4 research outputs found
The homotopy type of the loops on -connected -manifolds
For we compute the homotopy groups of -connected closed
manifolds of dimension . Away from the finite set of primes dividing
the order of the torsion subgroup in homology, the -local homotopy groups of
are determined by the rank of the free Abelian part of the homology.
Moreover, we show that these -local homotopy groups can be expressed as a
direct sum of -local homotopy groups of spheres. The integral homotopy type
of the loop space is also computed and shown to depend only on the rank of the
free Abelian part and the torsion subgroup.Comment: Trends in Algebraic Topology and Related Topics, Trends Math.,
Birkhauser/Springer, 2018. arXiv admin note: text overlap with
arXiv:1510.0519
Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture
We prove that any countable discrete and torsion free subgroup of a general
linear group over an arbitrary field or a similar subgroup of an almost
connected Lie group satisfies the integral algebraic K-theoretic (split)
Novikov conjecture over \cpt and \S, where \cpt denotes the C^*-algebra of
compact operators and \S denotes the algebra of Schatten class operators. We
introduce assembly maps with finite coefficients and under an additional
hypothesis, we prove that such a group also satisfies the algebraic K-theoretic
Novikov conjecture over \bar{\mathbb{Q}} and \mathbb{C} with finite
coefficients. For all torsion free Gromov hyperbolic groups G, we demonstrate
that the canonical algebra homomorphism \cpt[G]\map C^*_r(G)\hat{\otimes}\cpt
induces an isomorphism between their algebraic K-theory groups.Comment: v2 Exposition improved; one lemma and grant acknowledgement added; v3
some terminology changed and details added, Theorems 4.5 and 4.7 in v3 need
an extra hypothesis; v4 abridged version accepted for publication in JHR