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Posted On April 20, 2018 at 5:40 pm by / Comments Off on Get Rational Homotopy Theory PDF

By Yves Felix, Steve Halperin, Jean-Claude Thomas

ISBN-10: 1461265169

ISBN-13: 9781461265160

ISBN-10: 146130105X

ISBN-13: 9781461301059

as good as through the record of open difficulties within the ultimate component to this monograph. The computational strength of rational homotopy idea is because of the invention by way of Quillen [135] and by way of Sullivan [144] of an specific algebraic formula. In each one case the rational homotopy kind of a topological area is equal to the isomorphism category of its algebraic version and the rational homotopy form of a continuing map is equal to the algebraic homotopy classification of the correspond­ ing morphism among versions. those versions make the rational homology and homotopy of an area obvious. additionally they (in precept, constantly, and in prac­ tice, occasionally) let the calculation of alternative homotopy invariants similar to the cup product in cohomology, the Whitehead product in homotopy and rational Lusternik-Schnirelmann class. In its preliminary part study in rational homotopy conception interested in the identi­ of those types. those integrated fication of rational homotopy invariants in phrases the homotopy Lie algebra (the translation of the Whitehead product to the homo­ topy teams of the loop area OX less than the isomorphism 11'+1 (X) ~ 1I.(OX», LS class and cone size. because then, although, paintings has targeting the homes of those in­ variations, and has exposed a few actually impressive, and formerly unsuspected phenomena. for instance • If X is an n-dimensional easily attached finite CW advanced, then both its rational homotopy teams vanish in levels 2': 2n, otherwise they develop exponentially.

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Note also that if X is locally compact and Y is a k-space then the k-space topology in X x Y is just the ordinary topology. • Quotients. A quotient space Y of X is a surjection p : X ~ Y such that U c Y is open if and only if p-1 (U) is open. We only consider quotients Y such that Y is Hausdorff and X is a k-space; in this case Y is automatically a k-space. The product p x p' : X x X' ~ Y x Y' of two such quotient maps is itself a quotient map. ) • Mapping spaces. If X and Yare topological spaces then Y x (as a set) is the set of continuous maps from X to Y.

J) : X V 3 Homotopy Theory • Suppose given a pair (Z, B) and a continuous map f : B ~ X. , f(b), bE B. It is called the adjunction space obtained by attaching Z to X along f. • Given A c X the based space (XIA, [AJ) is obtained by identifying the points of A to a single point [A], and giving X I A the quotient topology. Thus XIA = * Uj X, where f : A ~ * is the constant map. • The suspension, II (X x of a based space (X, xo) is the based space X x ~X, {O, I} U {xo} x I). , g) if there is a continuous map F : X x I ~ Y such that F(x, 0) = f(x) and F(x, 1) = g(x), x E X.

A pair of topological spaces, (X, A) is a space X and a subspace A (with the k-space topology described above). c X A map of pairs, 'P : (X, A) -+ (Y, B) is a continuous map 'Px : X -+ Y restricting to 'P A : A -+ B. If Z is another space then (X, A) x Z = (X x Z, A x Z). • Given continuous maps X -4 Y ? Z the fibre product X x Y Z C X x Z is the subspace of points (x, z) such that f(x) = g(z). Projection defines a commutative square X xy Z -- Z X --+-. Y f and any pair of continuous maps 'P : W ---7 X,1/;: W ---7 Z such that f'P = g1jJ defines a continuous map ('P,1jJ) : W ---7 X Xz Y.

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Rational Homotopy Theory by Yves Felix, Steve Halperin, Jean-Claude Thomas

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