By Yves Félix, John Oprea, Daniel Tanré
Rational homotopy is crucial 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 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 methods and hence want a basic and urban creation to rational homotopy. this can be basically a ebook of functions. Geodesics, curvature, embeddings of manifolds, blow-ups, complicated 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 imperative subject matter of the e-book 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 equipment.
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Additional info for Algebraic Models in Geometry
34). 11, using the existence of a universal bundle for a Lie group (see page 40). 45 Suppose α ∈ 2I (G). 30, if we can show that α = 0, then this will imply H 2 (G; R) = 0. Expressing the L-invariance of α for a left invariant vector ﬁeld X gives: 0 = L(X)α = (i(X)d + di(X))α = i(X)dα + di(X)α = di(X)α, since α is closed. Hence, i(X)α is a closed 1-form. 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 .
DLg (Yk ))(gx) = i(X)ω(Y1 , . . , Yk )(x). 6 Invariant forms Hence, i(X)ω is left invariant. The veriﬁcation of the other statements is similar. The previous result justiﬁes the following deﬁnition. 27 Let G be a Lie group and M be a left G-manifold. The invariant cohomology of M is the homology of the cochain complex ( L (M), d). We denote it by HL∗ (M). The main result is the following theorem. 28 Let G be a compact connected Lie group and M be a compact left G-manifold. Then HL∗ (M) ∼ = H ∗ (M; R).
Xr ) → H. Fact 2: The morphism φ is an isomorphism. By construction, φ is surjective so we are reduced to establishing its injectivity. Observe that the restriction of φ to ∧ (x1 ) is injective. We argue by induction and suppose that its restriction to ∧ x1 , . . , xk−1 is injective. Let a ∈ ∧ (x1 , . . , xk ) be such that φ(a) = 0. We decompose a into a = a1 + xk a2 , with a1 and a2 in ∧ x1 , . . , xk−1 . We denote by φ k the following composition ∧ (x1 , . . , xk ) φ /H µ∗ /H ⊗ H qk ⊗id / H ⊗ H.