By Yves Félix, John Oprea, Daniel Tanré
Rational homotopy is crucial software for differential topology and geometry. this article goals to supply graduates and researchers with the instruments helpful 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 in addition for geometers who're confronted with difficulties requiring topological techniques and hence want a basic and urban creation to rational homotopy. this is often basically a ebook of purposes. Geodesics, curvature, embeddings of manifolds, blow-ups, advanced and Kähler manifolds, symplectic geometry, torus activities, configurations and preparations are all lined. The chapters regarding those matters act as an creation to the subject, a survey, and a consultant to the literature. yet it doesn't matter what the actual topic is, the vital subject of the publication persists; specifically, there's a attractive connection among geometry and rational homotopy which either serves to unravel geometric difficulties and spur the improvement of topological equipment.
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Then the classifying space for G has for cohomology algebra a polynomial algebra, H ∗ (BG; Q) = Q[v1 , . . , vr ] with vi ∈ H 2ni (BG; Q). 34) that the cohomology algebra H ∗ (G; Q) is always an exterior algebra on odd degree generators for any compact connected Lie group G. 81. The ﬁrst one, due to Borel , consists of a careful analysis of a spectral sequence. In the second one, we keep track of spaces more carefully. As the reader will see, this process simpliﬁes the technical part of the argument.
5. 10 Principal bundles In this section, we deﬁne principal bundles with structure group a Lie group G (also called principal G-bundles). Since the pullback of a principal Gbundle is a principal G-bundle, we look for a principal G-bundle p : E → B which is universal in the following sense: any principal G-bundle over a space B can be obtained as a pullback of p along a map B → B. A characterization of such bundles is proved by using the notion of CWcomplexes. 12. 65 Let G be a Lie group. A principal bundle with structure group G (or principal G-bundle) is a locally trivial bundle p : E → B with ﬁber the Lie group G, together with a right action E × G → E, (x, g) → xg, of G on E and a trivializing open cover of B, (Ui , ϕi ), such that ϕi (x, g ·g ) = ϕi (x, g)g , for any x ∈ E, g, g ∈ G.
Since G is semisimple, we have H 1 (G; R) = 0, so i(X)α is an exact 1-form. That is, there exists a smooth function f : G → R such that i(X)α = df . 8 Simple and semisimple compact connected Lie groups we know that i(X)α is left invariant. 28, left invariant cohomology is isomorphic to ordinary cohomology, so i(X)α must be exact by a left invariant function as well. 8), f is constant and i(X)α = df = 0 for any left invariant vector ﬁeld X. We show now that this implies α = 0. For this, let g ∈ G and choose any two vectors vg , wg ∈ Tg (G).
Algebraic Models in Geometry by Yves Félix, John Oprea, Daniel Tanré