Simplicial complex
simplicial scheme, abstract simplicial complex
A set, whose elements are called vertices, in which a family of finite non-empty subsets, called simplexes or simplices, is distinguished, such that every non-empty subset of a simplex is a simplex, called a face of
, and every one-element subset is a simplex.
A simplex is called -dimensional if it consists of
vertices. The maximal dimension of its simplices (which may be infinite) is called the dimension
of a simplicial complex
. A simplicial complex is called locally finite if each of its vertices belongs to only finitely many simplices. A simplicial complex is called ordered if its vertices admit a partial ordering that is linear on every simplex.
Example. Let be a set and let
be a family of non-empty subsets of
. A non-empty finite subset
is called a simplex if the set
is non-empty. The resulting simplicial complex
is called the nerve of the family
(cf. Nerve of a family of sets).
A simplicial mapping of a simplicial complex into a simplicial complex
is a mapping
such that for every simplex
in
, its image
is a simplex in
. Simplicial complexes and their simplicial mappings form a category.
If a simplicial mapping is an inclusion, then
is called a simplicial subcomplex of
. All simplices of a simplicial complex
of dimension at most
form a simplicial subcomplex of
, which is written
and is called the
-dimensional (or
-) skeleton of
. A simplicial subcomplex
of a simplicial complex
is called full if every simplex in
whose vertices all belong to
is itself in
.
Every simplicial complex canonically determines a simplicial set
, whose simplices of dimension
are all
-tuples
of vertices of
with the property that there is a simplex
in
such that
for each
. The boundary operators
and the degeneracy operators
of
are given by the formulas
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where denotes the omission of the symbol beneath it. When
is ordered one can define a simplicial subset
, consisting of those simplices
for which
. The (co)homology groups of
are isomorphic to the (co)homology groups of
and called the (co)homology groups of
.
To every triangulation (simplicial space) corresponds a simplicial complex, whose vertices are the vertices of
and whose simplices are those non-empty finite sets of vertices which span a simplex in
. For every simplicial complex
there is a triangulation, unique up to an isomorphism, whose simplicial complex is
. It is called the geometric realization (or body, or geometric simplicial complex) of
, and is denoted by
. This yields the geometric model in the sense of Giever–Hu (see Simplicial set)
of the simplicial set
, and when
is ordered, the geometric model in the sense of Milnor
of the simplicial set
. The correspondence
is a covariant functor from the category of simplicial complexes to the category of cellular spaces. A topological space
homeomorphic to the body
of some simplicial complex
is called a polyhedron (or a triangulated space, cf. Polyhedron, abstract) and the pair
, where
is the homeomorphism, is called a triangulation of
.
The points of the topological space can be identified with the functions
for which the set
is a simplex in
and
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The number is called the
-th barycentric coordinate of
. The formula
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defines a metric on , but the corresponding metric topology is, in general, stronger than the original one. The set
equipped with this metric topology is written as
.
A simplicial complex is isomorphic to the nerve of the family of stars of vertices of the space
, that is, to the nerve of the family of open subsets
, where
.
The following statements are equivalent: 1) the simplicial complex is locally finite; 2) the space
is locally compact; 3)
; 4)
is metrizable; and 5)
satisfies the first axiom of countability. Moreover, the space
is separable (compact) if and only if
is at most countable (finite).
The cells of the complex are in one-to-one correspondence with the simplices of
, and the closure
of the cell corresponding to a simplex
is given by
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It is homeomorphic to the -dimensional closed ball, where
, so that the complex
is regular. In addition, each set
has a canonical linear (affine) structure, with respect to which it is isomorphic to the standard simplex
. Because of this, and the fact that
for all simplices
, it turns out that the space
can be mapped homeomorphically (can be imbedded) into
(where
may be transfinite), so that all closed cells
are (rectilinear) simplices. This means that the image of
in
is a simplicial space (a polyhedron), i.e. a union of closed simplices intersecting only on entire faces. This simplicial space is called a realization of the simplicial complex
in
.
A simplicial complex can only be realized in
for finite
when
is locally finite, at most countable and of finite dimension. Moreover, if
, then
can be realized in
. A simplicial complex consisting of
vertices every
-element subset of which is a simplex cannot be realized in
.
From any simplicial complex one can construct a new simplicial complex,
, whose vertices are the simplices of
and whose simplices are families
of simplices of
such that
.
is called the barycentric refinement (or subdivision) of
. The cellular spaces
and
are naturally homeomorphic (but not isomorphic). Under this homeomorphism, every vertex
of
(that is, the zero-dimensional cell corresponding to the vertex
of
) is mapped onto the centre of gravity (the barycentre) of the closed simplex
.
The simplicial complex is ordered in a natural way. If
is ordered, then the correspondence
(first vertex of
) defines a simplicial mapping
that preserves the ordering. It is called the canonical translation. Its geometric realization (which is a continuous mapping
) is homotopic to the natural homeomorphism
.
A simplicial mapping (or its geometric realization
) is called a simplicial approximation of a continuous mapping
if, for every point
, the point
belongs to the minimal closed simplex containing the point
, or, equivalently, if for every vertex
,
. The mappings
and
are homotopic.
The simplicial approximation theorem states that if a simplicial complex is finite, then for every continuous mapping
there is an integer
such that for all
there is a simplicial approximation
of
(regarded as a mapping
).
References
[1] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
[2] | P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) |
[3] | J.H.C. Whitehead, "Simplicial spaces, nuclei and $M$-groups" Proc. London Math. Soc. , 45 (1939) pp. 243–327 |
Comments
In the West, the concept described here is usually called an (abstract) simplicial complex; the term simplicial scheme would normally be understood to mean a simplicial object in the category of schemes (cf. Simplicial object in a category).
References
[a1] | C.R.F. Maunder, "Algebraic topology" , v. Nostrand (1972) |
[a2] | S. Lefshetz, "Topology" , Chelsea, reprint (1956) |
[a3] | K. Lamotke, "Semisimpliziale algebraische Topologie" , Springer (1968) |
Comments
A facet of an abstract simplicial complex is a maximal face. A complex is pure if all facets have the same dimension.
For a face $F$ of a simplicial complex $K$, we let $F^\Delta$ denote all faces contained in $F$. A shelling is a linear order $\sqsubseteq$ on the facets of $K$, such that for a facet $F$, $$ \bigcup_{G \sqsubset F} G^\Delta \cap F^\Delta $$ is a subcomplex generated by the codimension 1 faces of $F$. A complex is shellable if it is pure and possesses a shelling (some authors omit the requirement to be pure). If a complex is shellable then its face ring is Cohen–Macaulay.
References
[b1] | Ezra Miller, Bernd Sturmfels, "Combinatorial commutative algebra" Graduate Texts in Mathematics 227 Springer (2005) ISBN 0-387-23707-0 Zbl 1090.13001 |
[b2] | Richard P. Stanley, "Combinatorics and commutative algebra" , (2nd ed.)mBirkhäuser (1996) ISBN 0-81764-369-9 Zbl 1157.13302 Zbl 0838.13008 |
Simplicial complex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simplicial_complex&oldid=54481