Group simply connected

A continuous connected group may be simply connected or multiply connected, depending on the topology of the parameter space. A subset of the euclidean space Sn is said to be k-fold connected if there are precisely k distinct paths connecting any two points of the subset which cannot be brought into each other by continuous deformation without going outside the subset. A schematic of four-fold connected space is shown in the lower diagram. 

In this section, We assume X is simply-connected for simplicity. Let NS(X) be the Neron-Severi group of X. By the assumption this is a hnitely generated free abelian group. The intersection form dehnes a non-degenerate symmetric bilinear form, which we denote by ( , ). The Hodge index theorem (see e.g., [5]) says that its index is (1, n). 

We will be concerned in this article with the non-simply connected vacuum described by the group 0(3), the rotation group. The latter is defined [6] as follows. Consider a spatial rotation in three dimensions of the form... 

The group space of 0(3) is doubly connected (i.e., non-simply connected) and can therefore support an Aharonov-Bohm effect (Section V), which is described by a physical inhomogeneous term produced by a rotation in the internal gauge space of 0(3) [24]. The existence of the Aharonov-Bohm effect is therefore clear evidence for an extended electrodynamics such as 0(3) electrodynamics, as argued already. A great deal more evidence is reviewed in this article in favor of 0(3) over U(l). For example, it is shown that the Sagnac effect [25] can be described accurately with 0(3), while U(l) fails completely to describe it. 

The 0(3) group is homomorphic with the SU(2) group, that of 2 x 2 unitary matrices with unit determinant [6]. It is well known that there is a two to one mapping of the elements of SU(2) onto those of 0(3). However, the group space of SU(2) is simply connected in the vacuum, and so it cannot support an Aharonov-Bohm effect or physical potentials. It has to be modified [26] to SU(2)/Z2 SO(3). 

It is only on this level that the link between helicity and topological quantization [103] can be understood properly. The 0(3) group, like the U(l) group, is multiply connected. The group space of U(l) is a circle [6, p. 105]. As explained earlier in this review, this is not simply connected because a path that goes twice... 

Exercise 4.28 (For topology students) Show that the group SO(3) is not simply connected. 

In Mi and M2, the closed path is homotopic to a null path. In Mi, this cycle is the boundary of a face, while in M2, the closed path is the boundary of all faces put together. More generally, a plane graph and a finite plane graph minus a face are simply connected. But the closed path in M3 is not homotopic to a null path. Actually, this closed path is a generator of the fundamental group Jti(Mf) — Z. 

The group T (l, m, n) can be realized as a group of isometries of a simply connected surface X of constant curvature, where ... 

Dre87] A. W. M. Dress, Presentations of discrete groups, acting on simply connected manifolds, in terms of parametrized systems of Coxeter matrices — a systematic approach, Advances in Mathematics 63-2 (1987) 196-212. 

Skinny molecular range, [af, a< ) af is defined above, whereas is the maximum threshold at and below which all locally nonconvex domains on the surface of density domains are simply connected. In simpler terms, in the skinny molecular range all nuclei are found within a single density domain, but there are formal "neck regions on the surface of density domains. In the terminology of shape group analysis [2], rings of D) type can be found on the surface of density domains. 

The helicenic simply connected polyhexes (helicenes) have also been considered separately. Their small numbers for h = 6 and 7 (cf. Table 8) are obtainable from a study of the illustrations in Balaban and Harary [13] and Balaban [30]. But all the entries of Table 8 for h < 10 have been given by the Diisseldorf-Zagreb group [16,17],... 

Suppose a methylene group were connected to two of the substituent groups from Table 6.2. Could you predict the chemical shift of the methylene hydrogens If the two substituents (X and Y) exert their (de)shielding effects independently, then perhaps the chemical shift of the methylene group could be calculated by simply adding the substituent parameters of both substituents to the chemical shift of methane ... 

FVom the proof of Theorem 1.2.2, we see that the local Hamiltonian properties of the field v are equivalent to the closedness of the differential 1-form a = 5 —Yidpi + Xidgi on the manifold Af. For a field to be globally Hamiltonian, it is sufficient that this form be exact. For instance, this will always be the case on (the Poincari /emma). If, however, a symplectic manifold is not simply-connected, then closed but not exact 1-forms may exist on it. This will be so if a group of one-dimensional cohomologies (Af, R) is nonzero. In both our examples, we deal with a nonzero group Ar (Af, R), namely... 

Proposition 1.2.2. Let a symplectic manifold M have a zero Brst group of real cohomologies JI M,R) (for instance, this will always be the case with a simply-connected manifold). Then any locally Hamiltonian vector held on the manifold will be at the same time globally Hamiltonian. 

If a manifold is simply-connected, then we always have H (M,R) = 0. The converse is, generally, not true. There exist non-simply-connected manifolds for which H (M, R) = 0. In fact, in these cases, the fundamental group xi(Af) of the manifold M "is not very large in the sense that is factor by the commutant, ie., the group z i/Itti, 7Ti] (which coincides with the one-dimensional group of integer-valued homologies Hi(M, Z)) has no infinite-order elements. 

Corollary 6.33. A simply connected CW complex X whose homology groups Hi X Z) are all trivial is contractible. 

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