\(f\)-polynomials, \(h\)-polynomials, and \(l^2\)-Euler characteristics. (English) Zbl 1207.57033

Boros, Dan

Publ. Mat., Barc. 54, No. 1, 73-81 (2010).

Author’s abstract: We introduce a many-variable version of the \(f\)-polynomial and \(h\)-polynomial associated to a finite simplicial complex. In this context the \(h\)-polynomial is actually a rational function. We establish connections with the \(l^2\)-Euler characteristic of right-angled buildings. When \(L\) is a triangulation of a sphere we obtain a new formula for the \(l^2\)-Euler characteristic.

Reviewer: Elias Gabriel Minian (Buenos Aires)

MSC:

57Q05	General topology of complexes
57M15	Relations of low-dimensional topology with graph theory
20F55	Reflection and Coxeter groups (group-theoretic aspects)
52B05	Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)

Keywords:

\(h\)-polynomial; \(l^2\)-Euler Characteristic

References:

[1]	A. Brøndsted, “An introduction to convex polytopes” , Graduate Texts in Mathematics 90 , Springer-Verlag, New York-Berlin, 1983. · Zbl 0509.52001
[2]	R. Charney and M. Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39(3) (1991), 373\Ndash378. · Zbl 0729.20015
[3]	M. W. Davis, “The geometry and topology of Coxeter groups” , London Mathematical Society Monographs Series 32 , Princeton University Press, Princeton, NJ, 2008. · Zbl 1142.20020
[4]	M. W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62(2) (1991), 417\Ndash451. · Zbl 0733.52006
[5]	M. W. Davis and B. Okun, Vanishing theorems and conjectures for the \(\ell^ 2\)-homology of right-angled Coxeter groups, Geom. Topol. 5 (2001), 7\Ndash74 (electronic). · Zbl 1118.58300
[6]	J. Dymara, Thin buildings, Geom. Topol. 10 (2006), 667\Ndash694 (electronic). · Zbl 1166.20301