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 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 likewise for geometers who're confronted with difficulties requiring topological ways and hence want a uncomplicated and urban advent to rational homotopy. this is often basically a publication 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 with regards to those matters 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 primary 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 info for Algebraic Models in Geometry
Proof Let F be any positive deﬁnite bilinear form on g and set F(X, Z) = F(Ad(g)(X), Ad(g)(Z))dg, G where Ad(g)(X) = ((DRg )−1 ◦ (DLg ))(X). This bilinear form satisﬁes F(Ad(g −1 )(X), Z) = F(X, Ad(g)(Z)) for any g in G. 11). 8 Simple and semisimple compact connected Lie groups We now come to our ﬁrst concrete application: the vanishing of the second Betti number of a compact semisimple Lie group and the determination of the third Betti number of a simple Lie group. 50, one can, in fact, do better and prove that the second homotopy group is zero and the third homotopy group of a simple Lie group is Z.
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).
F([X, Y], Z) = F(X, [Y, Z]) for any triple (X, Y, Z) of elements of g. Such an F is said to be invariant. Proof Let F be any positive deﬁnite bilinear form on g and set F(X, Z) = F(Ad(g)(X), Ad(g)(Z))dg, G where Ad(g)(X) = ((DRg )−1 ◦ (DLg ))(X). This bilinear form satisﬁes F(Ad(g −1 )(X), Z) = F(X, Ad(g)(Z)) for any g in G. 11). 8 Simple and semisimple compact connected Lie groups We now come to our ﬁrst concrete application: the vanishing of the second Betti number of a compact semisimple Lie group and the determination of the third Betti number of a simple Lie group.