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SARD’S THEOREM ALEX WRIGHT Abstract. A proof of Sard’s Theorem is presented, and applica-tions to the Whitney Embedding and Immersion Theorems, the existence of Morse functions, and the General Position Lemma are given. Suppose f: Mm!Nnis a map from a m-dimensional manifold M to an n-dimensional manifold N. (All manifolds and maps are ...

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The principal curvature in the direction tangential to the surface as smoves is = f00=(1+f02)3=2. The sphere tangent to the surface from inside ft<f(s)gcentered on t= 0 has radius d. ssatisfying f=d. s= 1=(1+f02)1=2. by similar triangles, and the sphere tangent on the other side centered on s= 0 has radius d.

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A little about the proof. The general outline of the proof is to start with an immersion with transverse self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections.

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Jan 21, 2015 · The proof of the theorem uses Whitney stratification. Now, let be the -ball around and the -sphere. We thus have a non-zero map which we can think of as a map to the corresponding sphere bundle. By chapter 4.3 in Hatcher, the obstruction to lifting this map to a map from , which we'll denote , lies in the cohomology group .

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56ebea068f0d1685f3c8de1d24d3631b54ec.pdf - Free download as PDF File (.pdf), Text File (.txt) or read online for free.

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Whitney's original papers "The Singularities of ...", "The Self-intersections of ..." and his book Geometric Integration Theory give proofs of these statements though there is likely a more accessible account of the immersion theorem.

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lor’s Theorem, was proved by H. Whitney in 1934. (See [7, 10] for the proof, and [11, 12] for related problems.) It roughly says that if f: E!R, where Eis a closed subset of Rn, can be approximated by Taylor polynomials of degree min a certain uniform way (as entailed by Taylor’s Formula), then f can be extended to a Cm-function on Rn. A Cm-Whitney eld on E encodes the data relevant for such an approximation of f.

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Proof that immersions and submersions are open subsets in the Whitney topology. 1/10/08: Proof that embeddings form an open set in the Whitney topology.

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The Whitney–Graustein theorem states that regular homotopy classes of regular curves in the plane are completely classified by the degree, i.e., two regular curves in the plane are regularly homotopic if and only if their degrees are equal (see [a2]; cf. also Homotopy).

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Whitney proved a stronger version of this theorem. Theorem 19.1. (Whitney 1944) Any compact n­manifoldadmits an embedding into R2n.

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The proof of the reconstruction theorem follows from similar arguments to those developed in [Hai14]. Let us recall here that the reconstruction theorem was inspired by the sewing lemma of Gubinelli [Gub04]. The embedding theorems are more delicate. Even for classical Besov spaces, their proofs are rather sensitive to the definition one chooses.
Theorem 1.1 follows easily from the following lemma. Lemma 1.3 (Discrete Loomis and Whitney inequality, 1949) For any finite A ⊂ Zd, |A d−1 ≤ d i=1 |Pi(A)|. Isoperimetric Inequalities Markov Type of Metric Spaces The Cheeger constant and mixing time Harmonic functions and random walks Embeddi Proof of Theorem 1.1 Proof of Theorem 1.1.
roots in the implicit function theorem, the theory of ordinary differential equations, and the Brown-Sard Theorem. Some algebraic results in the form adapted for the purpose and collected in the appendix are used as well. Very little algebraic topology enters the picture at this stage.
A little about the proof. The general outline of the proof is to start with an immersion with transverse self-intersections. These are known to exist from Whitney's earlier work on the weak immersion theorem. Transversality of the double points follows from a general-position argument. The idea is to then somehow remove all the self-intersections.
General application to the "regular value theorem" (see the Transversality Theorem in section 2.3 of [GP] for the more general version which we'll cover later). Mon 2/25: Finished the application of Sard's theorem to the existence of Morse functions (1.7); Whitney embedding theorem, compact case (1.8). Wed 2/27: Partitions of unity.

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This is not really a restriction if one takes into account the Whitney theorem by which any class Cn for n 1 admits a homeomorphism of Cn on a manifold of class C1. Moreover, the method that will be exposed also applies to the Cclass varieties for n 2. Given any covering of V using relatively compact open sets, by the paracompactness of V we can
Characterization of 2-connected graphs Theorem. (Whitney,1932) Let Gbe a graph, n(G) 3. Then Gis 2-connected iff for every u;v2V(G) there exist two internally disjoint u;v-paths in G.