Abstract simplicial complexes

An abstract simplicial complex above a set ${\displaystyle V}$ is a system ${\displaystyle {\mathcal {T}}\subset {\mathcal {P}}(V)}$ of non-empty subsets such that:

• ${\displaystyle \{v_{0}\}\in {\mathcal {T}}}$ for each ${\displaystyle v_{0}\in V}$;
• if ${\displaystyle E\in {\mathcal {T}}}$ and ${\displaystyle \emptyset \neq F\subset E}$ ${\displaystyle \Rightarrow }$ ${\displaystyle F\in {\mathcal {T}}}$.

The elements of ${\displaystyle {\mathcal {T}}}$ are called simplices, the elements of ${\displaystyle V}$ are called vertices. A simplex with ${\displaystyle n+1}$ vertices has dimension ${\displaystyle n}$ by definition. The dimension of an abstract simplicial complex is defined as ${\displaystyle {\text{dim}}({\mathcal {T}})={\text{sup}}\;\{{\text{dim}}(F):F\in {\mathcal {T}}\}\in \mathbb {N} \cup \infty }$.^[1]

Abstract simplicial complexes can be thought of as geometrical objects too. This requires the term of geometric simplex.

Geometric simplices in dimension 1, 2 and 3

Geometric simplices

Let ${\displaystyle p_{0},...p_{n}}$ be ${\displaystyle n+1}$ affinely independent points in ${\displaystyle \mathbb {R} ^{n}}$, i.e. the vectors ${\displaystyle (p_{1}-p_{0}),(p_{2}-p_{0}),\dots (p_{n}-p_{0})}$are linearly independent. The set ${\textstyle \Delta ={\Bigl \{}x\in \mathbb {R} ^{n}\;{\Big |}\;x=\sum _{i=0}^{n}t_{i}p_{i}\;with\;0\leq t_{i}\leq 1\;and\;\sum _{i=0}^{n}t_{i}=1{\Bigr \}}}$ is said to be the simplex spanned by ${\displaystyle p_{0},...p_{n}}$. It has dimension ${\displaystyle n}$ by definition. The points ${\displaystyle p_{0},...p_{n}}$ are called the vertices of ${\displaystyle \Delta }$, the simplices spanned by ${\displaystyle n}$ of the ${\displaystyle n+1}$ vertices are called faces and the boundary ${\displaystyle \partial \Delta }$ is defined to be the union of its faces.

The ${\displaystyle n}$-dimensional standard-simplex is the simplex spanned by the unit vectors ${\displaystyle e_{0},...e_{n}}$^[2]

Geometric simplicial complexes

A geometric simplicial complex ${\displaystyle {\mathcal {S}}\subset \mathbb {R} ^{n}}$ is a union of geometric simplices such that

• If ${\displaystyle S}$ is a simplex in ${\displaystyle {\mathcal {S}}}$, then all its faces are in ${\displaystyle {\mathcal {S}}}$.
• If ${\displaystyle S,T}$ are two distinct simplices in ${\displaystyle {\mathcal {S}}}$, their interiors are disjoint.

The set ${\displaystyle {\mathcal {S}}}$ can be realized as a topological space ${\displaystyle |{\mathcal {S}}|}$ by choosing the closed sets to be ${\displaystyle {\Bigl \{}A\subset |{\mathcal {S}}|\;{\Big |}\;A\cap \Delta }$ is closed for all ${\displaystyle \Delta \in {\mathcal {S}}{\Bigr \}}}$. It should be mentioned, that in general, the simplicial complex won't provide the natural topology of ${\displaystyle \mathbb {R} ^{n}}$. In the case that each point in the complex lies only in finetly many simplices, both topologies coincide^[2]

Each geometric complex can be associated with an abstract complex by choosing as a ground set ${\displaystyle V}$ the set of vertices that appear in any simplex of ${\displaystyle {\mathcal {S}}}$ and as system of subsets the subsets of ${\displaystyle V}$ which correspond to vertex sets of simplices in ${\displaystyle {\mathcal {S}}}$.

