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Download e-book for kindle: An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu

By Liviu Nicolaescu

ISBN-10: 146141105X

ISBN-13: 9781461411055

This self-contained remedy of Morse thought specializes in functions and is meant for a graduate direction on differential or algebraic topology. The booklet is split into 3 conceptually precise elements. the 1st half comprises the rules of Morse idea. the second one half involves purposes of Morse thought over the reals, whereas the final half describes the fundamentals and a few functions of advanced Morse conception, a.k.a. Picard-Lefschetz theory.

This is the 1st textbook to incorporate subject matters equivalent to Morse-Smale flows, Floer homology, min-max thought, second maps and equivariant cohomology, and intricate Morse idea. The exposition is more suitable with examples, difficulties, and illustrations, and should be of curiosity to graduate scholars in addition to researchers. The reader is anticipated to have a few familiarity with cohomology idea and with the differential and crucial calculus on gentle manifolds.

Some positive aspects of the second one variation contain extra purposes, akin to Morse idea and the curvature of knots, the cohomology of the moduli area of planar polygons, and the Duistermaat-Heckman formulation. the second one variation additionally contains a new bankruptcy on Morse-Smale flows and Whitney stratifications, many new routines, and numerous corrections from the 1st variation.

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Additional resources for An Invitation to Morse Theory (2nd Edition) (Universitext)

Sample text

1T(X ) . x . T (/1x . T (a T(:() )) 11X . 1' (a T (X ) . ET(X ») aT( X) . E T ( X ) . T (ET(X») ' Analogou sly we ver ify (3T (x ) . E T ( X ) . 11 ~T(X ) = (3T(X) ' E T ( X ) < aT(X ), (3T( X) > is mo nic, t he seco nd T-al gebra ax iom • T (ET ( X ») ' Since follows. Finally, t he relation (2. 7) implies that aT(X ) and (3T(X ) are T-hom omorphism s. t . sp = and (cp , '1/;) E ~ (y,T(X)) ' Then Cl'T (X) . () and 'I/; = {3T( X) . () . 7), and obtain : cpo I' 11X . T(aT(X)) . 1' ({)) . I' aT(X ) .

Hence t he assertion is verified. 5 shows that. ip is uniquely determined by 'ljJ . s existence for granted we denote t he left adjoint morphism of 'ljJ by 'ljJ*. 6 Let (B , ::;8) - - (A, ::; A) be an isotone C-morphism , and let 'ljJ* -i 'r/J be a Galois connection. 'ljJ* (b) If 'ljJ is monic, then 'r/J* . 'r/J lAo = l B· P ROOF . Since 'I/J is isotone, the relat ion 'ljJ = 'ljJ . 'ljJ* . ry of ~ A' T hus t he assertions (a) and (b) are evident. s . ::::: ) : cop t----+ SET-PO by X E ICI X~ Y P CA.

X,A) is transit ive. 4 be a C-epim orph ism , A ====t: B be a pair of 'P2 aD C-morphisms, and let (B, :So ~ B) be a part ially ordered C- obj ect. Th en th e following implication holds: (CPI . E , CP2 • E) E ~(E ,B) P RO O F . ==> We form the pullback sq ua re A' ~ :Sa ,p : , A - - -..... B xB {'PI ,"' 2) 34 CHAPT ER 2. PARTIALLY ORDERED lv[ ONADS and conclude from (CPl ' e , CP2 . £.... A' making the following diagram commutative: E c A (cP 1,'1'2) ' B xB It is sufficient to show that 1/J is a C-isomorphism.

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An Invitation to Morse Theory (2nd Edition) (Universitext) by Liviu Nicolaescu

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