map on d-fold symmetric power of Σ and

χ(φ, d) = χ(PFH(φ, d)) = χ(ECH(T_φ, d)) = χ(SWF(T_φ, d))

is the Euler characteristic of the periodic Floer homology of degree d or the Euler characteristic of the embedded contact homology of degree d of the mapping torus T_φ or the Euler characteristic of the corresponding Seiberg-Witten-Floer cohomology of degree d. There is a strong indication that SWF(T_φ, d) homology give also a categorification of the Nielsen periodic point theory and corresponding minimal zeta function.

There are intriguing questions about categorifications of arithmetic L functions.

Date received: October 22, 2008