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 precious for using rational homotopy in geometry. Algebraic types 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 techniques and hence want a basic and urban advent to rational homotopy. this can be 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 coated. The chapters on the topic of those matters act as an creation to the subject, a survey, and a advisor to the literature. yet it doesn't matter what the actual topic is, the vital subject matter of the ebook persists; specifically, there's a appealing connection among geometry and rational homotopy which either serves to unravel geometric difficulties and spur the advance of topological equipment.

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Then H 1 (G; R) ∼ = Hom(Z (g), R), where Z (g) is the center of the Lie algebra g. Proof Let ω ∈ g∗ be a left invariant 1-form. 2) we have: dω(X, Y) = Xω(Y) − Yω(X) − ω([X, Y]). Since the form ω is left invariant, this is also true for the functions ω(Y) and ω(X). Since left invariant functions are constant, the previous formula reduces to dω(X, Y) = −ω([X, Y]). e. ω ∈ [g, g]⊥ ). Now recall the deﬁnition of the center of g: Z (g) = X ∈ g | [X, Y] = 0 for any Y ∈ g . 38). From F(X, [Y, Z]) = F([X, Y], Z), we see that X is in the F-orthogonal complement [g, g]⊥ of [g, g] if and only if X ∈ Z (g).

9 Homogeneous spaces as the homotopy lifting property and the existence of the long exact homotopy sequence. 10. In the context of Lie groups, the main examples of locally trivial ﬁber bundles come from the notion of homogeneous spaces that we introduce now. Let H be a closed subgroup of a Lie group G. We denote by G/H the set of left cosets of H; that is, G/H is the quotient of G by the equivalence relation x ∼ y if and only if x−1 y ∈ H. The elements of G/H are denoted by xH for x ∈ G. In particular, H is the class of the neutral element e of G.

Proof Let X be a left invariant vector ﬁeld. 14) the exponential map exp : Te (G) → G is deﬁned by θt (e) = exp(tXe ), where θ is the 1-parameter subgroup associated to X. There is also an exponential map at any g ∈ G obtained by requiring exp(tXg ) = Lg (exp(tXe )), where Lg denotes left translation by g. e. DLh (Xe ) = Xh for all h ∈ G) we have θt (g) = exp(tXg ) = Lg (exp(tXe )) = g · exp(tX). In conclusion, the ﬂow acts on g by right translation. 2). Therefore, the form ω is right invariant for the action of elements in the image of the exponential if and only if L(X)ω = 0 for any left invariant vector ﬁeld X, see [113, Proposition VI, page 126].