## HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS (708)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Goldbring708hilbert’sfifth,

author = {Isaac Goldbring},

title = {HILBERT’S FIFTH PROBLEM FOR LOCAL GROUPS},

year = {708}

}

### OpenURL

### Abstract

Abstract. We solve Hilbert’s fifth problem for local groups: every locally euclidean local group is locally isomorphic to a Lie group. Jacoby claimed a proof of this in 1957, but this proof is seriously flawed. We use methods from nonstandard analysis and model our solution after a treatment of Hilbert’s fifth problem for global groups by Hirschfeld. 1.

### Citations

60 |
Topological Transformation Groups, Interscience
- Montgomery, Zippin
- 1955
(Show Context)
Citation Context ...ubsection, we define the analogs of a few ordinary group theoretic notions in the local group setting. We will not need most of this material until Section 8. The presentation given here borrows from =-=[12]-=- and [16]. = (G′,1 ′,ι ′,p ′) are local groups with domain(ι) = Λ, domain(p) = Ω, domain(ι ′) = Λ ′ and domain(p ′) = Ω ′. A morphism from G to G ′ is a continuous function f : G → G ′ such that: (1) ... |

47 | Topological transformation groups
- Montgomery, Zippin
- 1974
(Show Context)
Citation Context ...ubsection, we define the analogs of a few ordinary group theoretic notions in the local group setting. We will not need most of this material until Section 8. The presentation given here borrows from =-=[12]-=- and [16]. = (G′,1 ′,ι ′,p ′) are local groups with domain(ι) = Λ, domain(p) = Ω, domain(ι ′) = Λ ′ and domain(p ′) = Ω ′. A morphism from G to G ′ is a continuous function f : G → G ′ such that: (1) ... |

38 | Applied Nonstandard Analysis - Davis - 1977 |

24 |
Topological Groups
- Pontrjagin
- 1958
(Show Context)
Citation Context ...red to as the Local H5): Is every locally euclidean local group locally isomorphic to a Lie group? In [8], Jacoby claims to have solved the Local H5 affirmatively. However, as pointed out by Plaut in =-=[16]-=-, Jacoby fails to recognize a subtlety in the way the associative law holds in local groups, as we will now explain. In our local groups, if the products xy,yz,x(yz), and (xy)z are all defined, then x... |

12 |
Groups without Small Subgroups
- Gleason
- 1952
(Show Context)
Citation Context ...ean topological group, does there always exist a C ω structure on G such that the group operations become C ω ? In 1952, Gleason, Montgomery, and Zippin answered this question in the affirmative; see =-=[4]-=- and [13]. (“C ω ” means “real analytic,” and by a C ω structure on a topological space X we mean a real analytic manifold, everywhere of the same finite dimension, whose underlying topological space ... |

11 | Foundations of non-standard analysis: A gentle introduction to non-standard extensions
- Henson
(Show Context)
Citation Context ...ld, many of the arguments have grown substantially in length due to the care needed in working with local groups. We assume familiarity with elementary nonstandard analysis; otherwise, consult [2] or =-=[5]-=-. We should also mention that McGaffey [11] has a completely different nonstandard approach to Hilbert’s fifth problem for local groups, using the connection between local Lie groups and finite-dimens... |

10 |
Lie algebras and locally compact groups
- Kaplansky
- 1971
(Show Context)
Citation Context ...ve. Proof. This is immediate from the well-known fact that a closed subgroup of the additive group of R different from {0} and R is of the form Zr with r ∈ R >0. □ Definition 9.3. Following Kaplansky =-=[9]-=-, we call a topological space feebly finite-dimensional if, for some n, it does not contain a homeomorphic copy of [0,1] n . Clearly locally euclidean local groups are feebly finite-dimensional. Lemma... |

10 | Non-associative local Lie groups
- Olver
- 1996
(Show Context)
Citation Context ...ds in local groups, as we will now explain. In our local groups, if the products xy,yz,x(yz), and (xy)z are all defined, then x(yz) = (xy)z. This condition is called “local associativity” by Olver in =-=[14]-=-. A much stronger condition for a local group to satisfy is “global associativity” in which, given any finite sequence of elements from the local group and two different ways of introducing parenthese... |

