Geodesic
Topological and Homological Properties of the Orbit Space of a Simple Three-Dimensional Compact Linear Lie Group
Matematičeskie zametki, Tome 113 (2023) no. 3, pp. 440-447.

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The question of whether the orbit space of a compact linear group is a topological manifold and a homology manifold is considered. The case of a simple three-dimensional group is considered. An upper bound is obtained for the sum of integral parts of the halved dimensions of irreducible components for a representation whose quotient is a homology manifold. This strengthens a similar result obtained previously, which gave such a bound in the case where the quotient of the representation is a smooth manifold. Most representations for which the obtained estimate holds have also been considered previously. The argument uses standard considerations of linear algebra and the theory of Lie groups and algebras and their representations.
Mots-clés : Lie group
Keywords: linear representation of a group, topological quotient of an action, topological manifold, homology manifold. 
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O. G. Styrt. Topological and Homological Properties of the Orbit Space of a Simple Three-Dimensional Compact Linear Lie Group. Matematičeskie zametki, Tome 113 (2023) no. 3, pp. 440-447. http://geodesic.mathdoc.fr/item/MZM_2023_113_3_a9/

[1] M. A. Mikhailova, “O faktorprostranstve po deistviyu konechnoi gruppy, porozhdennoi psevdootrazheniyami”, Izv. AN SSSR. Ser. matem., 48:1 (1984), 104–126 | MR | Zbl

[2] C. Lange, “When is the underlying space of an orbifold a manifold?”, Trans. Amer. Math. Soc., 372:4 (2019), 2799–2828 | MR

[3] O. G. Styrt, “O prostranstve orbit kompaktnoi lineinoi gruppy Li s kommutativnoi svyaznoi komponentoi”, Tr. MMO, 70 (2009), 235–287 | MR | Zbl

[4] O. G. Styrt, “O prostranstve orbit trekhmernoi kompaktnoi lineinoi gruppy Li”, Izv. RAN. Ser. matem., 75:4 (2011), 165–188 | MR | Zbl

[5] O. G. Styrt, “O prostranstve orbit neprivodimogo predstavleniya spetsialnoi unitarnoi gruppy”, Tr. MMO, 74:1 (2013), 175–199 | Zbl

[6] O. G. Styrt, “On the orbit spaces of irreducible representations of simple compact Lie groups of types $B$, $C$, and $D$”, J. Algebra, 415 (2014), 137–161 | DOI | MR

[7] O. G. Styrt, Topological and Homological Properties of the Orbit Space of a Compact Linear Lie Group with Commutative Connected Component, arXiv: math.AG/1607.06907

[8] O. G. Styrt, “Topologicheskie i gomologicheskie svoistva prostranstva orbit kompaktnoi lineinoi gruppy Li s kommutativnoi svyaznoi komponentoi”, Vest. MGTU im. N. E. Baumana. Ser. Est. nauki, 2018, no. 3, 68–81

[9] O. G. Styrt, “Topologicheskie i gomologicheskie svoistva prostranstva orbit kompaktnoi lineinoi gruppy Li s kommutativnoi svyaznoi komponentoi. Vyvody”, Vest. MGTU im. N. E. Baumana. Ser. Est. nauki, 2018, no. 6, 48–63

[10] G. Bredon, Vvedenie v teoriyu kompaktnykh grupp preobrazovanii, Nauka, M., 1980 | MR