Aura Lucina Kantún-Montiel

Extensorial and Homotopic Properties for Almost Connected Groups.

In this talk we will focus on the equivariant homotopy theory, dealing with the homotopic properties of G-spaces, which are topological spaces equipped with a continuous action of a topological group G. A G-fibrant space is the equivariant version of a fibrant space in the sense of F. Cathey [4], that is to say, a G-fibrant space is a G-space with the extension property with respect to G–SSDR-maps.

As well as in the non-equivariant case, G-fibrant spaces constitute a generalization of equivariant Absolute Neighborhood Extensors (G–ANE). That equivariant version was introduced in [2] by A. Bykov and M. Texis in order to construct the equivariant strong shape category (G–sSh\mathcal{CM}) [3], where G is a compact group, and proved that all compact metrizable groups are G-fibrant spaces.

Due to the extensorial properties for almost connected group actions presented in [1], we can extend the G–sSh\mathcal{CM} category to the non-compact case and show that the natural projections of almost connected groups are strong G-fibrations and, as a consequence, locally compact almost connected groups are G-fibrant spaces as well.

References

[1] S. Antonyan, Extensorial properties of orbit spaces of proper group actions , Topol. Appl. 98 (1999) 35—46.

[2] A. Bykov, M. Texis, Equivariant fibrant spaces, Glasnik Matematički 40 (60) (2005), 323—331.

[3] A. Bykov, M. Texis, Equivariant strong shape, Top. Appl. 154 (2007) 2026—2039.

[4] F. Cathey, Strong shape theory, in: Shape Theory and Geometric Topology, Lecture Notes in Math. 870, Springer, Berlin (1981) 216—239.