Geometric realization

The above analysis shows that the set of turns To Tn E, E-, i = 1,2,3, provides a geometric realization of the quaternion group and thus establishes the connection between the quaternion units and turns through rc/2, and hence rotations through % (binary rotations). This suggests that the whole set of turns might provide a geometric realization of the set of unit quaternions. Section 12.5 will not only prove this to be the case, but will also provide us with the correct parameterization of a rotation. 

It follows that a turn is the geometric realization of a normalized quaternion, that the addition of turns is the geometric realization of the multiplication of quaternions of unit norm, and that the group of turns is isomorphous with the group of normalized quaternions. Furthermore, eqs. (1) and (2) provide the correct parameterization of a normalized quaternion as... 

X is geometrically realizable An embedding of the atoms into 3D space exists, with X as POF. However, bond lengths, bond angles and torsion angles are not considered, i.e. the molecule is assumed to have unlimited flexibility. 

Tests excluding the chemically non-realizable but geometrically realizable abstract POFs have to be justified chemically. Depending on the problem given, a particular rule may or may not be acceptable. 

POFs Xo nd Xi correspond to the two enantiomers shown in Figure 4.3. POF X2 is not chemically realizable. However, it is geometrically realizable Any planar placement of the five atoms has the POF X2-b There are 3 = 729 abstract POFs for structure 4.2b, cyclohexane. Each of the 6 selected increasing quadruples can have the orientation plus, minus, or zero independently. However, cyclohexane has 12 graph automorphisms. Considering relabelings according to these permutations, some of the abstract POFs are isomor-... 

In some of the tests presented we made strong assumptions on the conformations allowed. However, one may not always want to accept such strong postulates. In this section, we present a special kind of rule that is valid for any geometrically realizable POF. 

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. 

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. 

Any topological space that is homeomorphic to the standard geometric realization of A is called the geometric realization of A, and is denoted by Z. ... 

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. 

It is possible to give a similar definition of the geometric realization for the infinite case. However, one would need a careful treatment of the resulting infinite-dimensional vector space and the topology involved. Instead, we use this as an opportunity to introduce a gluing process, which will be used to construct various classes of cell complexes. 

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. 

It is easy to check that (2.4) indeed defines a metric on our space hence one can take the topology induced by this metric. This is exactly the topology of the geometric realization. 

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. 

Fig. 2.1. Coordinate description of points of a geometric realization of an abstract simplicial complex.

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. 

Sometimes the embedding of the geometric realization in the ambient space is prescribed from the beginning and is of importance. For this reason, many texts in algebraic topology introduce the following concept. 

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. 

Example 10.5. Figure 10.2 shows the regular trisps that realize the nerves of previously considered acyclic categories. Note that in these examples, the nerves are not (geometric realizations of) abstract simphcial complexes. 

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

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