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- W2019804543 abstract "A locally compact group with compact boundary is a locally compact topological semigroup in which an open subgroup is dense and has a compact complement. These semigroups were studied in a previous paper whose results and notation are freely used in the present work [2]. The general assumption made in this study in addition to the hypotheses mentioned is the existence of transitive group of homeomorphisms on the space of the semigroup. We restrict ourselves essentially to the connected case although the hypothesis we use actually is that the component of the identity of G be not compact; the question of totally disconnected homogeneous groups with boundary is therefore unsettled. The first main result which we obtain exhibits the fact that in a connected locally compact homogeneous group with boundary the influence of the boundary on the topological structure of S is so strong that, if the boundary B is finite dimensional, all of S is finite dimensional. The method of the proof uses a local fibering of S with cosets of the subsemigroup K as fiber, where K is the set of all elements s such that se = e with the identity e of B, and with neighborhoods of B as bases. A group theoretical lemma which is of some interest in its own right has to be proved to obtain the theorem mentioned: If a locally compact group has a neighborhood of the identity whose homotopy groups relative to the identity vanish for all positive dimensions, then there are arbitrarily small neighborhoods which are topologically the product of an euclidean cell, a (not necessarily finite dimensional) solenoid and a totally disconnected compact subset of G; all small connected subgroups are solenoids. It is obvious what happens, then, if G is in addition locally connected. If the boundary B of G in S is not only finite dimensional but also locally connected, then S is a manifold, i.e. is locally euclidean; hence both G and B are Lie groups. Theorem II gives a full description of the structure of S in this case. It is either a direct product of the full multiplicative semigroup of real numbers with some compact group or else the boundary is contained in an Lsemigroup in the sense of Mostert and Shields whose results we use to dispose" @default.
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- W2019804543 date "1963-01-01" @default.
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- W2019804543 title "Homogeneous locally compact groups with compact boundary" @default.
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- W2019804543 doi "https://doi.org/10.1090/s0002-9947-1963-0144320-9" @default.
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