On the width of homotopies

Calder, Allan; Siegel, Jerrold

doi:10.1016/0040-9383(80)90008-7

Cited by 13 publications

(7 citation statements)

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“…An examination of the proof shows that the hi chosen in it satisfies ||Λi || < 2τr. Similarly, going back to [12] in Step 7 of this proof, we get \\h 2 \\, \\h 3 \\ < π. Therefore cel(tx) < 4π + 2arcsin(ε/2), using Proposition 2.4 of [24].…”

Section: Remarkmentioning

confidence: 99%

Factorization problems in the invertible group of a homogeneousC^∗-algebra

Phillips¹

1996

Pacific J. Math.

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Let X be a compact metric space of dimension d. In previous work, we have shown that for all sufficiently large n, every element of the identity component Uo(C(X) 0 M n ) of the unitary group U(C(X)®M n ) is a product of at most 4 exponentials of skewadjoint elements. On the other hand, if X is a manifold then some elements of Uo(C(X) (8> M n ) require at least about d/n 2 exponentials. Similar qualitative behavior (with different bounds: 5 and d/(2n 2 )) holds for the problem of factoring elements of the identity component invo(C(X) 0M n ) of the invertible group as products of exponentials of arbitrary elements of the algebra. In this paper, we identify the sets of finite products of 10 other types of elements of invo(C(X)(g)M n ), and we show that the minimum lengths of factorizations have the same qualitative behavior as the two exponential factorization problems above (after a suitable minor modification in 3 of the 10 cases). We obtain upper bounds for large n that range from 5 to 22, and lower bounds approximately of the form rd/n 2 with r ranging from 1/16 to 2. The classes of elements we consider all make sense in general unital C*-algebras. They are: unipotents, positive invertibles, selfadjoint invertibles, symmetries, *-symmetries, commutators of elements of invo(^4) and Uo(A), accretive elements, accretive unitaries, and positive-stable elements (real part of spectrum positive). The last three classes are the ones requiring the slight modification; without it, lengths of factorization behave like exponential length rather than exponential rank.Introduction.

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Section: Remarkmentioning

confidence: 99%

Factorization problems in the invertible group of a homogeneousC^∗-algebra

Phillips¹

1996

Pacific J. Math.

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“…Remark 1: Calder and Siegel (11) have shown that if Y is a finite complex with finite fundamental group, for each n, there is then a b, such that if X is n-dimensional and f, g: X → R are homotopic maps, there is a homotopy h t from f to g, such that the path {h t (x) j 0 ≤ t ≤ 1} is b-Lipschitz for every x ∈ X. In the case of our map R 2 → S 2 , such a homotopy can be obtained by lifting to S 3 via the Hopf map and contracting the image along geodesics emanating from a point not in the image of R 2 .…”

Section: Optimistic Possibilitymentioning

confidence: 99%

Quantitative algebraic topology and Lipschitz homotopy

Ferry

Weinberger

2013

Proc. Natl. Acad. Sci. U.S.A.

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We consider when it is possible to bound the Lipschitz constant a priori in a homotopy between Lipschitz maps. If one wants uniform bounds, this is essentially a finiteness condition on homotopy. This contrasts strongly with the question of whether one can homotop the maps through Lipschitz maps. We also give an application to cobordism and discuss analogous isotopy questions.amenable group | uniformly finite cohomology T he classical paradigm of geometric topology, exemplified by, at least, immersion theory, cobordism, smoothing and triangulation, surgery, and embedding theory is that of reduction to algebraic topology (and perhaps some additional pure algebra). A geometric problem gives rise to a map between spaces, and solving the original problem is equivalent to finding a nullhomotopy or a lift of the map. Finally, this homotopical problem is solved typically by the completely nongeometric methods of algebraic topology (e.g., localization theory, rational homotopy theory, spectral sequences).Although this has had enormous successes in answering classical qualitative questions, it is extremely difficult [as has been emphasized by Gromov (1)] to understand the answers quantitatively. One general type of question that tests one's understanding of the solution of a problem goes like this: Introduce a notion of complexity, and then ask about the complexity of the solution to the problem in terms of the complexity of the original problem. Other possibilities can involve understanding typical behavior or the implications of making variations of the problem.In this task, the complexity of the problem is often reflected, somewhat imperfectly, in the Lipschitz constant of the map. Indeed, one can often view the Lipschitz constant of the map as a measure of the complexity of the geometric problem.For concreteness, let us quickly review the classical case of cobordism, following Thom (2). Let M be a compact smooth manifold. The problem is: When is M n the boundary of some other compact smooth W n+1 ? There are many possible choices of manifolds in this construction, such as oriented manifolds, manifolds with some structure on their (stabilized) tangent bundles, Piecewise linear (PL) and topological versions, and so on. However, for now, we will confine our attention to this simplest version.Thom (2) embeds M m in a high-dimensional Euclidean space M ⊂ S m+N−1 ⊂ R m+N and then classifies the normal bundle by a map νM: νM → E(ξ N ↓ Gr(N, m + N)) from the normal bundle to the universal bundle of N-planes in m + N space. Including R m+N into S m+N via one-point compactification, we can think of νM as being a neighborhood of M in this sphere (via the tubular neighborhood theorem) and extend this map to S m+N if we include E(ξ N ↓ Gr(N, m + N)) into its one-point compactification E(ξ N ↓ Gr(N, m + N))^, the Thom space of the universal bundle. Let us call this mapThom (2) , extending the embedding of M into S m+N . Extending Thom's construction over this disk gives a nullhomotopy of Φ M . Conversely, one uses the nullhomotop...

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“…It was also shown in [2] that these numbers are nontrivial: in fact, limk^x(bk(M)) = oo. The numbers reflect subtle topological and geometrical properties of M. A good example of this is the fact that b2n_2(S") = 277 if and only if n = 2,4 or 8 (otherwise b2n_2(S") = 3m).…”

mentioning

confidence: 99%

“…The fundamental existence theorem is 0.2. Theorem [2]. For each integer k > 0 there exists a real number bk(M) such that if X is any normal space of dimension < k and f ~ g: X -* M, then w(f,g)b k(M).…”

mentioning

confidence: 99%

Variational Invariants of Riemannian Manifolds

Siegel¹,

Williams²

1984

Transactions of the American Mathematical Society

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Abstract. This paper treats higher-dimensional analogues to the minimum geodesic distance in a compact Riemannian manifold M with finite fundamental group. These invariants are based on the concept of homotopy distance in M. This defines a parametrized variational problem which is approached by globalizing the Morse theory of the spaces of paths between two points of M to the space of all paths in M. We develop machinery that we apply to calculate the invariants for numerous examples. In particular, we shall observe that knowledge of the invariants for the standard spheres determines the question of the existence of elements of Hopf invariant one.

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On the width of homotopies

Cited by 13 publications

References 9 publications

Factorization problems in the invertible group of a homogeneousC^∗-algebra

Factorization problems in the invertible group of a homogeneousC^∗-algebra

Quantitative algebraic topology and Lipschitz homotopy

Variational Invariants of Riemannian Manifolds

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On the width of homotopies

Cited by 13 publications

References 9 publications

Factorization problems in the invertible group of a homogeneousC∗-algebra

Factorization problems in the invertible group of a homogeneousC∗-algebra

Quantitative algebraic topology and Lipschitz homotopy

Variational Invariants of Riemannian Manifolds

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Product

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Factorization problems in the invertible group of a homogeneousC^∗-algebra

Factorization problems in the invertible group of a homogeneousC^∗-algebra