Simplicial complexes

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

Last updates: 14 June 2012

Simplicial complexes

Let $V=\{v_1,\dots ,v_n\}$ be a finite set.

• A simplicial complex with vertex set $V$ is a collection $\Sigma$ of finite subsets of $V$ such that
  1. (a) If $v\in V$ then $\left\{v\right\}\in \Sigma$.
  2. (b) If $\sigma \in \Sigma$ and $\sigma \prime \subseteq \sigma$ then $\sigma \prime \in \Sigma$.

Let $\Sigma$ be a simplicial complex. The simplicial complex $\Sigma$ is partially ordered by inclusion.

• 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 chamber is a maximal simplex.

Let $\Sigma$ be a simplicial complex with vertex set $V=\{v_1,\dots ,v_n$. Let $E=ℝ\text{-span}\{v_1,\dots ,v_n\}$.

• The exterior algebra of $E$ is $$\Lambda \left(E\right)=\underset{\ell =0}{\overset{n}{⨁}}{\Lambda }^{\ell }\left(E\right),\text{where}{\Lambda }^{\ell }\left(E\right)=\text{span}\{e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_{\mathrm{\ell }}}|1\le i_1<i_2<\cdots <i_{}\le n\}$$ with the relations $$x\wedge y=-y\wedge x\text{and}(ax+by)\wedge z=a(x\wedge z)+b(y\wedge z),$$ for $a,b\in ℝ$ and $x,y,z\in E$.
• The homology of $\Sigma$ is the homology of the complex $$\cdots ⟶C^{\ell }⟶C^{\ell -1}⟶\cdots \text{given by}{C}^{\ell }=\text{span}\{v_{i_1}\wedge v_{i_2}\wedge \cdots \wedge v_{i_{\mathrm{\ell }}}|\{v_{i_1},v_{i_2},\dots ,v_{i_{\mathrm{\ell }}}\}\in \Sigma \}$$ with boundary map $d:C^{\ell }\to C^{\ell -1}$ given by $$d(x_1\wedge \cdots \wedge x_{\ell })=\sum _{j=1}^{\ell }{\left(-1\right)}^{j-1}{x}_1\wedge \cdots \wedge {\hat{x}}_j\wedge \cdots \wedge x_n.$$

The standard $k$-simplex is $${\Delta }^k=\left\{\right(x_0,\dots ,x_k)\in {ℝ}_{\ge 0}^{k+1}|x_0+\cdots +x_k\le 1\}.$$ $$\text{PICTURE of}{\Delta }^1\text{and}{\Delta }^2.$$ More generally, let $E$ be a real vector space and let $e_0,\dots ,e_k$ be linearly independent vectors in $E$ and let $$\Delta (e_0,\dots e_k)=\{x_0{e}_0+\cdots +x_k{e}_k|x_0+\cdots +x_k\le 1\}.$$ 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]).

Notes and References

These notes are part of attempt to sort out the zoo of definitions of homology and cohomology of a space (and the various corresponding definitions of a "space"). The definition of homology of a simplicial complex is one of the traditional starting points in algebraic topology as found, for example in [Mu, Ch. 1 \S 5]

References

[BouAlg] N. Bourbaki, Algebra I, Chapter I, Section 9 No. 4, Springer-Verlag, Berlin 1989. MR?????

[BouTop] N. Bourbaki, General Topology, Chapter IV, Section 1, Springer-Verlag, Berlin 1989. MR?????

[Mu] J.R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. ix+454 pp. ISBN: 0-201-04586-9 MR0755006

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