Geometric Simplicial Complexes

Definition 2.36. For a geometric simplicial complex K, let K denote the union of all simplices of K. The topology on K is defined as follows every simplex a of K has the induced topology, and in general A C K is open if and only if AC a is open in a, for all simplices a K (equivalently, the word open could he replaced with the word closed ). 

In general, a geometric realization of an abstract simplicial complex is a geometric simplicial complex, while the combinatorial incidence structure of the geometric simplicial complex will give an abstract one. 

The notions of a subcomplex and simplicial maps are defined for the geometric simplicial complexes in full analogy with the abstract simpficial complexes. These notions then coincide under the described correspondence between the two families of simplicial complexes. Note that when two complexes K and L have dimension less than 2, i.e., can be viewed as graphs, the graph homomorphisms (see Definition 9.20) between K and L are simplicial maps, but not vice versa. 

Proposition 2.37. Let L he a suhcomplex of a geometric simplicial complex K then L is a closed suhspace of K. ... 

Most of the terminology, such as skeleton, subcomplex, join, carries over from the simplicial situation. One new property worth observing is that a direct product of two geometric polyhedral complexes is again a geometric polyhedral complex, whereas the same is not true for the geometric simplicial complexes. 

A geometric simplicial p-complex k(p) is a finite set of disjoint q-simplices of the n-dimensional Euclidean space "E, where q = 0, I,. .. p, such that, if... 

Until now, we have restricted ourselves to considering the finite abstract simplicial complexes. The natural question is, what happens if we drop the condition that the ground set should be finite. We invite the reader to check that all the definitions make sense, and that all the statements hold just as they do for the finite ones. For reasons that will become clear once we look at the notion of geometric realization, we keep the condition that the simplices have finite cardinality. We restate Definition 2.1 for future reference. 

It is now time to describe the geometric picture that is encoded by the combinatorial data of an abstract simplicial complex. 

Definition 2.27. Given a finite abstract simplicial complex A, we define its standard geometric realization to be the topological space obtained by taking the union of standard a-simplices in for all a A. 

Very often, for the sake of brevity, in case no confusion arises, we shall talk about the topological properties of the abstract simplicial complex A, always having in mind the properties of the topological space. 4. We note that the geometric realizations of the void and of the empty complexes are both empty sets. 

Given a nonempty abstract simplicial complex A, the constructive definition of the geometric realization of A goes as follows ... 

Yet another alternative to define the geometric realization of an abstract simplicial complex would be to give a direct description of the set of points together with topology. This is what we do next. 

The crucial observation is that the points of the geometric realization of an abstract simplicial complex A are in 1-to-l correspondence with the set of all convex combinations whose support is a simplex of A. In fact, the support of each convex combination tells us precisely to which simplex it belongs. 

Intuitively, one can say that the points that are near to a point in the geometric realization can be obtained by a small deformation of the coefficients of the corresponding convex combination. If the deformation is sufficiently small, then the nonzero coefficients will stay positive. However, even under a very small deformation it may happen that the zero coefficients become nonzero. This is allowed as long as the support set remains a simplex of the initial abstract simplicial complex. Geometrically, this corresponds to entering the interior of an adjacent higher-dimensional cell. An illustration is provided in Figure 2.1. 

It follows that an automorphism of an abstract simplicial complex induces a continuous automorphism of its geometric realization, and that the geometric realizations of two isomorphic abstract simplicial complexes are homeo-morphic, with homeomorphisms induced by the isomorphism maps. On the other hand, the existence of a homeomorphism between Z i and A2 does not imply the isomorphism of A and. 2. If a topological space allows a simplicial structure, then it allows infinitely many nonisomorphic simplicial structures. 

Since the open star of a simplex is not an abstract simplicial complex, one cannot take its geometric realization. However, sometimes one instead considers the open subspace of. 4 given by the union of the interiors of the geometric simplices corresponding to the simplices in the open star, where again the interior of a vertex is taken to be the vertex itself. 

The geometric realizations of the abstract simplicial complexes Bdzl and A are related in a fundamental way. 

Proof. The explicit point description of the geometric realization of an abstract simplicial complex tells us that the points of 2i are indexed by convex combinations aivi - - + agVs such that vi,v, .v C A, whereas the points of BdZi are indexed by convex combinations feioi + + btcrt such... 

It follows immediately from our description of topology on geometric realizations of abstract simplicial complexes that the maps / and g are continuous. We leave it as an exercise for the reader to verify that these two maps are actually inverses of each other. ... 

When the abstract simplicial complex A is finite, then we can take its standard geometric realization and take the barycentric subdivisions of the individual n-simplices as just described. This gives the geometric realization of the barycentric subdivision of A. When, on the other hand, the abstract simplicial complex A is infinite, we can take the barycentric subdivisions of its simplices before the gluing and then observe that the gluing process is compatible with the new cell structure hence we will obtain the geometric realization of the barycentric subdivision of A as well. 

Since the barycentric subdivision of the generahzed simplicial complex is a geometric realization of an abstract simplicial complex, we can be sure that after taking the barycentric subdivision twice, the trisp will turn into the geometric realization of an abstract simplicial complex. On the other hand, with many trisps, taking the barycentric subdivision once would not suffice for that purpose. 

Finally, let us consider the case C = ASC. On the one hand, we see already with the example of two intervals that a topological direct product of the geometric realizations of two abstract simplicial complexes does not have an a priori given simplicial structure. On the other hand, it may well be that there is a categorical product that is different from the topological one. 

For example, an action on an abstract simplicial complex induces an action on the underlying topological space this is a composition with the geometric realization functor. An action on a poset P induces an action on the order complex A P), which is defined in Chapter 9 this is a composition with the order complex functor. Furthermore, it induces a G-action on any given homology group Hi A P)] TZ), which, in case 7 . is a field, is the same as a linear representation of G over TZ. 

Proposition 6.18. LetX be the geometric realization of an arbitrary abstract simplicial complex. Then there exists a formal deformation from X to BdX. 

A number of abstract simplicial complexes used in discrete and computational geometry, as well as in computational topology have gained prominence in the recent years. These complexes are defined starting from some data in a metric space. Strictly speaking, in current applications the input data is not combinatorial, but rather geometric. However, due to a profusion of metric spaces in combinatorial contexts, we include the definitions of these families of abstract simplicial complexes. 

Mil57j J.W. Mihior, The geometric realization of semi-simplicial complex, Ann. of Math. 65 (1957), 357 362. 

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