Over a monoidal model category, under some mild assumptions, we equip the categories of colored PROPs and their algebras with projective model category structures. A BoardmanVogt style homotopy invariance result about algebras over cofibrant colored PROPs is proved. As an example, we define homotopy topological conformal field theories and observe that such structures are homotopy invariant.

LetG be a locally compact topological group and EG its universal space for the family of compact subgroups. We give criteria for this space to be G-homotopy equivalent to a d-dimensional G-CW -complex, a finite G-CW -complex or a G-CW -complex of finite type. Essentially we reduce these questions to discrete groups, and to the homological algebra of the orbit category of discrete groups with re...

In this paper, we introduce and study the notion of precompacttopological generalized groups and some new results are given.

We show that if U∗ is a hypercover of a topological space X then the natural map hocolim U∗→X is a weak equivalence. This fact is used to construct topological realization functors for the A1-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant. Mathematics Subject Classification (2000...

It is shown that the nth quasitopological homotopy group of a topological space isomorphic to (n − 1)th its loop space. By using this fact, we obtain some results about groups. Finally, with help long exact sequence based pair and fibration in qTop introduced by Brazas 2013, field.

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2 ⊂ Sn, n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial. It is well-known that if the c...