By Ethan D. Bloch

The individuality of this article in combining geometric topology and differential geometry lies in its unifying thread: the idea of a floor. With various illustrations, workouts and examples, the coed involves comprehend the connection among smooth axiomatic method and geometric instinct. The textual content is saved at a concrete point, 'motivational' in nature, averting abstractions. a few intuitively attractive definitions and theorems relating surfaces within the topological, polyhedral, and delicate circumstances are provided from the geometric view, and aspect set topology is particular to subsets of Euclidean areas. The remedy of differential geometry is classical, facing surfaces in R3 . the cloth here's available to math majors on the junior/senior point.

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In particular, Tid (Sympl(M, ω)) contains the space of exact 1-forms {µ = dh | h ∈ C ∞ (M )} C ∞ (M )/ locally constant functions . 6 Let (M, ω) be a compact symplectic manifold with HdeRham (M ) = 1 0. Then any symplectomorphism of M which is sufficiently C -close to the identity has at least two fixed points. Proof. Suppose that f ∈ Sympl(M, ω) is sufficiently C 1 -close to id. =⇒ Graph f closed 1-form µ on M . dµ = 0 1 HdeRham (M ) = 0 =⇒ µ = dh for some h ∈ C ∞ (M ) . M compact =⇒ h has at least 2 critical points.

By Lecture 3, we have Graph f is lagrangian ⇐⇒ µ is closed. 1 ♦ Conclusion. A small C -neighborhood of id in Sympl(M, ω) is homeomorphic to a C 1 -neighborhood of zero in the vector space of closed 1-forms on M . So: Tid (Sympl(M, ω)) {µ ∈ Ω1 (M ) | dµ = 0} . In particular, Tid (Sympl(M, ω)) contains the space of exact 1-forms {µ = dh | h ∈ C ∞ (M )} C ∞ (M )/ locally constant functions . 6 Let (M, ω) be a compact symplectic manifold with HdeRham (M ) = 1 0. Then any symplectomorphism of M which is sufficiently C -close to the identity has at least two fixed points.

36 6 PREPARATION FOR THE LOCAL THEORY Outline of the proof. • Case of M = Rn , and X is a compact submanifold of Rn . 6 (ε-Neighborhood Theorem) Let U ε = {p ∈ Rn : |p − q| < ε for some q ∈ X} be the set of points at a distance less than ε from X. , a unique q ∈ X minimizing |q −x|). π Moreover, setting q = π(p), the map U ε → X is a (smooth) submersion with the property that, for all p ∈ U ε , the line segment (1 − t)p + tq, 0 ≤ t ≤ 1, is in U ε . The proof is part of Homework 5. Here are some hints.