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On kk-stellated and kk-stacked spheres

Abstract

We introduce the class Ξ£k(d)\Sigma_k(d) of kk-stellated (combinatorial) spheres of dimension dd (0≀k≀d+10 \leq k \leq d + 1) and compare and contrast it with the class Sk(d){\cal S}_k(d) (0≀k≀d0 \leq k \leq d) of kk-stacked homology dd-spheres. We have Ξ£1(d)=S1(d)\Sigma_1(d) = {\cal S}_1(d), and Ξ£k(d)βŠ†Sk(d)\Sigma_k(d) \subseteq {\cal S}_k(d) for dβ‰₯2kβˆ’1d \geq 2k - 1. However, for each kβ‰₯2k \geq 2 there are kk-stacked spheres which are not kk-stellated. The existence of kk-stellated spheres which are not kk-stacked remains an open question. We also consider the class Wk(d){\cal W}_k(d) (and Kk(d){\cal K}_k(d)) of simplicial complexes all whose vertex-links belong to Ξ£k(dβˆ’1)\Sigma_k(d - 1) (respectively, Sk(dβˆ’1){\cal S}_k(d - 1)). Thus, Wk(d)βŠ†Kk(d){\cal W}_k(d) \subseteq {\cal K}_k(d) for dβ‰₯2kd \geq 2k, while W1(d)=K1(d){\cal W}_1(d) = {\cal K}_1(d). Let KΛ‰k(d)\bar{{\cal K}}_k(d) denote the class of dd-dimensional complexes all whose vertex-links are kk-stacked balls. We show that for dβ‰₯2k+2d\geq 2k + 2, there is a natural bijection M↦MΛ‰M \mapsto \bar{M} from Kk(d){\cal K}_k(d) onto KΛ‰k(d+1)\bar{{\cal K}}_k(d + 1) which is the inverse to the boundary map βˆ‚β€‰β£:KΛ‰k(d+1)β†’Kk(d)\partial \colon \bar{{\cal K}}_k(d + 1) \to {\cal K}_k(d).Comment: Revised Version. Theorem 2.24 is new. 18 pages. arXiv admin note: substantial text overlap with arXiv:1102.085

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