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Upper bound theorems for homology manifolds

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Abstract

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β k (Δ)⩽Σ{β i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.)

We prove an analog of the UBC for all other even-dimensional homology manifolds.

Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices,\(\left( { - 1} \right)^k \left( {\mathcal{X}\left( \Delta \right) - 2} \right) \leqslant \left( {\begin{array}{*{20}c} {n - k - 2} \\ {k + 1} \\ \end{array} } \right)/\left( {\begin{array}{*{20}c} {2k + 1} \\ k \\ \end{array} } \right)\). We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.

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  1. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904, Jerusalem, Israel

    Isabella Novik

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  1. Isabella Novik
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Correspondence to Isabella Novik.

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Novik, I. Upper bound theorems for homology manifolds. Israel J. Math. 108, 45–82 (1998). https://doi.org/10.1007/BF02783042

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  • DOI: https://doi.org/10.1007/BF02783042

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