The fundamental group of the orbit space of a discontinuous group

Armstrong, M. A.

doi:10.1017/s0305004100042845

Cited by 107 publications

(137 citation statements)

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“…If π : X → X/ denotes the canonical projection, then π • δ is a loop in X/ based at p 0 , and because X is simply connected, its homotopy class does not depend on the choice of γ and we can write [γ ] = ϕ(c) for a uniquely defined ϕ(c) ∈ π 1 (X/ , p 0 ). The theorem of Armstrong [1] (O, p 0 ), and that its kernel is equal to the smallest normal subgroup A of which contains all c ∈ such that…”

Section: Topology Of the Orbit Spacementioning

confidence: 99%

Topology of symplectic torus actions with symplectic orbits

Duistermaat

Pelayo

2010

Rev Mat Complut

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We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M/T is a very good orbifold with first Betti number b 1 (M/T ) = b 1 (M) − dim T .

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Section: Topology Of the Orbit Spacementioning

confidence: 99%

Topology of symplectic torus actions with symplectic orbits

Duistermaat

Pelayo

2010

Rev Mat Complut

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“…Results for simplicial and discontinuous groups have been given in [1] and [2]. The object of this note is to produce a theorem which can deal with both discontinuous and continuous actions.…”

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confidence: 99%

Calculating the Fundamental Group of an Orbit Space

Armstrong¹

1982

Proceedings of the American Mathematical Society

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Abstract.Suppose G acts effectively as a group of homeomorphisms of the connected, locally path connected, simply connected, locally compact metric space X. Let G denote the closure of G in Homeo^), and N the smallest normal subgroup of G which contains the path component of the identity of G and all those elements of G which have fixed points. We show that irx(X/G) is isomorphic to G/N subject to a weak path lifting assumption for the projection X -* X/G.Given a topological space X together with a group G of homeomorphisms of X, what can we say about the fundamental group of the orbit space X/G! Results for simplicial and discontinuous groups have been given in [1] and [2]. The object of this note is to produce a theorem which can deal with both discontinuous and continuous actions.We shall assume that A' is a connected, locally path connected, locally compact metric space. Let G be a group of homeomorphisms of X which acts effectively on X, so that we can think of G as a subgroup of the group Homeo(Ar) of all homeomorphisms of X endowed with the compact open topology.Under very reasonable hypotheses (see conditions A,B,C below) the answer to our question is as follows. Let G denote the closure of G in Homeo(Ar), and let N be the smallest normal subgroup of G which contains the path component of the identity of G and all those elements of G which have fixed points. Then if A' is simply connected the fundamental group of X/G is isomorphic to the quotient group G/N.Suppose X fails to be simply connected but has a universal covering space X. Each homeomorphism g: X -» X lifts to a homeomorphism of X, and any two lifts of the same g differ by a covering transformation. Therefore we have an action of an extension of itx(X) by G on X whose orbit space is homeomorphic to X/G, and we can apply our result in this setting. Details of the group extension and of its action on X can be found in [5] and [3].Here are some examples to illustrate a variety of situations in which the result can be used. Example 1. Take S2 X R for X and Sx X Z for G, the action being as follows.The circle acts on S2 by rotation leaving the north and south poles fixed, and acts trivially on R. The generator of Z reflects S2 in the equator and translates R along

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“…Thus, in [5,3,5], the subgroup φ~ι(G\) contains no element, except the identity, fixing a point of H 3 . Thus, by the Theorem in [1], φ~ι(G\) is the fundamental group of its orbit space, the Weber-Seifert manifold. From the given relations, it is easy to calculate that the quotient group of ^"^(Gi) over its derived group is isomorphic to C5 x C5 x C 5 , thus confirming the calculation at the end of [11].…”

Section: The Spherical Spacesmentioning

confidence: 99%

“…By the Theorem in [1], the groups φ" x {G\) and φ~ι{G 2 ) are the fundamental groups of these manifolds. The Reidemeister-Schreier process in CAYLEY shows that φ~ι(G\) is generated by p = bacbdacdbcda, q = adcbdcbabcdc, r = ababcdabcdba, s = cdcdcbdcbaba, and has defining relations, In the representation of [5,3,5] on the left cosets of φ~ι{G\), the representation of each of the stabilizers V, E, F and C is faithful and the image of φ~ι (G\) intersects their images, and the images of all their conjugates, trivially.…”

Section: The Spherical Spacesmentioning

confidence: 99%

Four dodecahedral spaces

Lorimer¹

1992

Pacific J. Math.

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The fundamental group of the orbit space of a discontinuous group

Cited by 107 publications

References 3 publications

Topology of symplectic torus actions with symplectic orbits

Topology of symplectic torus actions with symplectic orbits

Calculating the Fundamental Group of an Orbit Space

Four dodecahedral spaces

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