g-vectors of manifolds with boundary
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 887-911.

We extend several g-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain Kühnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion Δ ^ of a manifold with boundary Δ; it is obtained from Δ by coning off the boundary of Δ with a single new vertex. We show that despite the fact that Δ ^ has a singular vertex, its Stanley–Reisner ring shares a few properties with the Stanley–Reisner rings of homology spheres. We close with a discussion of a connection between three lower bound theorems for manifolds, PL-handle decompositions, and surgery.

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DOI: 10.5802/alco.121
Classification: 05E45, 05E40, 13F55, 52B05, 55U10, 57Q15
Keywords: Homology manifolds with boundary, $g$-numbers, $g$-theorem, Stanley-Reisner rings, local cohomology, socle, Gorenstein rings.

Novik, Isabella 1; Swartz, Ed 2

1 Department of Mathematics University of Washington Seattle, WA 98195-4350, USA
2 Department of Mathematics Cornell University Ithaca, NY 14853-4201, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Novik, Isabella; Swartz, Ed. $g$-vectors of manifolds with boundary. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 887-911. doi : 10.5802/alco.121. https://alco.centre-mersenne.org/articles/10.5802/alco.121/

[1] Adiprasito, Karim Combinatorial Lefschetz theorems beyond positivity (2018) (https://arxiv.org/abs/1812.10454)

[2] Athanasiadis, Christos A. Some combinatorial properties of flag simplicial pseudomanifolds and spheres, Ark. Mat., Volume 49 (2011) no. 1, pp. 17-29 | DOI | MR | Zbl

[3] Babson, Eric; Novik, Isabella Face numbers and nongeneric initial ideals, Electron. J. Combin., Volume 11 (2004/06) no. 2, Paper no. Research Paper 25, 23 pages | MR | Zbl

[4] Bagchi, Bhaskar The mu vector, Morse inequalities and a generalized lower bound theorem for locally tame combinatorial manifolds, European J. Combin., Volume 51 (2016), pp. 69-83 | DOI | MR | Zbl

[5] Bagchi, Bhaskar; Datta, Basudeb Lower bound theorem for normal pseudomanifolds, Expo. Math., Volume 26 (2008) no. 4, pp. 327-351 | DOI | MR | Zbl

[6] Bagchi, Bhaskar; Datta, Basudeb On stellated spheres and a tightness criterion for combinatorial manifolds, European J. Combin., Volume 36 (2014), pp. 294-313 | DOI | MR | Zbl

[7] Basak, Biplab; Swartz, Ed Three-dimensional normal pseudomanifolds with relatively few edges, Adv. Math., Volume 365 (2020), Paper no. 107035 | DOI | MR | Zbl

[8] Billera, Louis J.; Lee, Carl W. A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A, Volume 31 (1981) no. 3, pp. 237-255 | DOI | MR | Zbl

[9] Bing, R. H. Some aspects of the topology of 3-manifolds related to the Poincaré conjecture, Lectures on modern mathematics, Vol. II, Wiley, New York, 1964, pp. 93-128 | MR | Zbl

[10] Brehm, Ulrich; Kühnel, Wolfgang Combinatorial manifolds with few vertices, Topology, Volume 26 (1987) no. 4, pp. 465-473 | DOI | MR | Zbl

[11] Datta, Basudeb; Murai, Satoshi On stacked triangulated manifolds, Electron. J. Combin., Volume 24 (2017) no. 4, Paper no. Paper 4.12, 14 pages | DOI | MR | Zbl

[12] Gräbe, Hans-Gert The canonical module of a Stanley–Reisner ring, J. Algebra, Volume 86 (1984) no. 1, pp. 272-281 | DOI | MR | Zbl

[13] Kalai, Gil Rigidity and the lower bound theorem. I, Invent. Math., Volume 88 (1987) no. 1, pp. 125-151 | DOI | MR | Zbl

[14] Kühnel, Wolfgang Higher-dimensional analogues of Czászár’s torus, Results Math., Volume 9 (1986) no. 1-2, pp. 95-106 | DOI | Zbl

[15] Kühnel, Wolfgang Tight polyhedral submanifolds and tight triangulations, Lecture Notes in Mathematics, 1612, Springer-Verlag, Berlin, 1995, vi+122 pages | DOI | MR | Zbl

