The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic.

Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.

De Rham showed that all of these approaches were interrelated and that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham cohomology.

A new approach uses a functor from filtered spaces to crossed complexes defined directly and homotopically using relative homotopy groups ; a higher homotopy van Kampen theorem proved for this functor enables basic results in algebraic topology, especially on the border between homology and homotopy, to be obtained without using singular homology or simplicial approximation.

Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory.

An example is the complex defining the homology theory of a ( finite ) simplicial complex.

The singular homology groups H < sub > n </ sub >( X ) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex X.

Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.

so chains form a chain complex, whose homology groups ( cycles modulo boundaries ) are called simplicial homology groups.

For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices.

It is possible to define abstract homotopy groups for simplicial sets.

In the case of a finite simplicial complex the homology groups H < sub > k </ sub >( X, Z ) are finitely-generated, and so has a finite rank.

The simplicial, singular, Čech and Alexander – Spanier groups are all defined by first constructing a chain complex or cochain complex, and then taking its homology.

Ultimately, this approximation technique is a standard ingredient in the proof that simplicial homology groups are topological invariants.

Analogues of the Tits alternative have been proved for many other classes of groups, such as fundamental groups of 2-dimensional simplicial complexes of non-positive curvature.

If X has a triangulation compatible with the stratification, then simplicial intersection homology groups can be defined in a similar way, and are naturally isomorphic to the singular intersection homology groups.

Bass – Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees.

Here H < sub > i </ sub > might be the simplicial homology or more general singular homology theory: the result itself is a pure piece of homological algebra about chain complexes of free abelian groups.

Letting C be the category of groups or category of abelian groups, we obtain the categories sGrp of simplicial groups and sAb of simplicial abelian groups, respectively.

Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integral sequence, where f < sub > i </ sub > is the number of ( i − 1 )- dimensional faces of Δ ( by convention, f < sub > 0 </ sub > = 1 unless Δ is the empty complex ).

The simplicial structure of the affine and spherical buildings associated to SL < sub > n </ sub >( Q < sub > p </ sub >), as well as their interconnections, are easy to explain directly using only concepts from elementary algebra and geometry ( see ).

For the affine group, an apartment is just the simplicial complex obtained from the standard tessellation of Euclidean space E < sup > n-1 </ sup > by equilateral ( n-1 )- simplices ; while for a spherical building it is the finite simplicial complex

Let F be a field and let X be the simplicial complex with vertices the non-trivial vector subspaces of V = F < sup > n </ sup >.

For simplicial complexes one defines the relative barycentric subdivision of modulo that consists of those simplixes with vertices associated to a sequence < math > S_0 <

In Δ < sup > op </ sup >, there are two particularly important classes of maps called face maps and degeneracy maps which capture the underlying combinatorial structure of simplicial sets.

By the Yoneda lemma, the n-simplices of a simplicial set X are classified by natural transformations in hom ( Δ < sup > n </ sup >, X ).< ref >

The singular set of a topological space Y is the simplicial set defined by S ( Y ): n → hom (| Δ < sup > n </ sup >|, Y ) for each object n ∈ Δ, with the obvious functoriality condition on the morphisms.

* If the form is E < sub > 8 </ sub >, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.

These points of intersection and their barycentres ( corresponding to higher dimensional simplices intersecting X ) generate an n-dimensional simplicial subcomplex in R < sup > m </ sup >, lying wholly inside the tubular neighbourhood.

An integral vector ( f < sub > 0 </ sub >, f < sub > 1 </ sub >, … f < sub > d − 1 </ sub >) is the f-vector of some ( d − 1 )- dimensional simplicial complex if and only if

# Δ < sub > f </ sub > is a simplicial complex.

In fact, for every two n-simplices intersecting in an ( n – 1 )- simplex or panel, there is a unique period two simplicial automorphism of A, called a reflection, carrying one n-simplex onto the other and fixing their common points.

Any simplicial automorphism φ of X defines a permutation π of Z / n Z such that label ( φ ( M ))

* If X is an n-dimensional simplicial complex such that every simplex is contained in an n-simplex and n − 1 simplex is contained in exactly two n-simplexes, then the underlying space of X is a topological pseudomanifold.

The two known counterexamples to the Dirac-Motzkin conjecture ( which states that any n-line arrangement has at least n / 2 ordinary points ) are both simplicial.

* A triangulation T of is a subdivision of into ( n + 1 )- dimensional simplices such that any two simplices in T intersect in a common face ( a simplex of any lower dimension ) or not at all, and any bounded set in intersects only finitely many simplices in T. That is, it is a locally finite simplicial complex that covers the entire space.

This is a special case of a simplicial space in which, for each n, the space of n-simplices is a manifold.

An n-dimensional building X is an abstract simplicial complex which is a union of subcomplexes A called apartments such that

Each building is a simplicial complex X which has to satisfy the following axioms:

Typically, the domain to be meshed is specified as a coarse simplicial complex ; for the mesh to be numerically stable, it must be refined, for instance by using Ruppert's algorithm.

When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.

* Abstract simplicial complex

The fundamental group of a ( finite ) simplicial complex does have a finite presentation.

In topology and combinatorics, it is common to “ glue together ” simplices to form a simplicial complex.

The associated combinatorial structure is called an abstract simplicial complex, in which context the word “ simplex ” simply means any finite set of vertices.

Face and facet can have different meanings when describing types of simplices in a simplicial complex.