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Soren Galatius

Publications and source records attributed to Soren Galatius.

31 records · Page 2Linked to original sources
arXiv2015

Abelian quotients of mapping class groups of highly connected manifolds

We compute the abelianisations of the mapping class groups of the manifolds $W_g^{2n} = g(S^n \times S^n)$ for $n \geq 3$ and $g \geq 5$. The answer is a direct sum of two parts. The first part arises from the action of the mapping class group on the middle homology, and takes values in the abelianisation of the automorphism group of the middle homology. The second part arises from bordism classes of mapping cylinders and takes values in the quotient of the stable homotopy groups of spheres by a certain subgroup which in many cases agrees with the image of the stable $J$-homomorphism. We relate its calculation to a purely homotopy theoretic problem.

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arXiv2013

Detecting and realising characteristic classes of manifold bundles

We apply our earlier work on the higher-dimensional analogue of the Mumford conjecture to two questions. Inspired by work of Ebert we prove non-triviality of certain characteristic classes of bundles of smooth closed manifolds. Inspired by work of Church-Farb-Thibault and Church-Crossley-Giansiracusa we investigate the dependence of characteristic classes of bundles on characteristic numbers of its fibre, total space and base space.

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arXiv2012

Stable moduli spaces of high dimensional manifolds

We prove an analogue of the Madsen-Weiss theorem for high dimensional manifolds. For example, we explicitly describe the ring of characteristic classes of smooth fibre bundles whose fibres are connected sums of g copies of S^n x S^n, in the limit as g tends to infinity. Rationally it is a polynomial ring in certain explicit generators, giving a high dimensional analogue of Mumford's conjecture. More generally, we study a moduli space N(P) of those null-bordisms of a fixed (2n-1)-dimensional manifold P which are highly connected relative to P. We determine the homology of N(P) after stabilisation using certain self-bordisms of P. The stable homology is identified with that of a certain infinite loop space.

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arXiv2012

Homological stability for moduli spaces of high dimensional manifolds

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises Harer's stability theorem for the homology of mapping class groups. Combined with previous work of the authors, it gives a calculation of the homology of these moduli spaces in a range of degrees.

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arXiv2010

The homotopy type of the cobordism category

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d=2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one.

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arXiv2009

Monoids of moduli spaces of manifolds

We study categories of d-dimensional cobordisms from the perspective of Tillmann and Galatius-Madsen-Tillmann-Weiss. There is a category $C_θ$ of closed smooth (d-1)-manifolds and smooth d-dimensional cobordisms, equipped with generalised orientations specified by a fibration $θ: X \to BO(d)$. The main result of GMTW is a determination of the homotopy type of the classifying space $BC_θ$. The goal of the present paper is a systematic investigation of subcategories $D$ of $C_θ$ having classifying space homotopy equivalent to that of $C_θ$, the smaller such $D$ the better. We prove that in most cases of interest, $D$ can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with $θ$-structure is the cohomology of the infinite loop space of a certain Thom spectrum. This was known for certain special $θ$, using homological stability results; our work is independent of such results and covers many more cases.

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arXiv2009

Madsen-Weiss for geometrically minded topologists

We give an alternative proof of Madsen-Weiss' generalized Mumford conjecture. Our proof is based on ideas similar to Madsen-Weiss' original proof, but it is more geometrical and less homotopy theoretical in nature. At the heart of the argument is a geometric version of Harer stability, which we formulate as a theorem about folded maps.

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arXiv2008

Stable homology of automorphism groups of free groups

Homology of the group Aut(F_n) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular, Aut(F_n) has no stable rational homology, settling a conjecture of Hatcher and Vogtmann.

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arXiv2008

Universal moduli spaces of surfaces with flat connections and cobordism theory

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.

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arXiv2007

Divisibility of the stable Miller-Morita-Mumford classes

We determine the sublattice generated by the Miller-Morita-Mumford classes $κ_i$ in the torsion free quotient of the integral cohomology ring of the stable mapping class group. We further decide when the mod p reductions $κ_i$ vanish.

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arXiv2005

Mod 2 homology of the stable spin mapping class group

We compute the mod 2 homology of spin mapping class groups in the stable range. In earlier work we computed the stable mod p homology of the oriented mapping class group, and the methods and results here are very similar. The forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd p but for p=2 it is far from being an isomorphism. We include a general discussion of tangential structures on 2-manifolds and their mapping class groups and then specialise to spin structures. As in Madsen and Weiss' solution to Mumford's conjecture, the stable homology is the homology of the zero space of a certain Thom spectrum.

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arXiv2004

Secondary Characteristic Classes of Surface Bundles

The Miller-Morita-Mumford classes associate to an oriented surface bundle $E\to B$ a class $κ_i(E) \in H^{2i}(B;\Z)$. In this note we define for each prime $p$ and each integer $i\geq 1$ a secondary characteristic class $λ_i(E) \in H^{2i(p-1)-2}(B;\Z)/\Zκ_{i(p-1)-1}$. The mod $p$ reduction $λ_i(E) \in H^*(B; \F_p)$ has zero indeterminacy and satisfies $pλ_i(E) = κ_{i(p-1)-1}(E) \in H^*(B;\Z/p^2)$.

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