$g$-vectors of manifolds with boundary

Novik, Isabella; Swartz, Ed

doi:10.5802/alco.121

g

-vectors of manifolds with boundary

Novik, Isabella; Swartz, Ed

Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 887-911.

Abstract

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 $\hat{Δ}$ 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 $\hat{Δ}$ 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.

Received: 2019-09-27
Revised: 2020-03-24
Accepted: 2020-03-27
Published online: 2020-08-20

DOI: https://doi.org/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.

@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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