TY - JOUR
T1 - Face numbers and the fundamental group
AU - Murai, Satoshi
AU - Novik, Isabella
N1 - Funding Information:
∗ Research is partially supported by JSPS KAKENHI 16K05102. ∗∗ Research is partially supported by NSF grant DMS-1361423. Received July 14, 2016 and in revised form September 13, 2016
Publisher Copyright:
© 2017, Hebrew University of Jerusalem.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - We resolve a conjecture of Kalai asserting that the g2-number of any (finite) simplicial complex Δ that represents a normal pseudomanifold of dimension d ≥ 3 is at least as large as (d+22)m(Δ), where m(Δ) denotes the minimum number of generators of the fundamental group of Δ. Furthermore, we prove that a weaker bound, h2(d+12)m(Δ), applies to any d-dimensional pure simplicial poset Δ all of whose faces of co-dimension ≥ 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Ψ all of whose vertex links satisfy Serre’s condition (Sr), we establish lower bounds on h1(Ψ),..,hr(Ψ) in terms of the μ-numbers introduced by Bagchi and Datta.
AB - We resolve a conjecture of Kalai asserting that the g2-number of any (finite) simplicial complex Δ that represents a normal pseudomanifold of dimension d ≥ 3 is at least as large as (d+22)m(Δ), where m(Δ) denotes the minimum number of generators of the fundamental group of Δ. Furthermore, we prove that a weaker bound, h2(d+12)m(Δ), applies to any d-dimensional pure simplicial poset Δ all of whose faces of co-dimension ≥ 2 have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset Ψ all of whose vertex links satisfy Serre’s condition (Sr), we establish lower bounds on h1(Ψ),..,hr(Ψ) in terms of the μ-numbers introduced by Bagchi and Datta.
UR - http://www.scopus.com/inward/record.url?scp=85034965030&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85034965030&partnerID=8YFLogxK
U2 - 10.1007/s11856-017-1591-y
DO - 10.1007/s11856-017-1591-y
M3 - Article
AN - SCOPUS:85034965030
VL - 222
SP - 297
EP - 315
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
SN - 0021-2172
IS - 1
ER -