A natural question is if vice versa, any abstract simplicial complex corresponds to a geometric complex. In general, the geometric construction as mentioned here is not flexible enough: Consider for instance abstract simplicial complex of infinite dimension. However, the following more abstract construction provides a topological space for any kind of abstract simplicial complex:

Let ${\displaystyle {\mathcal {T}}}$ be an abstract simplicial complex above a set ${\displaystyle V}$. Choose a union of simplices ${\displaystyle (\Delta _{F})_{F\in {\mathcal {T}}}}$, but each in ${\displaystyle \mathbb {R} ^{N}}$ of dimension sufficiently large, such that the geometric simplex ${\displaystyle \Delta _{F}}$ is of dimension ${\displaystyle n}$ if the abstract geometric simplex ${\displaystyle F}$ has dimension ${\displaystyle n}$. If ${\displaystyle E\subset F}$, ${\displaystyle \Delta _{E}\subset \mathbb {R} ^{N}}$can be identified with a face of ${\displaystyle \Delta _{F}\subset \mathbb {R} ^{M}}$ and the resulting topological space is the gluing ${\displaystyle \Delta _{E}\cup _{i}\Delta _{F}}$ Effectuating the gluing for each inclusion, one ends up with the desired topological space.

A 2-dimensional geometric simplicial complex with vertex V, link(V), and star(V) are highlighted in red and pink.

As in the previous construction, by the topology induced by gluing, the closed sets in this space are the subsets being closed in the subspace topology of each simplex ${\displaystyle \Delta _{F}}$.

The simplicial complex ${\displaystyle {\mathcal {T_{n}}}}$, which consists of all simplices ${\displaystyle {\mathcal {T}}}$ of dimension ${\displaystyle \leq n}$ is called the ${\displaystyle n}$-th skeleton of ${\displaystyle {\mathcal {T}}}$.

A natural neighborhood of a vertex ${\displaystyle V}$ of a simplicial complex ${\displaystyle {\mathcal {S}}}$ is considered to be the star ${\displaystyle star(K)={\Big \{}L\in {\mathcal {S}}\;|\;K\subset L{\Big \}}}$ of a simplex, its boundary is the link

${\displaystyle lk(K)={\Big \{}M\in {\mathcal {S}}\;|\;M\cap K=\emptyset {\Big \}}}$.

Simplicial maps

The maps considered in this category are simplicial maps: Let ${\displaystyle {\mathcal {K}}}$, ${\displaystyle {\mathcal {L}}}$ be abstract simplicial complexes above sets ${\displaystyle V_{K}}$, ${\displaystyle V_{L}}$. A simplicial map is a function ${\displaystyle f:V_{K}\rightarrow V_{L}}$ which maps each simplex in ${\displaystyle {\mathcal {K}}}$ onto a simplex in ${\displaystyle {\mathcal {L}}}$. By affine-linear extension on the simplices, ${\displaystyle f}$ induces a map between the geometric realizations of the complexes.^[2]

Examples

• Let ${\displaystyle W=\{a,b,c,d,e,f\}}$ and let ${\displaystyle {\mathcal {T}}={\Big \{}\{a\},\{b\},\{c\},\{d\},\{e\},\{f\},\{a,b\},\{a,c\},\{a,d\},\{a,e\},\{a,f\}{\Big \}}}$. The associated geometric complex is a star with center ${\displaystyle \{a\}}$.
• Let ${\displaystyle V=\{A,B,C,D\}}$ and let ${\displaystyle {\mathcal {S}}={\mathcal {P}}(V)}$. Its geometric realization ${\displaystyle |{\mathcal {S}}|}$ is a tetrahedron.
• Let ${\displaystyle V}$ as above and let ${\displaystyle {\mathcal {S}}'=\;{\mathcal {P}}({\mathcal {V}})\setminus \{A,B,C,D\}}$. The geometric simplicial complex is the boundary of a tetrahedron ${\displaystyle |{\mathcal {S'}}|=\partial |{\mathcal {S}}|}$.

