By Yves Félix, John Oprea, Daniel Tanré

Rational homotopy is the most important instrument for differential topology and geometry. this article goals to supply graduates and researchers with the instruments useful for using rational homotopy in geometry. Algebraic versions in Geometry has been written for topologists who're interested in geometrical difficulties amenable to topological equipment and in addition for geometers who're confronted with difficulties requiring topological methods and hence desire a easy and urban advent to rational homotopy. this is often basically a publication 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 concerning 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 valuable subject of the ebook persists; specifically, there's a attractive connection among geometry and rational homotopy which either serves to unravel geometric difficulties and spur the advance of topological tools.

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**Extra info for Algebraic Models in Geometry**

**Example text**

In other words, any compact connected Lie group admits a ﬁnite sheeted covering group which is the product of a torus and of simple Lie groups. For instance, any connected compact Lie group with a trivial center is isomorphic to a product of simple compact Lie groups. This is the beginning of the root system construction. The second part is the classiﬁcation of compact connected simple Lie groups. 53 The following groups are simple: 1. 2. 3. 4. the special unitary groups SU(n1 ); the special orthogonal groups SO(n2 ); the symplectic groups Sp(n3 ); the exceptional Lie groups G2 , E6 , E7 , E8 , F4 ; and any compact connected simple Lie group is isomorphic to one group of this list.

A left action is called • effective if gx = x for all x ∈ M implies g = e; • free if gx = x for any x ∈ M implies g = e. For right actions and right G-manifolds, we ask for a smooth map M × G → M, (x, g) → xg, such that x(g ·g ) = (xg)g and xe = x for any x ∈ M, g ∈ G, g ∈ G. 24 Let G be a Lie group. The Lie multiplication gives to G the structure of a • left G-manifold, with L : G × G → G, L(g, g ) = Lg (g ) = g · g ; • right G-manifold, with R : G × G → G, R(g , g) = Rg (g ) = g · g. Let G be a Lie group.

32 Let G be a Lie group. Denote by µ : G × G → G the multiplication map and by ν : G → G the inverse map. Then we have: Dµ(g,g ) = DLg + DRg and Dνg = − DLg −1 ◦ DRg−1 . 7 Cohomology of Lie groups Proof With the identiﬁcation T(G × G) ∼ = T(G) ⊕ T(G), we write a vector ﬁeld on G × G as: ((g, g ), (X, X )) = (g, (g , X )) + ((g, X), g ) ∈ g ×Tg (G) ⊕ Tg (G)× g . Therefore, we have: Dµ(g,g ) (X, X ) = DLg (X ) + DRg (X). From this formula and the equality µ(g, ν(g)) = e, we deduce: 0 = Dµ(g,g−1 ) (X, Dνg (X)) = DLg ◦ Dνg (X) + DRg−1 (X) and Dνg = − DLg −1 ◦ DRg−1 .