4 |
Cancellative semigroups on manifolds, Semigroup Forum 35
- Brown, Houston
- 1987
(Show Context)
Citation Context ...e suggests that this would be a worthwhile endeavor, as, for example, the solution of Hilbert’s fifth problem for cancellative semigroups on manifolds heavily relies on the truth of the Local H5; see =-=[1]-=- and [7]. Rather than reworking Jacoby’s paper, we have decided to mimic the nonstandard treatment of Hilbert’s fifth problem given by Hirschfeld [6]. Whereas nonstandard methods simplified the soluti... |

4 | More on cancellative semigroups on manifolds, Semigroup Forum 37 - Hofmann, Weiss - 1988 |

3 |
The nonstandard treatment of Hilbert’s fifth problem
- Hirschfeld
- 1990
(Show Context)
Citation Context ...s heavily relies on the truth of the Local H5; see [1] and [7]. Rather than reworking Jacoby’s paper, we have decided to mimic the nonstandard treatment of Hilbert’s fifth problem given by Hirschfeld =-=[6]-=-. Whereas nonstandard methods simplified the solution of Hilbert’s fifth problem, such methods are even more natural in dealing with the Local H5. The nonstandard proof of Hilbert’s fifth problem invo... |

3 |
Some theorems on the structure of locally compact local groups
- Jacoby
- 1957
(Show Context)
Citation Context ... section.) It is thus natural to pose a local version of Hilbert’s fifth problem (henceforth referred to as the Local H5): Is every locally euclidean local group locally isomorphic to a Lie group? In =-=[8]-=-, Jacoby claims to have solved the Local H5 affirmatively. However, as pointed out by Plaut in [16], Jacoby fails to recognize a subtlety in the way the associative law holds in local groups, as we wi... |

2 |
Small subgroups of finite dimensional groups
- Montgomery, Zippin
- 1952
(Show Context)
Citation Context ...logical group, does there always exist a C ω structure on G such that the group operations become C ω ? In 1952, Gleason, Montgomery, and Zippin answered this question in the affirmative; see [4] and =-=[13]-=-. (“C ω ” means “real analytic,” and by a C ω structure on a topological space X we mean a real analytic manifold, everywhere of the same finite dimension, whose underlying topological space is X.) Ho... |

2 | Associativity and the local version of Hilbert’s fifth problem - Plaut - 1993 |

1 |
den Dries, Unpublished notes
- van
- 1981
(Show Context)
Citation Context ...n G which is ruled by local 1-parameter subgroups in the sense that every element in this open neighborhood lies on some local 1-ps of G. In the rest of this section, we do not follow [6], but rather =-=[3]-=-, which contains the following series of lemmas in the global setting. Lemma 6.6. Suppose σ > N and a,b ∈ G(σ). Then [Xa] + [Xb] = [Xab]. Proof. From Theorem 6.5, we have S([Xa] + [Xb]) = S([Xa]) · S(... |

1 |
On local euclidean groups satisfying certain conditions
- Kuranishi
- 1950
(Show Context)
Citation Context ...have X(s)·X(δ) = X(δ)·X(s) and thus X(r+s) = X(1)·(X(δ)·X(s)) = (X(1)·X(δ))·X(s) = X(r)·X(s). □ 4. Consequences of NSS In this section we assume G is NSS. Special Neighborhoods The next lemma is from =-=[10]-=-, and we repeat its proof, with a more detailed justification of some steps. Lemma 4.1. There is a neighborhood V of 1 such that V ⊆ U2 and for all x,y ∈ V , if x 2 = y 2 , then x = y.20 ISAAC GOLDBR... |

1 |
A partial solution to a generalization of Hilbert’s local fifth problem: the standard part of a locally euclidean local nonstandard Lie group is an analytic Lie group
- McGaffey
(Show Context)
Citation Context ...antially in length due to the care needed in working with local groups. We assume familiarity with elementary nonstandard analysis; otherwise, consult [2] or [5]. We should also mention that McGaffey =-=[11]-=- has a completely different nonstandard approach to Hilbert’s fifth problem for local groups, using the connection between local Lie groups and finite-dimensional real Lie algebras to reduce to a prob... |

1 |
One parameter subgroups and nonstandard analysis
- Singer
- 1976
(Show Context)
Citation Context ...o the identity; therefore, much of [6] goes through in the local setting, since the infinitesimals form an actual group. These infinitesimals are used to generate local one-parameter subgroups, as in =-=[17]-=- and [6]. While our solution of the Local H5 is along the lines of the proof given by Hirschfeld, many of the arguments have grown substantially in length due to the care needed in working with local ... |