Beyond Sperner's lemma

Nikolai V. Ivanov

Home/Archive/arXiv

arXiv · 1902.00827

Beyond Sperner's lemma

Nikolai V. Ivanov

View original source ↗Read source document ↗Export metadata ↓

Abstract

In 1967 Herbert Scarf suggested a new proof of Brouwer's fixed point theorem based on a combinatorial analogue of Sperner's lemma. Scarf presented his arguments in very geometric language, even purely combinatorial ones. Recently H. Petri and M. Voorneveld published an almost geometry-free version of Scarf's proof. Their version eliminated even only implicitly geometric aspects of Scarf's proof, namely, the structure of an abstract simplicial complex behind the combinatorial arguments. The present paper is devoted to a proof of Scarf's analogue of Sperner's lemma in the abstract setting of a collection of linear orders on a finite set. This proof partially follows the proof by Petri and Voorneveld, but restores the implicit geometry to its rightful place. We also deduce Brouwer's fixed point theorem from this analogue and discuss various versions of Scarf's proof.

Explore related subjects

math.AT

Keep this discovery

BibTeX RIS

Nikolai V. Ivanov. 2019-02-03. Beyond Sperner's lemma. https://arxiv.org/abs/1902.00827

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

math.AT↗

arXiv2026

The homotopy types of directed path and trace spaces

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

math.AT↗

arXiv2026

Moduli spaces of geometric functorial field theories

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

math.AT↗