By Michael Farber
This monograph is an creation to the interesting box of the topology, geometry and dynamics of closed one-forms. the topic was once initiated by means of S. P. Novikov in 1981 as a research of Morse kind zeros of closed one-forms. the 1st chapters of the booklet, written in textbook sort, supply a close exposition of Novikov concept, which performs a basic function in geometry and topology. next chapters of the booklet current a number of subject matters the place closed one-forms play a crucial function. the main major effects are the next: the answer of the matter of exactness of the Novikov inequalities for manifolds with the limitless cyclic primary team. the answer of an issue raised by means of E. Calabi approximately intrinsically harmonic closed one-forms and their Morse numbers. the development of a common chain advanced which bridges the topology of the underlying manifold with information regarding zeros of closed one-forms. This advanced implies many fascinating inequalities together with Bott-type inequalities, equivariant inequalities, and inequalities concerning von Neumann Betti numbers. the development of a unique Lusternik-Schnirelman-type thought for dynamical platforms. Closed one-forms seem in dynamics throughout the inspiration of a Lyapunov one-form of a circulation. As is proven within the ebook, homotopy conception can be utilized to foretell the life of homoclinic orbits and homoclinic cycles in dynamical structures (""focusing effect"")
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Extra resources for Topology of closed one-forms
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A, deren Projektionen [al],[a2],... E A/U k eine ~ / 2 - Basis bilden. 10 Lemma: Es sei k m 0 , 1 , 2 oder 4 mod8 . Dann bilden die Projektionen der fol- genden Elemente yon A eine Basis yon (A/Uk)q: 1 k 0 2 3 Sq I 4 5 Sq 4 Sq I Sq2Sq I Sq 4 Sq 5 6 7 Sq 6 Sq 7 Sq 6 Sq 7 Sq6Sq 1 Sq4Sq2Sq I Sq I Sq 2 Sq2Sq I Sq 4 Sq 5 Sq3Sq I Sq 6 _ Sq 7 Sq4Sq 2 Sq6Sq I Sq4Sq2Sq I Sq 2 Sq 3 Sq 4 Sq Sq 6 4 Sq 7 Sq5Sq 2 Beweis: Durch Reehnen mit Adem- Relationen zeigt man, dab die Hbrigen zul~ssigen Monome Sq I sieh in A/U k schon als Linearkombination der angegebenen 28 Monome schreiben lassen.
T(~)/T(~I@M)) den relativen Thom- Isomorphismus, und wir schreiben ~/gl~M >M/WM fNr das NormalenbNndel yon M, dividiert durah die Trivialisierung ~[ ~ M" 18 Insbesondere erh~it man fiir M = Dm mit der vertwisteten Rahmung hg des Randes das Element yon ~(sm+8i), mus das [hi unter dem Isomorphis- ~m_l(SO) =~ ~ ( S m) --__~~(sm+8i) entspricht. D. Stong, fNr ungerade Primzahlen auf solche yon W. Giambalvo zurNckgreifen (siehe [Stong 2], ~Singer], [Giambalvo] ). Wir benutzen diese Resultate, um Kern und Kokern der induzierten Abbildung C * : H~'ibo
Es sei bemerkt, dab fHr ~k X = 0 die Abbildung qk+l: X