In this note we show that sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces are homeomorphic to 3-manifolds. As a corollary we obtain the validity of the Borel conjecture for sufficiently collapsed, irreducible and aspherical Alexandrov 3-spaces.

Let Γ be a discrete group. Assuming rational injectivity of the Baum–Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz’ positive scalar curvature sequence for BΓ. The lower bounds are formulated in terms of the part of degree up to 2 in the group homology of Γ with coefficients in the CΓ-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the c...

We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and recover known results when the twisting comes from finite order elements in discrete torsion.

Let A be an Artin algebra over a commutative Artinian ring k. If M is a finitely generated left A-module, let f:PM→M denote the minimal projective cover of M and let Ω(M)=kerf. In this paper the following interesting result is proven: if M,N are finitely generated left A-modules such that Ext1A(M,M)=Ext1A(N,N)=0, then M≅N if and only if M/radM≅N/radN and Ω(M)≅Ω(N). The proof is not easy and depends on obtaining results about "lift categories'' and "categories of complexes''. A corollary of th...

We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.

We define Twisted Equivariant K-homology groups using geometric cycles. We compare them with analytical approaches using Kasparov KK-theory and (twisted) C∗-algebras of groups and groupoids.

this paper deals with the representation theory of strongly, locally finite quivers. We first study some properties of the finitely presented or co-presented representations, and then construct in the category of locally finite-dimensional representations some almost split sequences which start with a finitely co-presented representation and end with a finitely presented representation. Furthermore, we obtain a general description of the shapes of the Auslander–Reiten components of the catego...