Topology

Algebraic Models in Geometry by Yves Félix, John Oprea, Daniel Tanré PDF

By Yves Félix, John Oprea, Daniel Tanré

ISBN-10: 019920652X

ISBN-13: 9780199206520

ISBN-10: 1435656393

ISBN-13: 9781435656390

Rational homotopy is the most 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 versions in Geometry has been written for topologists who're attracted to geometrical difficulties amenable to topological tools and likewise for geometers who're confronted with difficulties requiring topological ways and hence desire a easy and urban creation to rational homotopy. this can be primarily a publication of functions. Geodesics, curvature, embeddings of manifolds, blow-ups, advanced and Kähler manifolds, symplectic geometry, torus activities, configurations and preparations are all lined. The chapters with regards to those topics 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 booklet persists; particularly, there's a appealing connection among geometry and rational homotopy which either serves to unravel geometric difficulties and spur the improvement of topological tools.

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Example text

As we will see, the group SO(n) is not simply connected if n ≥ 2. The universal cover of SO(n) is called the nth-spinor group and denoted by Spin(n). 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.

Let ω ∈ AkDR (M) and x ∈ M be fixed. As a function f , we take G → ∧Tx (M)∗ , g → g ∗ ω(x). We get a differential form ρ(ω) on M defined by: ρ(ω)(X1 , . . , Xk )(x) = = G G g ∗ ω(X1 , . . , Xk )(x) dg (Lg )∗ ω(X1 , . . , Xk )(x) dg. We have thus built a map ρ : ADR (M) → ADR (M) and we now analyze its properties. Fact 1: ρ(ω) ∈ L (M). Let g ∈ G be fixed. The map DLg : Tx (M) → Tg x (M) induces a map ∗ ∧ DLg : ∧ Tg x (M)∗ → ∧Tx (M)∗ . 6 = G Lg ∗ Invariant forms ω(x) dg = ρ(ω)(x). L (M) Fact 2: If ω ∈ If Lg ω(x).

DRg )−1 Xn ). 10, the homomorphism of Lie groups Ad : G → Gl(g). As direct consequences of the definitions, we have (L∗g αR )h (X1 , . . , Xn ) = (αR )gh (DLg (X1 ), . . , DLg (Xn )) = α((DRgh )−1 ◦ (DLg )(X1 ), . ) = α((DRh )−1 ◦ (DRg )−1 ◦ DLg (X1 ), . ) = (αR )h ((DRg )−1 ◦ DLg (X1 ), . ) = (det(Ad(g))(αR )h (X1 , . . , Xn ). The composition det ◦Ad : G → R has for image a compact subgroup of R; that is, {1} or {−1, 1}. Since the group G is connected, we get det(Ad(g)) = 1, for any g ∈ G, and αR is a bi-invariant volume form.

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Algebraic Models in Geometry by Yves Félix, John Oprea, Daniel Tanré


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