The complex of end reductions of a contractible open 3-manifold: constructing 1-dimensional examples

End reductions and covering spaces of contractible open 3-manifolds

This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of W. Whenever W covers another 3-manifold the group of covering translations is isomorphic to a fixed point free group of automorphisms of S(W). We apply this result to give uncountably many examples of R^2-irreducible such W which cover 3-manifolds with infinite cyclic fundamental…

Shellability is NP-complete

We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial complex is $k$-decomposable is NP-hard. For $d \ge 3$, both problems remain NP-hard when restricted to contractible pure $d$-dimensional complexes. Another simple corollary of our result is that it is NP-hard to decide whether a given poset is CL-shellable.

Bestvina度量与树简化用于$K(π,1)$猜想

Bestvina metric and tree reduction for $K(π,1)$-conjecture

本文通过构造Artin群在Bestvina复形上的作用，将$K(π,1)$猜想简化为Coxeter图为树的Artin群性质，推导出新的满足该猜想的Artin群类别。