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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

Additional resources for A Course of Pure Geometry

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Transition functions and chart representation of higher order tangent maps. For a good understanding of the structure of T k M we will need to know the structure of the transition functions T k ϕij of T k M , and more generally, the structure of higher order tangent maps T k f . For k = 1 , recall that, for a smooth map f : V ⊃ U → W (which represents a smooth map M → N with respect to charts), we have T f (x + εv) = f (x) + εdf (x)v. 6 (iv)), we may calculate d(T f ) and hence T T f . One gets T 2 f (x + ε1 v1 + ε2 v2 + ε1 ε2 v12 ) = f (x) + ε1 df (x)v1 + ε2 df (x)v2 + ε1 ε2 df (x)v12 + d2 f (x)(v1 , v2 ) .

3 by checking the definition of differentiability over K[ε] directly, leads to fairly long and involved calculations. ) 42 DIFFERENTIAL GEMETRY OVER GENERAL BASE FIELDS AND RINGS 7. Scalar extensions. 1. Higher order tangent bundles. For any manifold M , the second order tangent bundle is T 2 M := T (T M ) , and the k -th order tangent bundle is inductively defined by T k M := T (T k−1 M ) . 2. The k -th order tangent bundle T k M is, in a canonical way, a manifold over the ring of iterated dual numbers T k K = K[ε1 ] .

2. If M is a manifold of class C k+1 over K, modelled on V , then T M is, in a canonical way, a manifold of class C k over K[ε], modelled on the scalar extended K[ε] -module V ⊕ εV . If f : M → N is of class C k+1 over K, then T f : T M → T N is of class C k over K[ε] . Thus the tangent functor can be characterized as the “functor of scalar extension” from manifolds over K into manifolds over K[ε] agreeing on open submanifolds of K-modules with the algebraic scalar extension functor. Proof. Change of charts on T M is described by the transition maps T ϕij (x, v) = (ϕij (x), dϕij (x)v) .