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 $\widehat{\Delta}$ of a manifold with boundary $\Delta $; it is obtained from $\Delta $ by coning off the boundary of $\Delta $ with a single new vertex. We show that despite the fact that $\widehat{\Delta}$ 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.

Revised:

Accepted:

Published online:

Keywords: Homology manifolds with boundary, $g$-numbers, $g$-theorem, Stanley-Reisner rings, local cohomology, socle, Gorenstein rings.

^{1}; Swartz, Ed

^{2}

@article{ALCO_2020__3_4_887_0, author = {Novik, Isabella and Swartz, Ed}, title = {$g$-vectors of manifolds with boundary}, journal = {Algebraic Combinatorics}, pages = {887--911}, publisher = {MathOA foundation}, volume = {3}, number = {4}, year = {2020}, doi = {10.5802/alco.121}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.121/} }

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. http://www.numdam.org/articles/10.5802/alco.121/

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

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

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

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

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

[7] Three-dimensional normal pseudomanifolds with relatively few edges, Adv. Math., Volume 365 (2020), 107035 | DOI | MR | Zbl

[8] 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] 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] Combinatorial manifolds with few vertices, Topology, Volume 26 (1987) no. 4, pp. 465-473 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[25] A duality in Buchsbaum rings and triangulated manifolds, Algebra Number Theory, Volume 11 (2017) no. 3, pp. 635-656 | DOI | MR | Zbl

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

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

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

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

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

[31] Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, Berlin-New York, 1982, viii+123 pages (Reprint) | MR

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

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

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

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

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

[38] Skeletal rigidity of simplicial complexes. I, European J. Combin., Volume 16 (1995) no. 4, pp. 381-403 | DOI | MR | Zbl

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

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

*Cited by Sources: *