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Centrally Symmetric Spheres Versus Centrally Symmetric Polytopes

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Centrally Symmetric Spheres Versus Centrally Symmetric Polytopes FALL SECTIONAL SAMPLER A map 휙 ∶ 푉 → 푉 induces a simplicial map on Δ if Face Numbers: 휙 takes faces to faces. A simplicial complex Δ is centrally symmetric (cs) if it is equipped with an involution 휙 on Centrally Symmetric 푉 that induces a free simplicial involution on Δ. In other words, for every nonempty face 퐹 of Δ the following holds: Spheres versus 휙(퐹) is also a face of Δ, 휙(퐹) ≠ 퐹, but 휙(휙(퐹)) = 퐹. For instance, the boundary complex of any cs simplicial Centrally Symmetric polytope 푃 is a cs simplicial sphere with the involution induced by the map 휙(푣) = −푣 on the vertex set of 푃. Polytopes The celebrated Upper Bound Theorem of McMullen (for polytopes) and Stanley (for spheres) asserts that in the class of all (푑 − 1)-dimensional simplicial spheres with Isabella Novik 푛 vertices, the boundary complex of a certain polytope (called the cyclic polytope) simultaneously maximizes all We start by introducing the main players. A polytope 푃 is the 푑 푑 the face numbers. One amazing property of the - convex hull of a set of finitely many points in ℝ . For in- dimensional cyclic polytope on 푛 vertices is that its [푑/2]- stance, a (geometric) simplex is the convex hull of a set of skeleton coincides with that of an (푛 − 1)-dimensional affinely independent points. We say that 푃 is centrally sym- simplex (for all values of 푛 ≥ 푑 + 1). In particular, ev- metric (cs, for short) if it is symmetric about the origin, that ery two vertices of the cyclic polytope of dimension 푑 ≥ 4 is, if 푃 = −푃. An important example of a cs polytope is 푘 ∗ are connected by an edge! (Here and below by -skeleton the 푑-dimensional cross-polytope 퐶푑 = conv(±푒1, ±푒2, of a simplicial polytope 푃 we mean the 푘-skeleton of the … , ±푒푑), where 푒1, 푒2, … , 푒푑 are the endpoints of the boundary complex of 푃.) 푑 standard basis in ℝ . The cross-polytope is a simplicial The existence of cyclic polytopes and the statement of polytope, meaning that all of its (proper) faces are geomet- the Upper Bound Theorem along with recently discovered ric simplices. tantalizing connections (initiated by Donoho) between cs Our other main player is a centrally symmetric (or cs) sim- polytopes with many faces and seemingly unrelated areas plicial complex. Recall first that a simplicial complex Δ on of error-correcting codes and sparse signal reconstruction a (finite) vertex set 푉 is a collection of subsets of 푉 that motivate the following questions in the centrally symmet- is closed under inclusion. The elements of Δ are called ric world. For a fixed 푑 ≥ 4 and 푁 ≥ 푑, what is the largest faces. The dimension of a face 퐹 is one less than the size number of 푖-faces that a cs simplicial sphere of dimension of 퐹, and the dimension of Δ is the maximum dimension 푑 − 1 with 2푁 vertices can have? What is the largest num- of its faces. The 푘-skeleton of Δ is the subcomplex of Δ ber of 푖-faces that a cs simplicial polytope of dimension consisting of all faces of dimension ≤ 푘. We refer to 푖- 푑 with 2푁 vertices can have? Do there exist cs simplicial dimensional faces as 푖-faces; their number is usually de- spheres of dimension 푑 − 1 with 2푁 vertices (for arbitrar- noted by 푓푖 and is called the 푖th face number. ily large 푁) whose [푑/2]-skeleton coincides with that of Each simplicial complex Δ admits a geometric realiza- the 푁-dimensional cross-polytope? What about cs simpli- tion |Δ| that contains a geometric 푖-simplex for each 푖- cial polytopes with the same parameters? Are the answers face of Δ. This allows us to talk about simplicial spheres— for cs spheres and cs polytopes the same or (drastically) simplicial complexes whose geometric realizations are different? homeomorphic to a sphere. For instance, if 푃 is a sim- We will discuss these and other questions, summariz- plicial polytope, then the empty set along with the col- ing some of the surprising and fascinating developments lection of vertex sets of proper faces of 푃 is a simplicial in this field over the last fifty years. Our discussion willin- sphere called the boundary complex of 푃. However, as clude results from a very recent joint project with Hailun works of Kalai, and Pfeifle and Ziegler show, the boundary Zheng of the University of Michigan. complexes of simplicial polytopes of dimensions 푑 ≥ 4 (푑 − 1) form only a tiny fraction of -dimensional simpli- Credits cial spheres. Author photo is courtesy of Lev Novik. Isabella Novik is Robert R. & Elaine F. Phelps Professor of Mathematics at the University of Washington. She is supported by NSF grant DMS-1664865. Her email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti1966 Isabella Novik 1698 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 10.
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