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: 2020-03-24

Accepted: 2020-03-27

Published online: 2020-08-20

Classification: 05E45, 05E40, 13F55, 52B05, 55U10, 57Q15

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

@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 = {alco.centre-mersenne.org/item/ALCO_2020__3_4_887_0/} }

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/item/ALCO_2020__3_4_887_0/

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