Simplicial complex

A simplicial complex is a term used in algebraic topology . In a simplicial is a purely combinatorial writable object by means of which the key properties of certain, as triangulated called topological spaces can be algebraically characterized. In particular, simplicial complexes are used to define invariants for the underlying topological space .

The idea of ​​the simplicial complex is to investigate a topological space in that - if possible - a set is constructed in the d- dimensional Euclidean space by joining simplices , which is homeomorphic to the given topological space. The “instructions for assembling” the simplices, that is, the information about how the simplices are put together, is then characterized purely algebraically in the form of a sequence of group homomorphisms .

Definitions

Abstract simplicial complex

A three-dimensional simplicial complex

An abstract simplex is a finite non-empty set. An element of an abstract simplex is called a corner of , a non-empty subset of is again an abstract simplex and is called a facet (or side ) of . ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

An abstract or combinatorial simplicial is a set of simplices with the property that every facet of a simplex again heard so . The union of all the corners of simplices of the simplicial will vertex set or Eckpunktbereich called and called. ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \sigma '\subseteq \sigma }$ ${\displaystyle \sigma \in {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \sigma '\in {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle V({\mathcal {K}})}$

The dimension of an abstract simplex containing corners is defined as, and the dimension of the simplicial complex is defined as the maximum of the dimension of all simplices. If the dimension of the simplices is not restricted, then is called infinite-dimensional. ${\displaystyle k+1}$ ${\displaystyle k}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$

The simplicial complex is called finite if it is a finite set, and locally finite if every vertex only belongs to a finite number of simplices. ${\displaystyle {\mathcal {K}}}$

The -Skeleton of a simplicial complex is the set of all its simplices of the dimension . ${\displaystyle n}$ ${\displaystyle {\mathcal {K}}_{n}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \leq n}$

Geometrical simplicial complex

A geometric simplicial complex is a set of simplices in a Euclidean space having the property that each facet of a simplex again belongs, and that for all the simplices of the average of either empty or a common facet of and is. The union of all simplices of the geometric complex is designated with. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \mathbb {R} ^{d}}$ ${\displaystyle \sigma '\subseteq \sigma }$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \sigma ,\tau \in {\mathcal {S}}}$ ${\displaystyle \sigma \cap \tau }$ ${\displaystyle \sigma }$ ${\displaystyle \tau }$ ${\displaystyle |{\mathcal {S}}|}$

Geometric realization

A geometrical simplicial complex , the corners of which correspond to a given abstract simplicial complex , is called a geometrical realization of the simplicial complex . It is designated with . All geometric realizations of an abstract simplicial complex are homeomorphic to one another . ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle \vert {\mathcal {K}}\vert }$

At one point there is a clear simplex that lies inside . This simplex is referred to as the carrier simplex of . ${\displaystyle x\in \vert {\mathcal {K}}\vert }$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle x}$ ${\displaystyle x}$

Triangulation

A topological space is called triangulable if it is homeomorphic to a geometrical simplicial complex.

Simplicial illustrations

A simplicial mapping is a mapping between the sets of corners , in which for each simplex its corners are mapped to the corners of a simplex in below the mapping . ${\displaystyle f\colon {\mathcal {K}}\to {\mathcal {L}}}$ ${\displaystyle f\colon V({\mathcal {K}})\to V({\mathcal {L}})}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {L}}}$

A simplicial mapping induces a continuous mapping . For this purpose, an affine linear continuation is constructed inside each geometric simplex . ${\displaystyle f\colon {\mathcal {K}}\to {\mathcal {L}}}$ ${\displaystyle \vert f\vert \colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }$

Conversely, a continuous mapping according to a finite number of barycentric subdivisions can be approximated by a simplicial mapping , see simplicial approximation theorem . Here stands for the barycentric subdivision. ${\displaystyle g\colon \vert {\mathcal {K}}\vert \to \vert {\mathcal {L}}\vert }$ ${\displaystyle f\colon \operatorname {Bd} ^{m}({\mathcal {K}})\to {\mathcal {L}}}$ ${\displaystyle \operatorname {Bd} }$

A simplicial mapping that is bijective , i.e. the inverse mapping is also a simplicial mapping, is called a simplicial isomorphism .

The Simplizialkomplex as a chain complex

Let be a finite simplicial complex. The -th simplicial group of is the free Abelian group , which is generated by the set of simplices with dimension , it is noted with . The elements of the group are called simplicial chains. If one chooses a total order for all vertices that lie in any simplex of , one also obtains an order for each individual simplex through restriction . A boundary operator is then defined by ${\displaystyle {\mathcal {K}}}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle p}$ ${\displaystyle C_{p}^{\Delta }({\mathcal {K}})}$ ${\displaystyle p}$ ${\displaystyle {\mathcal {K}}}$ ${\displaystyle p}$ ${\displaystyle \partial \colon C_{p}^{\Delta }({\mathcal {K}})\to C_{p-1}^{\Delta }({\mathcal {K}})}$

${\displaystyle \partial (\langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle ):=\sum _{i=0}^{p}(-1)^{i}\langle v_{k_{0}},\ldots ,v_{k_{i-1}},v_{k_{i+1}},\ldots ,v_{k_{p}}\rangle ,}$

where the group element generated from the corners means. The boundary operator applies to all simplicial chains . Therefore it is a chain complex and homology can be explained on this in the usual way . This homology is called simplicial homology . ${\displaystyle \langle v_{k_{0}},\ldots ,v_{k_{p}}\rangle }$ ${\displaystyle \partial (\partial c)=0}$ ${\displaystyle p}$ ${\displaystyle c}$ ${\displaystyle (C_{p}^{\Delta }({\mathcal {K}}),\partial )}$

history

Triangulations and an equivalent formulated in matrix notation for the chain complex formed from them were investigated by Henri Poincaré towards the end of the nineteenth century. Simplicial mapping was first used by Brouwer in 1912 . In the 1920s, the perspective that led to the concept of the chain complex emerged.

See also

• Simplicial amount
• Simplicial homology
• CW complex - a more general term that includes simplicial complexes as a special case.
• Flag complex

Individual evidence

1. ↑ H. Hopf , P. Alexandroff : Topologie , Berlin, 1935, p. 158 ( online )
2. ↑ H. Hopf, P. Alexandroff: Topologie , Berlin, 1935, p. 172 ( online )
3. ^ Jean Dieudonné: A History of Algebraic and Differential Topology 1900-1960 , pp. 4–6, Boston 1989, Reprint 2009, ISBN 978-0-8176-4906-7 , doi : 10.1007 / 978-0-8176-4907- 4th

swell

• John M. Lee: Introduction to Topological Manifolds. Springer-Verlag, New York NY et al. 2000, ISBN 0-387-98759-2 ( Graduate Texts in Mathematics 202), pages 96, 323-324
• SN Malygin & MM Postnikov: Simplicial complex . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Web links

• Jörg Bewersdorff : Algebraic topology and fixed points . Introductory overview article (PDF file; 179 kB).
• Jie Wu: Lecture Notes on Algebraic Topology: Simplicial Complexes (PDF; 713 kB).