Examples

• Simplicial complexes can be triangulated by identity.
• Let ${\displaystyle {\mathcal {S}},{\mathcal {S'}}}$ be as in the examples seen above. The closed unit ball ${\displaystyle \mathbb {D} ^{3}}$ is homeomorphic to a tetrahedron so it admits a triangulation, namely the homeomorphism ${\displaystyle t:|{\mathcal {S}}|\rightarrow \mathbb {D} ^{3}}$. Restricting ${\displaystyle t}$ to ${\displaystyle |{\mathcal {S}}'|}$ yields a homeomorphism ${\displaystyle t':|{\mathcal {S}}'|\rightarrow \mathbb {S} ^{2}}$.

The 2-dimensional sphere and a triangulation

• The torus ${\displaystyle \mathbb {T} ^{2}=\mathbb {S} ^{1}\times \mathbb {S} ^{1}}$ admits a triangulation. To see this, consider the torus as a square where the parallel faces are glued together. This square can be triangulated as shown below:

A two dimensional torus, represented as the gluing of a square via the map g, identifying its opposite sites

• The projective plane ${\displaystyle \mathbb {P} ^{2}}$ admits a triangulation (see CW-complexes)
• One can show that differentiable manifolds admit triangulations.^[3]

Homology

Via triangulation, one can assign a chain complex to topological spaces that arise from its simplicial complex and compute its simplicial homology. Compact spaces always admit finite triangulations and therefore their homology groups are finitely generated and only finitely many of them do not vanish. Other data as Betti- Numbers or Euler characteristic can be derived from homology.

Topological invariance

To use these invariants for the classification of topological spaces up to homeomorphism one needs invariance of the characteristics regarding homeomorphism.

A famous approach to the question was at the beginning of the 20th century the attempt to show that any two triangulations of the same topological space admit a common subdivision. This assumption is known as Hauptvermutung ( German: Main assumption). Let ${\displaystyle |{\mathcal {L}}|\subset \mathbb {R} ^{N}}$ be a simplicial complex. A complex ${\displaystyle |{\mathcal {L'}}|\subset \mathbb {R} ^{N}}$ is said to be a subdivision of ${\displaystyle {\mathcal {L}}}$ iff:

• every simplex of ${\displaystyle {\mathcal {L'}}}$ is contained in a simplex of ${\displaystyle {\mathcal {L}}}$ and
• every simplex of ${\displaystyle {\mathcal {L}}}$ is a finite union of simplices in ${\displaystyle {\mathcal {L'}}}$ .^[2]

Those conditions ensure that subdivisions does not change the simplicial complex as a set or as a topological space. A map ${\displaystyle f:{\mathcal {K}}\rightarrow {\mathcal {L}}}$ between simplicial complexes is said to be piecewise linear if there is a refinement ${\displaystyle {\mathcal {K'}}}$ of ${\displaystyle {\mathcal {K}}}$ such that ${\displaystyle f}$ is piecewise linear on each simplex of ${\displaystyle {\mathcal {K}}}$. Two complexes that correspond to another via piecewise linear bijection are said to be combinatorial isomorphic. In particular, two complexes that have a common refinement are combinatorially equivalent. Homology groups are invariant to combinatorial equivalence and therefore the Hauptvermutung would give the topological invariance of simplicial homology groups. In 1918, Alexander introduced the concept of singular homology. Henceforth, most of the invariants arising from triangulation were replaced by invariants arising from singular homology. For those new invariants, it can be shown that they were invariant regarding homeomorphism and even regarding homotopy equivalence.^[6] Furthermore it was shown that singular and simplicial homology groups coincide.^[6] This workaround has shown the invariance of the data to homeomorphism. Hauptvermutung lost in importance but it was initial for a new branch in topology: The piecewise linear (short PL- topology) topology examines topological properties of topological spaces.^[7]

Reidemeister-Torsion

To disprove the Hauptvermutung it is helpful to use combinatorial invariants which are not topological invariants. A famous example is Reidemeister-Torsion. It can be assigned to a tuple ${\displaystyle (K,L)}$ of CW- complexes: If ${\displaystyle L=\emptyset }$ this characteristic will be a topological invariant but if ${\displaystyle L\neq \emptyset }$ in general not. An approach to Hauptvermutung was to find homeomorphic spaces with different values of Reidemeister-Torsion. This invariant was used initially to classify lens- spaces and first counterexamples to the Hauptvermutung were built based on lens- spaces:^[8]

