Twisted complexes and simplicial homotopies

New methods for constructing shellable simplicial complexes

A clutter $mathcal{C}$ with vertex set $[n]$ is an antichain of subsets of $[n]$, called circuits, covering all vertices. The clutter is $d$-uniform if all of its circuits have the same cardinality $d$. If $mathbb{K}$ is a field, then there is a one-to-one correspondence between clutters on $V$ and square-free monomial ideals in $mathbb{K}[x_1,ldots,x_n]$ as follows: To each clutter $mathcal{C}...

Completions and Simplicial Complexes

We first introduce the notion of a completion. Completions are inductive properties which may be expressed in a declarative way and which may be combined. We show that completions may be used for describing structures or transformations which appear in combinatorial topology. We present two completions, 〈CUP〉 and 〈CAP〉, in order to define, in an axiomatic way, a remarkable collection of acyclic...

Simplicial Complexes and Spaces

The notion of simplicial complexes is usually introduced at the graduate level, as part of the study of algebraic topology. So the main objective of this paper is to organize and present materials found in the literature so that it is accessible to students at the undergraduate level. We first generalize linear algebra to affine linear algebra. Then using these results, we define, and subsequen...

On Isomorphism of Simplicial Complexes and Their Related Algebras

Grothendieck Groups of Complexes with Null-homotopies

This paper is an addendum to [FH]. We reprove the main theorem of that paper by using Grothendieck groups of modules over certain DGAs. While the changes to the argument in [FH] are largely cosmetic, our approach shortens the proof from 25 pages down to 10 and greatly clarifies the overall structure. The authors of [FH] describe their proof as both “tedious” and “cumbersome”, whereas the approa...