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Thomas Church

Publications and source records attributed to Thomas Church.

25 records · Page 2Linked to original sources
arXiv2012

On the geometric nature of characteristic classes of surface bundles

Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2 surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) := in Z. We prove that when n is odd the number e_n^#(E -> M) depends only on the diffeomorphism type of E, not on g, M, or the map E -> M. More generally, we prove that e_n^#(E -> M) depends only on the cobordism class of E. Recent work of Hatcher implies that this stronger statement is false when n is even. If E -> M is a holomorphic fibering of complex manifolds, we show that for every n the number e_n^#(E -> M) only depends on the complex cobordism type of E. We give a general procedure to construct manifolds fibering as surface bundles in multiple ways, providing infinitely many examples to which our theorems apply. As an application of our results we give a new proof of the rational case of a recent theorem of Giansiracusa--Tillmann that the odd MMM classes e_{2i-1} vanish for any surface bundle which bounds a handlebody bundle. We show how the MMM classes can be seen as obstructions to low-genus fiberings. Finally, we discuss a number of open questions that arise from this work.

math.GT↗
arXiv2011

Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles

Characteristic classes of fibre bundles $E^{d+n}\to B^n$ in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base. Church, Farb and Thibault [CFT] raised the question of which generalised Miller-Morita-Mumford classes have the property that the associated characteristic number is independent of the fibering and depends only on the cobordism class of the total space $E$. Here we determine a complete answer to this question in both the oriented category and the stably almost complex category. An MMM class has this property if and only if it is a fibre integral of a vector bundle characteristic class that satisfies a certain approximate version of the additivity of the Chern character.

math.AT↗
arXiv2011

Homological stability for configuration spaces of manifolds

Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B_n(M) satisfies classical homological stability: for each i, H_i(B_n(M); Q) is isomorphic to H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural examples where rational homological stability holds even though integral homological stability fails. To prove the main theorem, we introduce the notion of monotonicity for a sequence of S_n--representations, which is of independent interest. Monotonicity provides a new mechanism for proving representation stability using spectral sequences. The key technical point in the main theorem is that certain sequences of induced representations are monotone.

math.AT↗
arXiv2011

Parametrized Abel-Jacobi maps and abelian cycles in the Torelli group

Let I_g,* denote the (pointed) Torelli group. This is the group of homotopy classes of homeomorphisms of the genus g >= 2 surface S_g with a marked point, acting trivially on H := H_1(S_g). In 1983 Johnson constructed a beautiful family of invariants tau_i: H_i(I_g,*) -> /\^{i+2} H for 0 <= i <= 2g-2, using a kind of Abel-Jacobi map for families, in order to detect nontrivial cycles in I_g,*. Johnson proved that tau_1 is an isomorphism rationally, and asked if the same is true for tau_i with i > 1. The goal of this paper is to introduce various methods for computing tau_i; in particular we prove that tau_i is not injective (even rationally) for any 2 <= i < g, and that tau_2 is surjective. For g >= 3, we find enough classes in the image of tau_i to deduce that H_i(I_g,*, Q) is nonzero for each 1 <= i < g, in contrast with mapping class groups. Many of our classes are stable, so we can deduce that H_i(I_infty,1, Q) is infinite-dimensional for each i >= 1. Finally, we conjecture a new kind of "representation-theoretic stability" for the homology of the Torelli group, for which our results provide evidence.

math.GT↗
arXiv2011

Separating twists and the Magnus representation of the Torelli group

The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism r: I_{g,1} -> GL_{2g}(Z[H]). Here H is the first homology group of the surface. This representation is not faithful; in particular, Suzuki previously described precisely when the commutator of two Dehn twists about separating curves is in the kernel of r. Using the trace of the Magnus representation, we apply a new method of showing that two endomorphisms generate a free group to prove that the images of two positive separating multitwists under the Magnus representation either commute or generate a free group, and we characterize when each case occurs.

math.GT↗
arXiv2009

Infinite generation of the kernels of the Magnus and Burau representations

Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of "Johnson-type" homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of Bur_n, we do this with the assistance of a computer calculation.

math.GR↗