By Yves Félix, John Oprea, Daniel Tanré

ISBN-10: 019920652X

ISBN-13: 9780199206520

ISBN-10: 1435656393

ISBN-13: 9781435656390

Rational homotopy is crucial software for differential topology and geometry. this article goals to supply graduates and researchers with the instruments precious for using rational homotopy in geometry. Algebraic versions in Geometry has been written for topologists who're interested in geometrical difficulties amenable to topological tools and likewise for geometers who're confronted with difficulties requiring topological methods and therefore want a basic and urban creation to rational homotopy. this is often primarily a ebook of functions. Geodesics, curvature, embeddings of manifolds, blow-ups, complicated and Kähler manifolds, symplectic geometry, torus activities, configurations and preparations are all coated. The chapters relating to those matters act as an advent to the subject, a survey, and a advisor to the literature. yet it doesn't matter what the actual topic is, the important topic of the publication persists; particularly, there's a appealing connection among geometry and rational homotopy which either serves to resolve geometric difficulties and spur the advance of topological equipment.

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**Extra resources for Algebraic Models in Geometry**

**Example text**

49. In fact, however, it is possible to prove that π2 (G) = 0, in the semisimple compact case, and π3 (G) = Z in the simple case. 17, page 335]. 82 with a more homotopical argument. 30, that H 3 (G; R) is isomorphic to the vector space of invariant 3-forms on G. We follow the proofs of [107, Problem IV-B] and [41, Section V-12]. The idea is to transform the problem of ﬁnding an invariant 3-form on G into the problem of ﬁnding an invariant symmetric bilinear form on g. 38, we will get the ﬁrst part of the statement.

Un ) with un = un , un un−1 = un−1 − un−1 , un un , un−1 − un−1 , un un ... Therefore, we obtain a section σ : U → O(n) of the canonical projection deﬁned by σ (v1 , . . , vk ) = GS(e1 , . . , en−k , v1 , . . , vk ) = (e1 , . . , en−k , v1 , . . , vk ). More generally, in the case of a closed subgroup of a Lie group, there always exists a local section (see [60, 12, Proposition 1]). We do not give 29 30 1 : Lie groups and homogeneous spaces the proof here. In every concrete example, a local section can be easily constructed as above.

From the surjectivity part of the hypothesis, we now get the following morphism of principal G-bundles Dk+1 × G / E0 Dk+1 ψ p0 / B0 Using the injectivity part again gives the following sequence of morphisms of principal G-bundles where the maps Sk → Dk+1 and Sk ×G → Dk+1 ×G 35 36 1 : Lie groups and homogeneous spaces are the canonical injections: Sk × G / Dk+1 × G / E0 / Dk+1 Sk ψ p0 / B0 The restriction of to Dk+1 × {e} is an extension of f : Sk → E which implies that f is nullhomotopic.

### Algebraic Models in Geometry by Yves Félix, John Oprea, Daniel Tanré

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