By Phillip A. Griffiths, John W. Morgan

This thoroughly revised and corrected model of the well known Florence notes circulated via the authors including E. Friedlander examines simple topology, emphasizing homotopy conception. incorporated is a dialogue of Postnikov towers and rational homotopy conception. this can be then through an in-depth examine differential types and de Tham’s theorem on simplicial complexes. additionally, Sullivan’s effects on computing the rational homotopy sort from kinds is presented.

New to the second one variation:

*Fully-revised appendices together with an accelerated dialogue of the Hirsch lemma

*Presentation of a ordinary facts of a Serre spectral series consequence

*Updated content material in the course of the publication, reflecting advances within the zone of homotopy theory

With its smooth strategy and well timed revisions, this moment version of *Rational Homotopy concept and Differential Forms* could be a necessary source for graduate scholars and researchers in algebraic topology, differential kinds, and homotopy concept.

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**Example text**

It also deals with the existence question of constructing maps from X to Y. In both cases, we work inductively over the skeleta. nC1/ I ! Y. n/ ! nC1/ ! Y. We will deal with the second case first. 1. Suppose that Y is simply connected. 1. Y/ ! ŒSn ; Y; where ŒSn ; Y means the free homotopy classes of maps of Sn into Y (with no restrictions on the image of base point). P. Griffiths and J. 1007/978-1-4614-8468-4__6, © Springer Science+Business Media New York 2013 53 54 6 Obstruction Theory 2.

Dn . We may extend F to a map FW en I ! Dn (this is a little exercise). Thus, f extends to a map QfW Sn f0g [ N ! Sn such that QfW @N ! @Dn Sn . Since Sn n int Dn is contractible, we can extend Qf to a map on the rest of Sn I, sending the complement of N to Dn . This map will have p as a regular value with preimage the arc A. The resulting map on Sn f1g will miss p and hence is homotopic to a constant. p/ has more than two points and n 2, then this argument allows us to deform f by a homotopy to “cancel” two of the points with opposite local degree.

X/ ! X; A/ ! : : : As one consequence of this and the calculations above, we see that excision is false for the homotopy groups. D2 ; S1 / ! S2 / ¤ 0. Of course, where excision to hold for homotopy groups, then they would satisfy all the axioms for homology. If that were true, then the argument in Chap. X; A//. It would follow that the homology and homotopy groups were isomorphic. 2. Let X and Y be CW complexes with base points, x0 and y0 , being 0-cells. X; x0 / ! X; x0 / ! Y; y0 / for all n 0 Suppose that Y is connected.