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Note that this is some kind of must-know book for quantum mechanics - at least over here in Europe. Many professors base their lectures on this book and recommend reading it for a better understanding of quantum mechanics because they don't have time to cover all the subjects covered in the book...
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As the authors explain eloquently, the (computational) power of rational homotopy theory comes from its algebraic formulation, which was first discussed by Sullivan and the mathematician Daniel Quillen, and involves the use of graded objects with both an algebraic structure and a "differential". What is fascinating about the role of the differential is its connection with homotopy theory, and not just in homology and cohomology theory as encountered in first-year graduate courses in algebraic topology. The authors deal with three different graded categories with a differential in the book, namely modules over a differential graded algebra (R, d), commutative cochain algebras, and differential graded Lie algebras. In analogy to the free resolution of an arbitrary module over a ring, associated with these three cases is a modeling construction that in the first case is a semi-free resolution of a module over (R, d), in the second case a "Sullivan model" which is a commutative cochain algebra which is free as a commutative graded algebra, and in the third case a free Lie model, which is free as a graded Lie algebra.
The first case arises topologically when considering continuous maps between spaces and the singular cochain algebras via the induced cochain map. When the map is a fibration, the authors compute the cohomology of the fiber using a semifree resolution. The first case also arises in considering the action of a topological monoid over a space and the singular chains. When the action is a principal G-fibration X-> Y, the authors compute the homology of Y using semifree resolutions. The authors then give a proof of the Whitehead-Serre theorem using this result. The proof of this follows their plan to avoid diagram-chasing techniques as much as possible: they do not use spectral sequences.
The second case involves a generalization of the classical commutative cochain algebra of smooth differential forms on a manifold. The authors construct a "Sullivan functor" from topological spaces to commutative cochain algebras, the Sullivan model, and the Sullivan "realization functor", the latter of which converts a Sullivan algebra into a rational topological space. The rational homotopy types of a space are then in bijective correspondence to isomorphism classes of "minimal" Sullivan algebras, and the homotopy classes of maps between rational spaces are in bijective correspondence to homotopy classes of maps between minimal Sullivan algebras. The characterization of a Sullivan algebra as being "minimal" comes from the fact that for such algebras there is a natural isomorphism between the vector space on which the Sullivan algebra is modeled and integral homomorphisms of the homotopy group to the ground field.
The third case involves the use of differential graded Lie algebras. The authors construct the "homotopy Lie algebra" of a simply connected topological space, which is the homotopy group of the loop space tensored with the ground field, and the homotopy Lie algebra of a minimal Sullivan algebra. The latter is interesting in that it involves using the quadratic part of the differential in order to obtain the Lie bracket. These two constructions of homotopy Lie algebras are the same for the Sullivan algebra over the space. In this context, the authors consider "free Lie models" for differential graded Lie algebras, which can be thought of as an assignment of generators to each single n-cells of a CW complex. The authors give many helpful examples of free Lie models that illustrate this, such as for the sphere, adjunction spaces, projective spaces, and homotopy fibers.
Since rational and ordinary homotopy are different in terms of their information content, it is perhaps not surprising that the Lusternik-Schnirelmann category makes its appearance in this book. The rational LS category is the LS category of a rational CW complex in the rational homotopy type of the space, and the authors calculate it in terms of Sullivan models, verifying that the rational case is much easier to compute than the general case. As further verification, the authors show that the Postnikov fibers in a Postnikov decomposition of a simply connected finite CW complex all have finite rational LS category, which is not true in the integral case. Even further, they show that the rational LS category of a product is the sum of the products, contrary to the ordinary LS category which is not well-behaved for products and fibrations.
The authors also discuss various applications at the end of the book, involving how to break up n-dimensional simply connected finite CW complexes into two groups: those whose rational homotopy groups vanish in degrees greater than or equal to 2n, and those where they grow exponentially. The former are called "rationally elliptic" and the latter "rationally hyperbolic". This classification can be determined, as they show, from a calculation of the "Betti numbers" of the loop group of the space over the rationals. A collection of unsolved problems for the ambitious reader ends the book.
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For the multi language learner this is a must