By Andrei Pajitnov
Within the early Nineteen Twenties M. Morse came across that the variety of serious issues of a gentle functionality on a manifold is heavily on the topic of the topology of the manifold. This grew to become a kick off point of the Morse idea that is now one of many easy elements of differential topology.
Circle-valued Morse conception originated from an issue in hydrodynamics studied by way of S. P. Novikov within the early Nineteen Eighties. these days, it's a always transforming into box of latest arithmetic with functions and connections to many geometrical difficulties similar to Arnold's conjecture within the idea of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta services, and the idea of knots and hyperlinks within the third-dimensional sphere.
the purpose of the ebook is to offer a scientific therapy of geometric foundations of the topic and up to date learn effects. The ebook is obtainable to first yr graduate scholars focusing on geometry and topology.
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Extra resources for Circle-Valued Morse Theory
If t, t ∈ R+ , ||w|| ||v|| + 1, |t − t| 2(||v|| + 1) , then (2) ||γ(x , t ; w) − γ(x , t; w)|| |t − t | · ||w|| For a vector ﬁeld w ∈ Vect1b (Rm ) satisfying ||w −v|| curve y(τ ) = γ(x , τ ; w) with τ ∈ [0, t] and is an α-approximate solution of the equation y (t) = h(y(t)) · v(y(t)). α . 2 1, every integral ||x − x || 1 Section 1. Manifolds without boundary 21 Indeed, ||y (t) − h(y(t)) · v(y(t))|| = ||w(y(t)) − h(y(t)) · v(y(t))|| α since the integral curve lies entirely in the domain where h equals 1.
J=k+1 This bilinear form is non-degenerate and its index equals k. The classical Morse lemma (, Lemma 4) asserts that the preceding example is essentially the only one. 6 (Morse lemma). Let p ∈ M be a non-degenerate critical point of a C ∞ function f : M → R, and let k = ind p. Then there is a chart Ψ : U → V deﬁned in a neighbourhood U of p such that (MC) Ψ(p) = 0, and f ◦ Ψ−1 = Qk + f (p). 7. A chart Ψ : U → V around p, which satisﬁes the condition (MC) is called a Morse chart for f at p.
Denote by Lξ the set of all closed 1-forms with de Rham cohomology class equal to ξ. Observe that Lξ is an aﬃne subspace of Ω1 (M ), and we endow Lξ with the induced topology. 24. Let ω be a closed 1-form on a manifold M , and X ⊂ M . We say, that ω is regular on X, or X-regular if every zero p ∈ X of ω is non-degenerate. 25. Let M be a manifold, and ξ ∈ H 1 (M, R). Let X ⊂ M be a compact subset. The subset of all X-regular 1-forms is open and dense in Lξ . Proof. We begin with the following lemma.