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See detailOn the Cohomological Crepant Resolution Conjecture for the complexified Bianchi orbifolds
Perroni, Fabio; Rahm, Alexander UL

in Algebraic and Geometric Topology (in press)

We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is ... [more ▼]

We give formulae for the Chen--Ruan orbifold cohomology for the orbifolds given by a Bianchi group acting on complex hyperbolic 3-space. The Bianchi groups are the arithmetic groups PSL_2(A), where A is the ring of integers in an imaginary quadratic number field. The underlying real orbifolds which help us in our study, given by the action of a Bianchi group on real hyperbolic 3-space (which is a model for its classifying space for proper actions), have applications in physics. We then prove that, for any such orbifold, its Chen-Ruan orbifold cohomology ring is isomorphic to the usual cohomology ring of any crepant resolution of its coarse moduli space. By vanishing of the quantum corrections, we show that this result fits in with Ruan's Cohomological Crepant Resolution Conjecture. [less ▲]

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See detailClassifying spaces of algebras over a prop
Yalin, Sinan UL

in Algebraic and Geometric Topology (2014), 14(5), 2561--2593

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy ... [more ▼]

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop leads us to introduce new methods to overcome this difficulty. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes. [less ▲]

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