Simplicial Complexes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 18 November 2009

The concept

The standard $k$-simplex is

$${\Delta }^k=\left\{\right(x_0,\dots ,x_k)\in {ℝ}_{\mathrm{\ge 0}}^{k+1}\mid x_0+\cdots +x_k\le 1\}$$.

PICTURE of ${\Delta }^1$ and ${\Delta }^2$. More generally, take $v_0,\dots ,v_k$ linearly independent and let

$$\Delta (v_0,\dots ,v_k)=\{x_0{v}_0+\cdots +x_k{v}_k\mid x_0+\cdots +x_k=1\}\text{.}$$

A simplicial complex with vertex set $V$ is a collection $\Sigma$ of finite subsets of $V$ such that

1. If $v\in V$ then $\left\{v\right\}\in \Sigma$.
2. If $\sigma \in \Sigma$ and ${\sigma }^{\prime }\subseteq \sigma$ then ${\sigma }^{\prime }\in \Sigma$,

A simplex is an element of $\Sigma$. A vertex is a simplex with one element. A face of a simplex $\sigma$ is a subset of $\sigma$. A simplicial complex $\Sigma$ is partially ordered by inclusion. A chamber is a maximal simplex.

A geometric realization of $\Sigma$ is a topological space $X$ whose structure is completely controlled by the simplicial complex $\Sigma$ where each $k$-simplex in $\Sigma$ corresponds to a standard $k$-simplex in $X$. It is a bit challenging to make precise sense of the "completely controlled by" in sufficient generality so it is better to ignore this problem and use simplicial complexes for examples but avoid simplicial complexes in general theory (see also the discussion in [Hatcher, p.107]). Another historical solution is to use simplicial sets (see [Gelfand-Manin Ch. 1 Sec. 2.1.2]).

Let $V=\text{span}\{v_1,\dots ,v_n\}$. The exterior algebra of $V$ is

$$\Lambda \left(V\right)={⨁}_{i=0}^n{\Lambda }^{\ell }\left(V\right)\text{,}\text{where}{\Lambda }^{\ell }\left(V\right)=\text{span}\{v_{i_1}\wedge v_{i_2}\wedge \cdots \wedge v_{i_{\ell }}\mid 1<i_1<i_2<\cdots <i_{\ell }\le n\}$$

and the relation $x\wedge y=–y\wedge x$.

The homology of a simplicial complex $X$ is the homology of the complex

$$\cdots ⟶C^{\ell }⟶C^{\ell –1}⟶\cdots \text{given by}{C}^{\ell }=\text{span}\left\{v_{i_1}\wedge v_{i_2}\wedge \cdots \wedge v_{i_{\ell }}\right|\{v_{i_1},v_{i_2},\dots ,v_{i_{\ell }}\}\in X\}$$

with boundary map $d:C^{\ell }\to C^{\ell –1}$ given by

$$d\left(x_1\wedge \cdots \wedge x_{\ell }\right)=\sum _{j=1}^{\ell }(–1{)}^{j–1}{x}_1\wedge \cdots \wedge {\widehat{x}}_j\wedge \cdots \wedge x_n$$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)