Ivancevic_Applied-Diff-Geom

314 Applied Differential Geometry: A Modern IntroductionIn geometrical situations it is often unrealistic to suppose that one cancalculate the index precisely, but as we shall see it is often possible to givenlower bounds for the index. As an example, note that if M is not simply–connected, then Ω mp is not connected. Each curve of minimal length inthe path components is a geodesic from m to p which is a local minimumfor the arc–length functional. Such geodesics evidently have index zero. Inparticular, if one can show that all geodesics, except for the minimal onesfrom m to p, have index > 0, then the manifold must be simply–connected.We will apply Morse theory in biodynamics/robotic in section (3.13.5.2)below.3.10.5.2 (Co)Bordism Theory on Smooth Manifolds(Co)bordism appeared as a revival of Poincaré’s unsuccessful 1895 attemptsto define homology using only manifolds. Smooth manifolds (withoutboundary) are again considered as ‘negligible’ when they are boundariesof smooth manifolds–with–boundary. But there is a big difference, whichkeeps definition of ‘addition’ of manifolds from running into the difficultiesencountered by Poincaré; it is now the disjoint union. The (unoriented)(co)bordism relation between two compact smooth manifolds M 1 , M 2 ofsame dimension n means that their disjoint union ∂W = M 1 ∪ M 2 is theboundary ∂W of an (n + 1)D smooth manifold–with–boundary W . Thisis an equivalence relation, and the classes for that relation of nD manifoldsform a commutative group N n in which every element has order 2. Thedirect sum N • = ⊕ n≥0 N n is a ring for the multiplication of classes deducedfrom the Cartesian product of manifolds.More precisely, a manifold M is said to be a (co)bordism from A to Bif exists a diffeomorphism from a disjoint sum, ϕ ∈ diff(A ∗ ∪ B, ∂M). Two(co)bordisms M(ϕ) and M ′ (ϕ ′ ) are equivalent if there is a Φ ∈ diff(M, M ′ )such that ϕ ′ = Φ ◦ ϕ. The equivalence class of (co)bordisms is denoted byM(A, B) ∈ Cob(A, B) [Stong (1968)].Composition c Cob of (co)bordisms comes from gluing of manifolds [Baezand Dolan (1995)]. Let ϕ ′ ∈ diff(C ∗ ∪ D, ∂N). One can glue (co)bordismM with N by identifying B with C ∗ , (ϕ ′ ) −1 ◦ ϕ ∈ diff(B, C ∗ ). We get theglued (co)bordism(M ◦ N)(A, D) and a semigroup operation,c(A, B, D) : Cob(A, B) × Cob(B, D) −→ Cob(A, D).A surgery is an operation of cutting a manifold M and gluing to cylin-