Searcharxiv⌕ Search

arXiv subjects

Karim Adiprasito

Publications and source records attributed to Karim Adiprasito.

At least 19 recordsLinked to original sources
arXiv2026

Subadditivity of shifts, Eilenberg-Zilber shuffle products and homology of lattices

We show that the maximal shifts in the minimal free resolution of the quotients of a polynomial ring by a monomial ideal are subadditive as a function of the homological degree. This answers a question that has received some attention in recent years. To do so, we define and study a new model for the homology of posets, given by the so called synor complex. We also introduce an Eilenberg-Zilber type shuffle product on the simplicial chain complex of lattices. Combining these concepts we prove that the existence of a nonzero homology class for a lattice forces certain nonzero homology classes in lower intervals. This result then translates into properties of the minimal free resolution. In particular, it yields a strengthening of the original subadditivity statement.

math.AC↗
arXiv2025

$p$-anisotropy on the moment curve for homology manifolds and cycles

We prove that the Gorensteinification of the face ring of a cycle is totally $p$-anisotropic in characteristic $p$. In other words, given an appropriate Artinian reduction, it contains no nonzero $p$-isotropic elements. Moreover, we prove that the linear system of parameters can be chosen corresponding to a geometric realization with points on the moment curve. In particular, this implies that the parameters do not have to be chosen very generically.

math.CO↗
arXiv2025

On graphs, homology bases, and triangulated homology spheres

We describe a construction that takes as input a graph and a basis for its first homology, and returns a triangulation of a 3-dimensional homology sphere. This makes precise an idea of M. Gromov and A. Nabutovski. The immediate application, essentially described by Gromov, is to translate problems about asymptotics of homology sphere triangulations to asymptotic counting problems for constant-degree graphs with "short" homology bases. We construct families of 3- sphere triangulations with dual graphs that are expanders, answering a relaxation of a question asked by G. Kalai. Our results also imply that if the number of d-dimensional triangulated homology spheres with n facets is superexponential in n for some d then the same holds for d = 3.

math.CO↗
arXiv2024

Sparse handlebody decompositions and non-finiteness of $g_3=0$

We prove that a PL manifold admits a handle decomposition into handles of index $\le k$ if and only if $M$ is $k$-stacked, i.e., it admits a PL triangulation in which all $(d-k-1)$-faces are on $\partial M$. We use this to solve a problem posed in 2008 by Kalai: In any dimension higher than four, there are infinitely many homology-spheres with $g_3 =0$.

math.GT↗
arXiv2024

Normal crossing immersions, cobordisms and flips

We study various analogues of theorems from PL topology for cubical complexes. In particular, we characterize when two PL homeomorphic cubulations are equivalent by Pachner moves by showing the question to be equivalent to the existence of cobordisms between generic immersions of hypersurfaces. This solves a question and conjecture of Habegger and Funar.

math.GT↗
arXiv2024

All triangulations have a common stellar subdivision

We address two longstanding open problems, one originating in PL topology, another in birational geometry. First, we prove the weighted version of Oda's \emph{strong factorization conjecture} (1978), and prove that every two birational toric varieties are related by a common iterated blowup (at rationally smooth points). Second, we prove that every two PL homeomorphic polyhedra have a common stellar subdivisions, as conjectured by Alexander in~1930.

math.CO↗
arXiv2021

A subexponential size triangulation of $\mathbb{R}P^n$

We address a long-standing and long-investigated problem in combinatorial topology, and break the exponential barrier for triangulations of real projective space, constructing a trianglation of $\mathbb{RP}^n$ of size $e^{(\frac{1}{2}+o(1))\sqrt{n}{\log n}}$.

math.CO↗
arXiv2021

Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitrary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.

math.CO↗
arXiv2021

A complete characterization of $(f_0, f_1)$-pairs of 6-polytopes

We completely characterize the first two entries, namely the $(f_0, f_1)$-vector pairs, for $6$-dimension polytopes. We also find the characterization for $7$-dimension polytopes with excess degree greater than $11$ and, we conjecture bounds fulfilled by $(f_0, f_1)$-vector pairs for any $d$-polytope having an excess degree greater than $3d-10$.

math.CO↗
arXiv2021

The Partition Complex: an invitation to combinatorial commutative algebra

We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge of algebra and topology. On the other hand, we also develop new techniques and results using this approach. In particular, we provide - A novel, self-contained method of establishing Reisner's theorem and Schenzel's formula for Buchsbaum complexes. - A simple new way to establish Poincaré duality for face rings of manifolds, in much greater generality and precision than previous treatments. - A "master-theorem" to generalize several previous results concerning the Lefschetz theorem on subdivisions. - Proof for a conjecture of Kühnel concerning triangulated manifolds with boundary.

math.CO↗
arXiv2021

Lefschetz and Lower Bound theorems for Minkowski sums

This note provides a Lefschetz theorem for Minkowski sums of polytopes, and conclude lower bound theorems for Minkowski sums of polytopes. It is written as an appendix to arXiv:1405.7368, so notation and references follow that paper.

math.CO↗
arXiv2021

Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums

In this paper we settle long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed faces of Minkowski sums. This has a wide range of applications and generalizes the classical the Upper Bound Theorems of McMullen and Stanley. Our main tool is relative Stanley--Reisner theory, a powerful generalization of the algebraic theory of simplicial complexes inaugurated by Hochster, Reisner, and Stanley. A key feature of our theory is the ability to accomodate topological as well as combinatorial restrictions. We illustrate this by providing several simplicial isoperimetric and reverse isoperimetric inequalities.

math.CO↗
arXiv2019

On the realization space of the cube

We consider the realization space of the $d$-dimensional cube, and show that any two realizations are connected by a finite sequence of projective transformations and normal transformations. We use this fact to define an analog of the connected sum construction for cubical $d$-polytopes, and apply this construction to certain cubical $d$-polytopes to conclude that the rays spanned by $f$-vectors of cubical $d$-polytopes are dense in Adin's cone. The connectivity result on cubes extends to any product of simplices, and further, it shows the respective realization spaces are contractible.

math.CO↗
arXiv2019

Rigidity with few locations

Graphs triangulating the $2$-sphere are generically rigid in $3$-space, due to Gluck-Dehn-Alexandrov-Cauchy. We show there is a \emph{finite} subset $A$ in $3$-space so that the vertices of each graph $G$ as above can be mapped into $A$ to make the resulted embedding of $G$ infinitesimally rigid. This assertion extends to the triangulations of any fixed compact connected surface, where the upper bound obtained on the size of $A$ increases with the genus. The assertion fails, namely no such finite $A$ exists, for the larger family of all graphs that are generically rigid in $3$-space and even in the plane.

math.CO↗
arXiv2019

Unstable blueprints can be shared

This expository note illustrates toric perturbation and biased pairing theory to show that Artinian reductions of face rings of $2$-spheres that do not satisfy the Lefschetz property can be cut along a flat equator. This complements classical work of Bricard and Connelly, and exhibits a fundamental symmetry in non-rigid triangulations of spheres.

math.CO↗
arXiv2019

Collapsibility of CAT(0) spaces

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All CAT(0) cube complexes are collapsible. (2) Any triangulated manifold admits a CAT(0) metric if and only if it admits collapsible triangulations. (3) All contractible d-manifolds ($d \ne 4$) admit collapsible CAT(0) triangulations. This discretizes a classical result by Ancel--Guilbault.

math.MG↗