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**1 - 5**of**5**### DIFFERENTIAL TOPOLOGY 46 YEARS LATER

"... field which was then young but growing very rapidly. During the intervening years, many problems in differential and geometric topology which had seemed totally impossible have been solved, often using drastically new tools. The following is a ..."

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field which was then young but growing very rapidly. During the intervening years, many problems in differential and geometric topology which had seemed totally impossible have been solved, often using drastically new tools. The following is a

### BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)

"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."

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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental

### Figure 1).

"... 1. The recognition problem for topological manifolds. 1.1. DEFINITION. Let E n denote the collection of «-tuples x * (xv..., xn), xt real. Define d(x,y) « Ç2(xt- y) 2) x/2. Then E " becomes a metric space with metric d and is called «-dimensional Euclidean space. A (topological) «-manifold M i ..."

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1. The recognition problem for topological manifolds. 1.1. DEFINITION. Let E n denote the collection of «-tuples x * (xv..., xn), xt real. Define d(x,y) « Ç2(xt- y) 2) x/2. Then E " becomes a metric space with metric d and is called «-dimensional Euclidean space. A (topological) «-manifold M is a separable metric space locally homeomorphic with E n (see