The Piecewise-Linear Structure of Euclidean Space

Stallings, John R.

doi:10.1017/s0305004100036756

Cited by 204 publications

(91 citation statements)

References 9 publications

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“…In §l we explain our notation and terminology and also introduce some preliminary facts that are used (often without specific reference) later on in the paper. In §2 we prove an equivariant engulfing theorem, Theorem 2.4, which is an equivariant analogue of Stallings' engulfing theorem [16]. The main result in §3 is Proposition 3.2 which gives necessary and sufficient conditions for an equivariant p.l.…”

Section: ) Each Codimension 2 Fixed-point Situation (Lh L(>h2») Is mentioning

confidence: 99%

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Recognition of linear actions on spheres

Illman¹

1982

Trans. Amer. Math. Soc.

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ABSTRACT. Let G be a finite group acting smoothly on a homotopy sphere ~m. We wish to establish necessary and sufficient conditions for the given G-action on ~ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a G-homeomorphism y: ~ -. sm(p), where sm(p) c Rm+l(p) denotes the unit sphere in an orthogonal representation space Rm+l(p) for G. In order for a G-action on ~ to be topologically equivalent to a linear action it is clearly necessary that:(i) For each subgroup H of G the fixed-point set ~H is homeomorphic to a sphere, or empty.(ii) For any subgroups Hand H <; Hi' I .;; i .;; k, of G the pair (~Il, U7= l~Hi) is homeomorphic to a standard pair (sn, U7=lS,"i), where each Sri, I.;; i';; k, is a standard n i-subsphere of sn.In this paper we consider the case where the fixed-point set ~G is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need:(ii)' For any subgroups H and H <; Hi' I .;; i .;; p, of G such that dim ~H, = dim ~H ~ 2, the pair (~Il, U;= l~Hi) is homeomorphic to a standard pair (Sn, U;= lSin-2), where each Sr-2 , I .;; i';; p, is a standard (n ~ 2)-subsphere of sn.Our main results are then that, in the case when G is abelian, conditions (i) and (ii)' are necessary and sufficient for a given G-action on ~ to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension I and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.Let G be a finite group acting smoothly on a homotopy sphere ~m. (A homotopy sphere ~m is a smooth manifold which is homeomorphic to sm.) We wish to establish necessary and sufficient conditions for the given G-action on ~ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a G-homeomorphism y: ~ --> sm(p), where sm(p) c R m + I(p) denotes the unit sphere in an orthogonal representation space Rm+ I(p) for G. In order for a given G-action on ~ to be topologically equivalent to a linear action it is clearly necessary that the following two conditions hold:(i) For each subgroup H of G the fixed-point set ~H is homeomorphic to a sphere, or empty.

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Section: ) Each Codimension 2 Fixed-point Situation (Lh L(>h2») Is mentioning

confidence: 99%

“…The equivariant engulfing theorem. In this section we establish an equivariant analogue of Stallings' engulfing theorem [16]. First we consider equivariant collapsing and prove the easy but crucial Proposition 2.3 below.…”

Section: ) Each Codimension 2 Fixed-point Situation (Lh L(>h2») Is mentioning

confidence: 99%

Recognition of linear actions on spheres

Illman¹

1982

Trans. Amer. Math. Soc.

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“…The key technical tool developed in the book is that of engulfing [Sta1]. Engulfing is a method designed to stretch an open set, using regular neighborhood theory, to surround a critical subcomplex.…”

Section: To Determine Which Set-theoretic Structures Have a Connectiomentioning

confidence: 99%

Book Review: Embeddings in manifolds

Cannon¹

2011

Bull. Amer. Math. Soc.

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“…It is known to be true for manifolds of dimension three or less ([13], [14]) and for many high dimensional compact manifolds ( [20], [21]), but the only high dimensional result for open (i.e. noncompact with empty boundary) manifolds is that it is true in case the manifolds are topologically Fn («^5) [19].…”

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confidence: 99%