[16] Kühnel, Wolfgang; Lassmann, Gunter Permuted difference cycles and triangulated sphere bundles, Discrete Math., Volume 162 (1996) no. 1-3, pp. 215-227 | DOI | MR | Zbl

[17] Lutz, Frank H Triangulated Manifolds with Few Vertices: Combinatorial Manifolds (2005) (http://arxiv.org/pdf/math/0506372v1.pdf)

[18] McMullen, Peter The numbers of faces of simplicial polytopes, Israel J. Math., Volume 9 (1971), pp. 559-570 | DOI | MR | Zbl

[19] Munkres, James  R. Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984, ix+454 pages | MR | Zbl

[20] Murai, Satoshi Algebraic shifting of strongly edge decomposable spheres, J. Combin. Theory Ser. A, Volume 117 (2010) no. 1, pp. 1-16 | DOI | MR | Zbl

[21] Murai, Satoshi Tight combinatorial manifolds and graded Betti numbers, Collect. Math., Volume 66 (2015) no. 3, pp. 367-386 | DOI | MR | Zbl

[22] Murai, Satoshi; Nevo, Eran On r-stacked triangulated manifolds, J. Algebraic Combin., Volume 39 (2014) no. 2, pp. 373-388 | DOI | MR | Zbl

[23] Murai, Satoshi; Novik, Isabella Face numbers and the fundamental group, Israel J. Math., Volume 222 (2017) no. 1, pp. 297-315 | DOI | MR | Zbl

[24] Murai, Satoshi; Novik, Isabella Face numbers of manifolds with boundary, Int. Math. Res. Not. IMRN (2017) no. 12, pp. 3603-3646 | DOI | MR | Zbl

[25] Murai, Satoshi; Novik, Isabella; Yoshida, Ken-ichi A duality in Buchsbaum rings and triangulated manifolds, Algebra Number Theory, Volume 11 (2017) no. 3, pp. 635-656 | DOI | MR | Zbl

[26] Novik, Isabella Upper bound theorems for homology manifolds, Israel J. Math., Volume 108 (1998), pp. 45-82 | DOI | MR | Zbl

[27] Novik, Isabella; Swartz, Ed Applications of Klee’s Dehn–Sommerville relations, Discrete Comput. Geom., Volume 42 (2009) no. 2, pp. 261-276 | DOI | MR | Zbl

[28] Novik, Isabella; Swartz, Ed Gorenstein rings through face rings of manifolds, Compos. Math., Volume 145 (2009) no. 4, pp. 993-1000 | DOI | MR | Zbl

[29] Novik, Isabella; Swartz, Ed Socles of Buchsbaum modules, complexes and posets, Adv. Math., Volume 222 (2009) no. 6, pp. 2059-2084 | DOI | MR | Zbl

[30] Novik, Isabella; Swartz, Ed Face numbers of pseudomanifolds with isolated singularities, Math. Scand., Volume 110 (2012) no. 2, pp. 198-222 | DOI | MR | Zbl

[31] Rourke, Colin Patrick; Sanderson, Brian Joseph Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982, viii+123 pages (Reprint) | MR

[32] Schenzel, Peter On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z., Volume 178 (1981) no. 1, pp. 125-142 | DOI | MR | Zbl

[33] Stanley, Richard P. The upper bound conjecture and Cohen–Macaulay rings, Studies in Appl. Math., Volume 54 (1975) no. 2, pp. 135-142 | DOI | MR | Zbl

[34] Stanley, Richard P. The number of faces of a simplicial convex polytope, Adv. Math., Volume 35 (1980) no. 3, pp. 236-238 | DOI | MR | Zbl

[35] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR | Zbl

[36] Swartz, Ed Face enumeration—from spheres to manifolds, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 3, pp. 449-485 | DOI | MR | Zbl

[37] Swartz, Ed Thirty-five years and counting (2014) (http://arxiv.org/pdf/1411.0987.pdf)

[38] Tay, Tiong-Seng; White, Neil; Whiteley, Walter Skeletal rigidity of simplicial complexes. I, European J. Combin., Volume 16 (1995) no. 4, pp. 381-403 | DOI | MR | Zbl

[39] Walkup, David W. The lower bound conjecture for 3- and 4-manifolds, Acta Math., Volume 125 (1970), pp. 75-107 | DOI | MR | Zbl

[40] Whiteley, Walter La division de sommet dans les charpentes isostatiques, Structural Topology (1990) no. 16, pp. 23-30 (Dual French-English text) | MR | Zbl

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