By Dennis Barden

An advent to simple rules in differential topology, according to the numerous years of educating event of either authors. one of the themes coated are delicate manifolds and maps, the constitution of the tangent package and its affiliates, the calculation of genuine cohomology teams utilizing differential varieties (de Rham theory), and purposes similar to the Poincare-Hopf theorem referring to the Euler variety of a manifold and the index of a vector box. each one bankruptcy comprises workouts of various trouble for which options are supplied. distinct good points comprise examples drawn from geometric manifolds in measurement three and Brieskorn forms in dimensions five and seven, in addition to specified calculations for the cohomology teams of spheres and tori.

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**Additional resources for An Introduction to Differential Manifolds**

**Example text**

Xn = O}. 1W To ensure that 1rv is an immersion we need -=/- v for each tangent vector z to M, since then (1rv)*(z)-=/- 0. Clearly it suffices to consider the unit vectors z, which form a submanifold u(M) of r(M). Let f : u(M) ----+ sn-l be the parallel translation to the sphere of unit tangent vectors at the origin. Since the tangent bundle to ]Rm is trivial, the map f may be thought of as the composition of the inclusion in M x ]Rn followed by projection on the second factor. So f is smooth and, by Sard's theorem, since dim(u(M)) = 2m-1 < n-1, f(u(M)) c sn-l has measure zero.

As with all of Chapter 3, this section may be omitted at a first reading. 1. ) A compact manifold Mm of dimension m may be immersed in IR 2 m and embedded in JR 2 m+ 1 . 4. Proof. Assume that the compact manifold Mm is embedded in ]Rn, which we denote by M ~ IRn, and let JRn-l = {x E IRnlxn = O}. For each unit vector v in ]Rn\ IRn-l we consider the projection, 1l"v, of M into JRn-l parallel to v. We claim that, provided n > 2m + 1, it is possible to choose v so that 1rv is an injective immersion and hence, since M is compact, an embedding.

JR -----, JR-, t f-4 { 0 exp(-1/t2 ) if t :( 0, ift>O, and check that it is smooth. (r _ t) E [O, 1]. (r - t) = 0, that is, if and only if t ~ r. Next, define 'I/Jr: JR----, JR; t f-4 1- r(ltl - r). Then 'I/Jr is differentiable since it is constant near t posite of smooth functions. Finally, the function = 0 and otherwise a com- has the properties that we require. 4. • Using bump functions, we may establish the following special case of Whitney's Embedding Theorem. 3. Let Mm be a compact manifold.