An Alexander-type duality for valuations
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- by Karim A. Adiprasito and Raman Sanyal PDF
- Proc. Amer. Math. Soc. 143 (2015), 833-843 Request permission
Abstract:
We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller–Reiner (2005). We give a generalization of Brion’s theorem for this relative situation, and we discuss the topology of the possible subcomplexes for which the duality relation holds.References
- Alexander Barvinok, A course in convexity, Graduate Studies in Mathematics, vol. 54, American Mathematical Society, Providence, RI, 2002. MR 1940576, DOI 10.1090/gsm/054
- Matthias Beck, Christian Haase, and Frank Sottile, Formulas of Brion, Lawrence, and Varchenko on rational generating functions for cones, Math. Intelligencer 31 (2009), no. 1, 9–17. MR 2480796, DOI 10.1007/s00283-008-9013-y
- Matthias Beck and Sinai Robins, Computing the continuous discretely, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. MR 2271992
- A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
- Michel Brion, Points entiers dans les polyèdres convexes, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 653–663 (French). MR 982338
- E. Ehrhart, Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux, J. Reine Angew. Math. 226 (1967), 1–29 (French). MR 213320, DOI 10.1515/crll.1967.226.1
- E. Ehrhart, Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires, J. Reine Angew. Math. 227 (1967), 25–49 (French). MR 217010, DOI 10.1515/crll.1967.227.25
- Michel A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72. MR 253347, DOI 10.1090/S0002-9947-1969-0253347-3
- I. G. Macdonald, Polynomials associated with finite cell-complexes, J. London Math. Soc. (2) 4 (1971), 181–192. MR 298542, DOI 10.1112/jlms/s2-4.1.181
- P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. MR 448239, DOI 10.1112/plms/s3-35.1.113
- Ezra Miller and Victor Reiner, Reciprocal domains and Cohen-Macaulay $d$-complexes in $\Bbb R^d$, Electron. J. Combin. 11 (2004/06), no. 2, Note 1, 9. MR 2120111
- James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
- James R. Munkres, Topological results in combinatorics, Michigan Math. J. 31 (1984), no. 1, 113–128. MR 736476, DOI 10.1307/mmj/1029002969
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744
- Richard P. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253. MR 411982, DOI 10.1016/0001-8708(74)90030-9
- J. H. C. Whitehead, Simplicial Spaces, Nuclei and m-Groups, Proc. London Math. Soc. (2) 45 (1939), no. 4, 243–327. MR 1576810, DOI 10.1112/plms/s2-45.1.243
- E. C. Zeeman, The generalised Poincaré conjecture, Bull. Amer. Math. Soc. 67 (1961), 270. MR 124906, DOI 10.1090/S0002-9904-1961-10578-8
Additional Information
- Karim A. Adiprasito
- Affiliation: Institut des Hautes Études Scientifiques, Paris, France
- MR Author ID: 963585
- Email: adiprasito@ihes.fr, adiprssito@math.fu-berlin.de
- Raman Sanyal
- Affiliation: Fachbereich Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany
- MR Author ID: 856938
- Email: sanyal@math.fu-berlin.de
- Received by editor(s): April 13, 2013
- Published electronically: October 28, 2014
- Additional Notes: The first author has been supported by the DFG within the research training group “Methods for Discrete Structures” (GRK1408) and by the Romanian NASR, CNCS — UEFISCDI, project PN-II-ID-PCE-2011-3-0533.
The second author has been supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^\mathrm {o}$ 247029. - Communicated by: Jim Haglund
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 143 (2015), 833-843
- MSC (2010): Primary 52B45, 57Q99, 52C07, 55U30
- DOI: https://doi.org/10.1090/S0002-9939-2014-12366-5
- MathSciNet review: 3283669