A Conjecture of Kozlov from the 1998 Proceedings of the American Mathematical Society: Non-Evasive Order Complexes and Generalizations of Non-Complemented Lattices

TL;DR

This paper proves Kozlov's 1998 conjecture that the order complex of a finite poset with specific lattice-like properties is non-evasive, advancing understanding in algebraic topology and combinatorics.

Contribution

It confirms Kozlov's conjecture, showing that certain posets have non-evasive order complexes, a significant result in topological combinatorics.

Findings

Proved Kozlov's conjecture on non-evasiveness.

Established conditions under which order complexes are non-evasive.

Contributed to the theory of posets and their topological properties.

Abstract

Let $P$ be a finite poset with an element $s$ such that (1) for all $x\in P$, either $s\vee x$ or $s\wedge x$ exists; and (2) for all $x,y\in P$ such that $x<y$, if $s\wedge x$ does not exist but $s\wedge y$ does exist, then $(s\wedge y)\vee x$ exists. Kozlov, the winner of the 2005 European Prize in Combinatorics ("for deep combinatorial results obtained by algebraic topology and particularly for the solution of a conjecture of Lov\'asz"), conjectured in the 1998 Proceedings of the American Mathematical Society that the order complex of $P$ is non-evasive. We prove this conjecture.