Topology in Molecular Biology
172 V.M. Buchstaber Fig. 9.8.Mμ^2 =1:realprojectiveplaneRP^2 Fig. 9.9.Mμ^2 =2: Klein bottleK^2 M1 # M2 Fig. 9.10.Connected sum 9 ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 173 f 1 (K)≤ ( m 2 ) . For 2-dim triangulations we obtain: 6(m ...
174 V.M. Buchstaber For eachUαthere is fixed a homeomorphismφα:Uα→Rnproviding thelocal coordinatesx^1 α,...,xnαinUα. Therefore, ...
9 Euler, Dehn–Sommerville Characteristics, and Their Applications 175 vector at a pointx∈Mcan be written as ann-tuple (ξαj). The ...
176 V.M. Buchstaber An (autonomous)dynamic systemon a manifoldMis a smooth vector field ξonM. In terms of the local coordinates{ ...
10 Hopf Fibration and Its Applications M. Monastyrsky Summary.In this chapter we deal with Hopf fibration – one of the key exam- ...
178 M. Monastyrsky Fori≥3wehave 0 →πi(S^3 )→πi(S^2 )→ 0. In particularπ 3 (S^2 )=Z. The homotopy classes of mapsS^3 →S^2 are cha ...
10 Hopf Fibration and Its Applications 179 (See the proof in [1] or [2].) We now investigate different properties of Hopf fibrat ...
180 M. Monastyrsky In the nonorientable case Ind(P◦Q) is defined as the residue module 2 of the number of intersection points. P ...
10 Hopf Fibration and Its Applications 181 h(f 0 )=h(f 1 ) it suffices to show that the deformationft:S^2 n−^1 ×I → Sn connectin ...
182 M. Monastyrsky 10.3.1 Generalized Linking Number Consider the following problem: What kind of topological invariants would m ...
10 Hopf Fibration and Its Applications 183 ∫ B 1 u 1 ∧u 2 =− ∫ B 2 u 2 ∧u 1 =k(l 1 ,l 2 ). (10.9) Let us write the linking coeff ...
184 M. Monastyrsky Proposition 7.The integrals ∫ B 1 u ̃ 123 =− ∫ B 2 u ̃ 123 = ∫ S^3 ̃v′ 123 =k 2 (l) (10.11) have integer valu ...
10 Hopf Fibration and Its Applications 185 Proposition 8 (Formula Cˇalugˇareanu). k(γ,γv)=tw+Wr. (10.13) It is useful to compare ...
186 M. Monastyrsky For arbitrary genus it is extremely difficult and remains an unsolved prob- lem. There exist some partial res ...
10 Hopf Fibration and Its Applications 187 The article [13] has shown that there are infinitely many simple closed curves onS^2 ...
188 M. Monastyrsky W. Blaschke, G. Thomsen,Vorlesungen ̈uber Differential geometry III(Springer, Berlin Heidelberg New York, 19 ...
11 Multi-Valued Functionals, One-Forms and Deformed de Rham Complex ∗ D.V. Millionschikov Summary.We discuss some applications o ...
190 D.V. Millionschikov where byHtωp(M,R) we denote thep-th cohomology group of the de Rham complex (Λ∗(M),d+tω) with respect to ...
11 One-Forms and Deformed de Rham Complex 191 Sα(γ)= ∫ γ ( 1 2 gijx ̇ix ̇j−U+eAαkx ̇k ) dt, (11.1) where x^1 =θ, x^2 =φ, F 12 dθ ...
«
4
5
6
7
8
9
10
11
12
13
»
Free download pdf