By Yves Félix, John Oprea, Daniel Tanré

Rational homotopy is an important instrument for differential topology and geometry. this article goals to supply graduates and researchers with the instruments beneficial for using rational homotopy in geometry. Algebraic versions in Geometry has been written for topologists who're attracted to geometrical difficulties amenable to topological equipment and in addition for geometers who're confronted with difficulties requiring topological techniques and hence want a easy and urban creation to rational homotopy. this can be primarily a booklet of purposes. 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 topics act as an advent to the subject, a survey, and a consultant to the literature. yet it doesn't matter what the actual topic is, the relevant subject matter of the publication persists; specifically, 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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**Example text**

The orthogonal group O(n) is the prototype of Lie groups. 14, Chapter V]). If θ is a real number, we denote by R(θ) = cos θ sin θ − sin θ cos θ the rotation matrix corresponding to the rotation in R2 by angle θ . Let (θ1 , . . , θr ) be r real numbers. Denote by (R(θ1 ), . . , R(θr ), 1) the matrix having the R(θi ) and 1 along the diagonal and 0 entries otherwise. The group of matrices of the form (R(θ1 ), . . , R(θr ), 1) is the maximal torus of SO(2r + 1) and the rank of SO(2r + 1) is r. The Weyl group of SO(2r + 1) has 2r r!

35 One can determine the elements x2pi +1 from the structure law of G as follows. Denote by µ∗ : H ∗ (G; Q) → H ∗ (G; Q)⊗H ∗ (G; Q) the map induced by the multiplication µ : G × G → G. An element h ∈ H ∗ (G; Q) is called primitive if µ∗ (h) = 1 ⊗ h + h ⊗ 1. The set PG of primitive elements is a vector space and we have H ∗ (G; Q) ∼ = ∧PG . We will not use this in the sequel and we do not give a proof, instead referring to [113, Theorem IV, page 167], for instance, for an argument with coefﬁcients in R.

R ) → (±θ1 , . . , ±θr ). As for the Lie group SO(2r), its maximal torus consists of matrices (R(θ1 ), . . , R(θr )) and the rank of SO(2r) is r. Its Weyl group has 2r−1 r! elements acting on the maximal torus by a permutation of the coordinates 9 10 1 : Lie groups and homogeneous spaces composed with the substitutions (θ1 , . . , θr ) → (ε1 θ1 , . . , εr θr ), with εi = ±1 and ε1 · · · εr = 1. 2 Subgroups of the complex linear group Denote by Gl(n, C) the group of invertible n × n-matrices with entries in the complex numbers C.