By Andrei Pajitnov
Within the early Nineteen Twenties M. Morse chanced on that the variety of severe issues of a soft functionality on a manifold is heavily regarding the topology of the manifold. This grew to become a place to begin of the Morse thought that is now one of many uncomplicated elements of differential topology.
Circle-valued Morse thought originated from an issue in hydrodynamics studied via S. P. Novikov within the early Nineteen Eighties. these days, it's a consistently growing to be box of up to date arithmetic with functions and connections to many geometrical difficulties akin to Arnold's conjecture within the idea of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta services, and the speculation of knots and hyperlinks within the third-dimensional sphere.
the purpose of the e-book is to offer a scientific remedy of geometric foundations of the topic and up to date examine effects. The ebook is obtainable to first 12 months graduate scholars focusing on geometry and topology.
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Extra info for Circle-Valued Morse Theory
7. Let v be a downward normal ﬁeld. There is a neighbourhood V of ∂0 W , and a neighbourhood W of v in Vect1N (W ) such that for every (x, w) ∈ V × W the w-trajectory starting at x quits W , and τ (x, w) 2τ (x, v). Section 2. Cobordisms Proof. 27 Assuming that ∂0 W has a collar of height α with respect to v, let Φ : ∂0 W × [0, α[→ W, Φ(y, t) = γ(y, t; −v) be the corresponding map, and let V = Φ(∂0 W × [0, α/2]). The vector ﬁeld Φ−1 ∗ (v) on ∂0 W × [0, α/2] has coordinates (0, −1). Therefore there is a neighbourhood W of v in Vect1N (W ) such that for every w ∈ W the second coordinate of the ﬁeld Φ−1 ∗ (w) is between −2 and −1/2.
The same is true for the vector ﬁeld w on the cobordism V . 3. The evaluation map. Now we can establish the C 0 -continuity property for normal vector ﬁelds on cobordisms. Let W be a cobordism. Put M = W × R × Vect1N (W ). The set M has the natural topology of a direct product (recall that the space Vect1N (W ) is endowed with the C 0 -topology). Let D denote the set of all triples (x, t, v) ∈ M, such that the domain of deﬁnition of the maximal integral curve of v starting at x contains the interval [0, t].
A Morse function f on a cobordism W will be usually denoted as follows: f : W → [a, b]. Observe that for a Morse function f on a cobordism W every critical point is isolated, and since W is compact, the set of critical points is ﬁnite. The set of critical points of a Morse function f is denoted by S(f ), the set of critical points of index k by Sk (f ), and we put m(f ) = card S(f ), mk (f ) = card Sk (f ). 2. Examples of Morse functions. In this subsection we discuss several classical examples of Morse functions.
Circle-Valued Morse Theory by Andrei Pajitnov