Classification of lens- spaces

In its original formulation, Lens spaces are 3-manifolds, constructed as quotient spaces of the 3-sphere: Let ${\displaystyle p,q}$ be natural numbers, such that ${\displaystyle p,q}$ are coprime . The lens space ${\displaystyle L(p,q)}$ is defined to be the orbit space of the free group action

${\displaystyle \mathbb {Z} /p\mathbb {Z} \times S^{3}\to S^{3}}$

${\displaystyle (k,(z_{1},z_{2}))\mapsto (z_{1}\cdot e^{2\pi ik/p},z_{2}\cdot e^{2\pi ikq/p})}$.

For different tuples ${\displaystyle (p,q)}$, Lens spaces will be homotopy- equivalent but not homeomorphic. Therefore they can't be distinguished with the help of classical invariants as the fundamental group but by the use of Reidemeister-Torsion.

Two Lens spaces ${\displaystyle L(p,q_{1}),L(p,q_{2})}$are homeomorphic, if and only if ${\displaystyle q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}}$.^[9] This is the case iff two Lens spaces are simple-homotopy-equivalent. The fact can be used to construct counterexamples for the Hauptvermutung as follows. Suppose there are spaces ${\displaystyle L'_{1},L'_{2}}$ derived from non-homeomorphic Lens spaces ${\displaystyle L(p,q_{1}),L(p,q_{2})}$having different Reidemeister torsion. Suppose further that the modification into ${\displaystyle L'_{1},L'_{2}}$ does not affect Reidemeister torsion but such that after modification ${\displaystyle L'_{1}}$ and ${\displaystyle L'_{2}}$ are homeomorphic. The resulting spaces will disprove the Hauptvermutung.

Pachner Moves

One Pachner-move replaces two tetrahedra by three tetrahedra

Pachner moves are a way to manipulate triangulations: Let ${\displaystyle {\mathcal {S}}}$ be a simplicial complex. For two simplices ${\displaystyle K,L}$ the Join

${\displaystyle K*L={\Big \{}tk+(1-t)l\;|\;k\in K,l\in L\;t\in [0,1]{\Big \}}}$ are the points lying on straights between points in ${\displaystyle K}$ and in ${\displaystyle L}$. Choose ${\displaystyle S\in {\mathcal {S}}}$ such that ${\displaystyle lk(S)=\partial K}$ for any ${\displaystyle K}$ lying not in ${\displaystyle {\mathcal {S}}}$. A new complex ${\displaystyle {\mathcal {S'}}}$, can be obtained by replacing ${\displaystyle S*\partial K}$ by ${\displaystyle \partial S*K}$. This replacement is called a Pachner move. The theorem of Pachner states that whenever two triangulated manifolds are PL- equivalent, there is a series of Pachner moves transforming both into another.^[17]

Classification of manifolds

By triangulating 1-dimensional manifolds, one can show that they are always homeomorphic to disjoint copies of the real line and the unit sphere ${\displaystyle \mathbb {S} ^{1}}$. Moreover, surfaces, i.e. 2-manifolds, can be classified completely: Let ${\displaystyle S}$ be a compact surface.

• If ${\displaystyle S}$ is orientable, it is homeomorphic to a 2-sphere with ${\displaystyle n}$ tori of dimension ${\displaystyle 2}$ attached, for some ${\displaystyle n\geq 0}$.
• If ${\displaystyle S}$ is not orientable, it is homeomorphic to a Klein Bottle with ${\displaystyle n}$ tori of dimension ${\displaystyle 2}$ attached, for some ${\displaystyle n\geq 0}$.

To prove this theorem one constructs a fundamental polygon of the surface: This can be done by using the simplicial structure obtained by the triangulation.^[19]

Maps on simplicial complexes

Giving spaces the structure of a simplicial structure might help to understand maps defined on the spaces. The maps can often be assumed to be simplicial maps via the simplicial approximation theorem: