Polyhedral complex

Recent developments in the chemistry of polyhedral complexes derived from transition metals and carboranes... 

Munich, Germany, 1967 Plenary lectures Pure Appl. Chem. 17, 179-272 (1968) and Butterworth, London (1968). Of relevance to Main Group chemistry M. F. Hawthorne Recent advances in the chemistry of polyhedral complexes derived from transition metals and carboranes, pp. 195-210 (27). 

Although the silicon atom has the same outer electronic structure as carbon its chemistry shows very little resemblance to that of carbon. It is true that elementary silicon has the same crystal structure as one of the forms of carbon (diamond) and that some of its simpler compounds have formulae like those of carbon compounds, but there is seldom much similarity in chemical or physical properties. Since it is more electro-positive than carbon it forms compounds with many metals which have typical alloy structures (see the silicides, p. 789) and some of these have the same structures as the corresponding borides. In fact, silicon in many ways resembles boron more closely than carbon, though the formulae of the compounds are usually quite different. Some of these resemblances are mentioned at the beginning of the next chapter. Silicides have few properties in common with carbides but many with borides, for example, the formation of extended networks of linked Si (B) atoms, though on the other hand few silicides are actually isostructural with borides because Si is appreciably larger than B and does not form some of the polyhedral complexes which are peculiar to boron and are one of the least understood features of boron chemistry. 

Some induction schemes for ct, n, and 5 orbital basis sets on C y sites of polyhedral complexes are to be found in Appendix D. In addition to the Frobenius theorem, there is also a stronger result for induction theory based on the concept of a fiber bundle. This requires the coupling of representations and will be considered in Sect. 6.9. 

Table 3. Valence orbitals for a range of spherical or approximately spherical polyhedral complexes.

Definition 2.38. A geometric polyhedral complex P in R is a collection of convex polytopes in R such that... 

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. 

Definition 2.39. A topological space X is called a polyhedral complex if... 

An example of a polyhedral complex is shown on the left of Figure 2.3. Again, all the basic operations and terminology of the simphcial complex extend to the polyhedral ones. This includes the barycentric subdivision, since it can be done on the polyhedra before the gluing, and then one can observe that the obtained complexes will glue to a simphcial complex in a compatible way. 

Definition 2.40. Let X be a polyhedral complex, and let S he the set of some of its vertices. We let X[S ] denote the polyhedral complex that consists of all cells whose set of vertices is a subset of S. This complex is called the induced subcomplex. 

To start with, notice that every polyhedral complex is embeddable into... 

Once the complex is embedded into R", a usual way to define the link of a vertex v is to place a sphere of sufficiently small radius with its center in v. The intersection of this sphere with the complex is the link. To see that this is actually a polyhedral complex, it is enough to notice that up to face-preserving homeomorphism, the intersection of a small sphere with the polytope can be replaced by the intersection with a hyperplane. This hyperplane can be foimd as follows take the hyperplane H whose intersection with our poljdope is equal to the considered vertex, and consider the parallel translation of this hyperplane by a sufficiently small number in the direction of the poljdope. 

Finally, one can also define the link of an arbitrary face a of the polyhedral complex. To do that, take any point x in the interior of a. It has a small closed neighborhood that can be represented as a direct product S x Q, where d = dimo. S is a closed ball of dimension d, and Q is some polyhedral complex, which can be thought of as the transversal complex of a. The face a is replaced by the vertex x in Q, and we can take the link of x in Q. This is the link of a in our polyhedral complex. 

Definition 2.41. A polyhedral complex whose cells are simplices is called a generalized simplicial complex. ... 

This is a class of polyhedral complexes that contains the generalized simplicial complexes and is closed under direct products. An example of a prod-simphcial complex is shown on the right of Figure 2.3. 

The prodsimplicial complex Bip (G) is our first example of the so-called Hom(-, -)-construction, namely, as a polyhedral complex it is isomorphic to Horn K2, G). 

The simplicial collapse defined in Section 6.4 is a special case of Definition 11.12. We are now ready to formulate the central result of this section. For technical convenience, we restrict ourselves to considering cellular collapses in the setting of polyhedral complexes only. 

Remark 11. If - The converse of Theorem 11.13(a) is clearly true in the following sense if Ac is a subcomplex of A and if there exists a sequence of collapses from Z to. 4c, then the matching on the cells of 4 4c induced by this sequence of collapses is acyclic. In particular, a polyhedral complex 4 is collapsible if and only if the poset 1F A) 6 allows a complete acyclic matching. 

Theorem 18.3. For an arbitrary graph G, the neighborhood complex M G) and the polyhedral complex Bip (G) have the same simple homotopy type. 

The discussion in Section 6.5 implies now that the polyhedral complex of all bipartite subgraphs of G, Bip(G), and the neighborhood complex Af G), have the same simple homotopy type, and yields an explicit formal deformation between these two complexes. ... 

We note that for an arbitrary polyhedral complex K such that all faces of K are direct products of simplices, and a vertex x of K, the link of x, Ik Ca ), is a simplicial complex. This follows from the fact that a link of any vertex in a hypercube is a simplex, and the identity lk( xS)(v,w ) = Ik (v) IkB(w), for arbitrary polyhedral complexes A and B (compare with identities (10.2) and (10.14)). 

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