Crossed modules as maps between connected components of topological groups. (English) Zbl 1388.18021
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The aim of this paper is to prove:
1) Suppose that \(\mathfrak{M}\) is a normal subgroup of the topological group \(\mathfrak{X}\). Then the induced map \(\pi_{0}(\mathfrak{M}) \rightarrow \pi_{0}(\mathfrak{X})\) is a normal map, with a crossed module structure induced by conjugation.
2) Conversely, given a normal map \(n: N \rightarrow G\) of discrete groups, there is a topological group \(\mathfrak{X}\) and a closed normal subgroup \(\mathfrak{M}\) of \(\mathfrak{X}\) with a commuting diagram and isomorphisms as follows: \[ \begin{tikzcd} \pi_0(\mathfrak M) \arrow[r, "\overline n"] \arrow[d, "\cong" left] & \pi_0(\mathfrak X) \arrow[d, "\cong"] \\ N \arrow[r, "n"] & G\end{tikzcd}, \] with \(\overline{n}\) being the naturally induced map of connected components.
3) The topological groups \(\mathfrak{M}\) and \(\mathfrak{X}\), in part 2), can be constructed naturally as homotopically discrete groups.
(Compare with [O. Mucuk et al., Appl. Categ. Struct. 23, No. 3, 415–428 (2015; Zbl 1316.18011)] and [K. Norrie, Bull. Soc. Math. Fr. 118, No. 2, 129–146 (1990; Zbl 0719.20018)]).
The aim of this paper is to prove:
1) Suppose that \(\mathfrak{M}\) is a normal subgroup of the topological group \(\mathfrak{X}\). Then the induced map \(\pi_{0}(\mathfrak{M}) \rightarrow \pi_{0}(\mathfrak{X})\) is a normal map, with a crossed module structure induced by conjugation.
2) Conversely, given a normal map \(n: N \rightarrow G\) of discrete groups, there is a topological group \(\mathfrak{X}\) and a closed normal subgroup \(\mathfrak{M}\) of \(\mathfrak{X}\) with a commuting diagram and isomorphisms as follows: \[ \begin{tikzcd} \pi_0(\mathfrak M) \arrow[r, "\overline n"] \arrow[d, "\cong" left] & \pi_0(\mathfrak X) \arrow[d, "\cong"] \\ N \arrow[r, "n"] & G\end{tikzcd}, \] with \(\overline{n}\) being the naturally induced map of connected components.
3) The topological groups \(\mathfrak{M}\) and \(\mathfrak{X}\), in part 2), can be constructed naturally as homotopically discrete groups.
(Compare with [O. Mucuk et al., Appl. Categ. Struct. 23, No. 3, 415–428 (2015; Zbl 1316.18011)] and [K. Norrie, Bull. Soc. Math. Fr. 118, No. 2, 129–146 (1990; Zbl 0719.20018)]).
Reviewer: Saïd Zarati (Tunis)
MSC:
| 18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |
| 55U10 | Simplicial sets and complexes in algebraic topology |
| 57T30 | Bar and cobar constructions |
| 55Q05 | Homotopy groups, general; sets of homotopy classes |
References:
| [1] | R. Brown and C. B. Spencer, G-groupoid, crossed modules and fundamental groupoid of topological groups, Indag. Math. 38 (1976), no. 4, 296-302.; Brown, R.; Spencer, C. B., G-groupoid, crossed modules and fundamental groupoid of topological groups, Indag. Math., 38, 4, 296-302 (1976) · Zbl 0333.55011 |
| [2] | E. D. Farjoun and Y. Segev, Crossed modules as homotopy normal maps, Topology Appl. 157 (2010), no. 2, 359-368.; Farjoun, E. D.; Segev, Y., Crossed modules as homotopy normal maps, Topology Appl., 157, 2, 359-368 (2010) · Zbl 1189.55006 |
| [3] | J. P. May, Simplicial Ojects in Algebraic Topology, University of Chicago Press, Chicago, 1992.; May, J. P., Simplicial Ojects in Algebraic Topology (1992) · Zbl 0769.55001 |
| [4] | O. Mucuk, T. Sahan and N. Alemdar, Normality and quotients in crossed modules and group-groupoids, Appl. Categ. Structures 23 (2015), 415-428.; Mucuk, O.; Sahan, T.; Alemdar, N., Normality and quotients in crossed modules and group-groupoids, Appl. Categ. Structures, 23, 415-428 (2015) · Zbl 1316.18011 |
| [5] | K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), no. 2, 129-146.; Norrie, K., Actions and automorphisms of crossed modules, Bull. Soc. Math. France, 118, 2, 129-146 (1990) · Zbl 0719.20018 |
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