Abstract: For a given group $G$ the category of $G$-spaces and $G$-equivariant maps admits a model structure in which the weak equivalences and fibrations are defined as $G$-maps that induce weak equivalences and fibrations on $H$-fixed point spaces for every $H \leq G$. In this model category the fibrant-cofibrant objects are $G$-$CW$-complexes. A weak equivalence between such objects is a $G$-homotopy equivalence; and thus, induces weak equivalences on $H$-orbits for every $H \leq G$. The converse, however, is not true. It is natural to ask what is needed for $X,Y$ so that maps $f:X\to Y$ inducing weak equivalences on $H$-orbits also induce weak equivalences on $H$-fixed point spaces. To provide an answer, we construct a new model structure on the category of $G$-spaces in which the weak equivalences and cofibrations are defined as maps inducing weak equivalences and cofibrations on $H$-orbits for each $H \leq G$. We show that a weak equivalence between objects that are fibrant in this new model structure is a weak equivalence in the fixed point model structure. This is a joint work with Aslı Güçlükan İlhan.
Fizik-Matematik Seminer Odası