Möbius transform, moment-angle complexes and Halperin-Carlsson conjecture
Xiangyu Cao
and Zhi Lü
DOI: 10.1007/s10801-011-0296-2
Abstract
The motivation for this paper comes from the Halperin-Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the Möbius transform of an abstract simplicial complex K on [ m]={1,\cdots , m} in terms of the Betti numbers of the Stanley-Reisner face ring k( K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z K (\mathbb D, \mathbb S) \mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z K \mathcal{Z}_{K} and the real moment-angle complex \mathbb R Z K \mathbb{R}\mathcal {Z}_{K} as special examples. We show that H *( Z K (\mathbb D, \mathbb S); k) H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor\thinspace k[ v]( k( K), k). Finally we show that the Halperin-Carlsson conjecture holds for Z K \mathcal{Z}_{K} (resp. \mathbb R Z K \mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T m -action on Z K \mathcal{Z}_{K} (resp. (\Bbb Z 2) m -action on \mathbb R Z K \mathbb{ R}\mathcal{Z}_{K}).
Pages: 121–140
Keywords: keywords Möbius transform; moment-angle complex; Halperin-Carlsson conjecture
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References
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2. Adem, A., Davis, J.F.: Topics in transformation groups. In: Handbook of Geometric Topology, pp. 1-
54. North-Holland, Amsterdam (2002)
3. Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups. Cambridge Studies in Advanced Mathematics, vol.
32. Cambridge University Press, Cambridge (1993)
4. Bahri, A., Bendersky, M., Cohen, F.R., Gitler, S.: Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces. Proc. Natl. Acad. Sci. USA 106, 12241-12244 (2009)
5. Buchstaber, V.M., Panov, T.E.: Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series, vol.
24. Amer. Math. Soc., Providence (2002)
6. Buchstaber, V.M., Panov, T.E.: Combinatorics of simplicial cell complexes and torus action. Proc. Steklov Inst. Math. 247, 33-49 (2004)
7. Carlsson, G.: On the non-existence of free actions of elementary abelian groups on products of spheres. Am. J. Math. 102, 1147-1157 (1980)
8. Carlsson, G.: On the rank of abelian groups acting freely on (Sn)k . Invent. Math. 69, 393-400 (1982)
9. Carlsson, G.: Free (Z/2)k -actions and a problem in commutative algebra. In: Transformation Groups, Poznań 1985, Lecture Notes in Math., vol. 1217, pp. 79-83. Springer, Berlin (1986)
10. Conner, P.E.: On the action of a finite group on Sn \times Sn. Ann. Math. 66, 586-588 (1957)
11. Davis, M.W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62, 417-451 (1991)
12. Denham, G., Suciu, A.: Moment-angle complexes, monomial ideals and Massey products. Pure Appl. Math. Q. 3, 25-60 (2007) J Algebr Comb (2012) 35:121-140
13. Halperin, S.: Rational homotopy and torus actions. In: Aspects of Topology, London Math. Soc. Lecture Note Ser., vol. 93, pp. 293-306. Cambridge University Press, Cambridge (1985)
14. Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Math., vol. 227.
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