92 research outputs found
From mapping class groups to automorphism groups of free groups
We show that the natural map from the mapping class groups of surfaces to the
automorphism groups of free groups, induces an infinite loop map on the
classifying spaces of the stable groups after plus construction. The proof uses
automorphisms of free groups with boundaries which play the role of mapping
class groups of surfaces with several boundary components.Comment: to appear in J. Lond. Math. So
Homological stability for mapping class groups of surfaces
We give a complete and detailed proof of Harer's stability theorem for the
homology of mapping class groups of surfaces, with the best stability range
presently known. This theorem and its proof have seen several improvements
since Harer's original proof in the mid-80's, and our purpose here is to
assemble these many additions.Comment: Proof of claim 3 in the spectral sequence argument corrected (see new
lemma 2.5, corollaries 2.6 and 2.7). To appear in the Handbook of Modul
Homological stability for the mapping class groups of non-orientable surfaces
We prove that the homology of the mapping class groups of non-orientable
surfaces stabilizes with the genus of the surface. Combining our result with
recent work of Madsen and Weiss, we obtain that the classifying space of the
stable mapping class group of non-orientable surfaces, up to homology
isomorphism, is the infinite loop space of a Thom spectrum build from the
canonical bundle over the Grassmannians of 2-planes in R^{n+2}. In particular,
we show that the stable rational cohomology is a polynomial algebra on
generators in degrees 4i--this is the non-oriented analogue of the Mumford
conjecture
Homological stability for classical groups
We prove a slope 1 stability range for the homology of the symplectic,
orthogonal and unitary groups with respect to the hyperbolic form, over any
fields other than , improving the known range by a factor 2 in the case of
finite fields. Our result more generally applies to the automorphism groups of
vector spaces equipped with a possibly degenerate form (in the sense of Bak,
Tits and Wall). For finite fields of odd characteristic, and more generally
fields in which -1 is a sum of two squares, we deduce a stability range for the
orthogonal groups with respect to the Euclidean form, and a corresponding
result for the unitary groups.
In addition, we include an exposition of Quillen's unpublished slope 1
stability argument for the general linear groups over fields other than ,
and use it to recover also the improved range of
Galatius-Kupers-Randal-Williams in the case of finite fields, at the
characteristic.Comment: v2: Revision. Now recovers the Galatius-Kupers-Randal-Williams
improved stability range for general linear groups over finite field
The homology of the Higman-Thompson groups
We prove that Thompson's group is acyclic, answering a 1992 question of
Brown in the positive. More generally, we identify the homology of the
Higman-Thompson groups with the homology of the zeroth component of
the infinite loop space of the mod Moore spectrum. As , we
can deduce that this group is acyclic. Our proof involves establishing
homological stability with respect to , as well as a computation of the
algebraic K-theory of the category of finitely generated free Cantor algebras
of any type .Comment: 49 page
Stabilization for mapping class groups of 3-manifolds
We prove that the homology of the mapping class group of any 3-manifold
stabilizes under connected sum and boundary connected sum with an arbitrary
3-manifold when both manifolds are compact and orientable. The stabilization
also holds for the quotient group by twists along spheres and disks, and
includes as particular cases homological stability for symmetric automorphisms
of free groups, automorphisms of certain free products, and handlebody mapping
class groups. Our methods also apply to manifolds of other dimensions in the
case of stabilization by punctures.Comment: v4: improvements in the exposition as well as improvements in the
main combinatorial theorem in the paper, concerning complexes built from
join
Homological stability for automorphism groups of RAAGs
We show that the homology of the automorphism group of a right-angled Artin
group stabilizes under taking products with any right-angled Artin group.Comment: final versio
- …