On some applications of infinite-dimensional manifolds to the theory of shape

Chapman, T. A.

doi:10.4064/fm-76-3-181-193

Cited by 81 publications

(43 citation statements)

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“…The proof of the Theorem above (as given in [9]) has the same broad outline as the proof in [6], but the details of the argument are largely different because of the lack of suitable finite-dimensional analogues of certain infinite-dimensional techniques.…”

Section: Statements Of Results For the Finite-dimensional Characterimentioning

confidence: 99%

“…In the proof of the infinite-dimensional characterization of [6], cited in §3 above, some recent results in infinite-dimensional manifolds modeled on Q were used. The proof of the Theorem above (as given in [9]) has the same broad outline as the proof in [6], but the details of the argument are largely different because of the lack of suitable finite-dimensional analogues of certain infinite-dimensional techniques.…”

Section: Statements Of Results For the Finite-dimensional Characterimentioning

confidence: 99%

“…This result confirms an informal conversational conjecture of Morton Brown concerning the geometric intuition about what shape ought to mean. It and a companion earlier result of the author [6] cited in §3 below give easily stated characterizations of shape. They also put the study of shape in the point-set (homeomorphism) domain as distinct from the homotopy domain of the definition of Borsuk and of the characterization of Mardesic and Segal cited in §3 below.…”

Section: Introductionmentioning

confidence: 98%

“…In [6] the author used a different infinite-dimensional approach to obtain necessary and sufficient conditions in order for two compacta to have the same shape. Since this characterization is similar to (and to some extent motivates) the finite-dimensional characterization cited in §4 below, we state it.…”

Section: Introductionmentioning

confidence: 99%

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Characterizing shapes of compacta

Chapman¹

1972

Bull. Amer. Math. Soc.

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ABSTRACT. In this note we announce a characterization of Borsuk's concept of shape for finite-dimensional compact metric spaces. Let X, Y be compact subsets of some Euclidean space E 1 such that (1) X lies in some Euclidean subspace of E" having codimension at least 2 dim X + 1 and (2) Y lies in some Euclidean subspace of E 1 having codimension at least 2 dim F+ 1. Then n can be chosen large enough (and dependent only on dim X, dim Y) such that the following is true: X and Y have the same shape iff E?\X and E n \ Y are homeomorphic. We also discuss the relationship of this result to previously known characterizations of shape.

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Section: Statements Of Results For the Finite-dimensional Characterimentioning

confidence: 99%

Section: Statements Of Results For the Finite-dimensional Characterimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Characterizing shapes of compacta

Chapman¹

1972

Bull. Amer. Math. Soc.

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“…Shapes and complements. The beautiful result of Chapman [16] that two Z-sets in Q have the same shape if and only if their complements in Q are homeomorphic has inspired a number of investigations concerning similar theorems for compacta in E n or S n . First, Chapman [17] proved a finite dimensional version of his theorem, but needed strong codimension requirements as well as a fairly complex embedding condition.…”

Section: Fundamental Dimension and Euclidean Coefficients The Euclidmentioning

confidence: 99%