Indira Chatterji, Yves Cornulier, Guido Mislin, Christophe Pittet
Let G be a connected Lie group, G^d the underlying discrete group, and BG, BG^d their classifying spaces. Let R denote the radical of G. We show that all classes in the image of the canonical map in cohomology H^*(BG,R)->H^*(BG^d,R) are bounded if and only if the derived group [R,R] is simply connected. We also give equivalent conditions in terms of stable commutator length and distortion.
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On flat manifold bundles and the connectivity of Haefliger's classifying spaces
该研究在叶状流形丛框架下探讨Haefliger-Thurston猜想，通过比较有限维李群G与Diff₀(G)、局部化平坦M-丛的和乐，分析低维流形上平坦M-丛的配边类，推进该猜想的相关研究。
Let G be a connected complex Lie group. We show that any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle over some finite covering space of the base space if and only if each real characteristic class of positive degree of G vanishes. A third equivalent condition is that the derived group of the radical of G is simply connected. As a corollary, the same conditions are equivalent if G is a connected amenable Lie group. In particular, if G is a connected compact Lie group then any flat principal G-bundle over any finite CW-complex pulls back to a trivial bundle …
Flat principal 2-group bundles and flat string structures
本文构造了可微层上平坦2-群丛的双范畴，并证明其可化为普通G-丛加2-格平凡化，从而刻画平坦弦结构。
We show that the Igusa-Klein topological torsion and the Bismut-Lott analytic torsion are equivalent for any flat vector bundle whose holonomy is a finite subgroup of $\mathrm{GL}_n(\mathbb{Q})$. Our proof uses Artin's induction theorem in representation theory to reduce the problem to the special case of trivial flat line bundles, which is a recent result of Puchol, Zhu and the second author. The idea of using Artin's induction theorem appeared in a paper of Ohrt on the same topic, of which our present work is an improvement.
We establish a generic counting formula for the Euler number of a flat vector bundle of rank $2n$ over a $2n$ dimensional closed manifold, in terms of vertices of transversal open coverings of the underlying manifold. We use the Mathai-Quillen formalism to prove our result.
Using methods originating in the theory of intersection spaces, specifically a de Rham type description of the real cohomology of these spaces by a complex of global differential forms, we show that the Leray-Serre spectral sequence with real coefficients of a flat fiber bundle of smooth manifolds collapses if the fiber is Riemannian and the structure group acts isometrically. The proof is largely topological and does not need a metric on the base or total space. We use this result to show further that if the fundamental group of a smooth aspherical manifold acts isometrically on a Riemannian …
We study the topology of the moduli space of flat SU(2)-bundles over a nonorientable surface X. This moduli space may be identified with the space of homomorphisms Hom(π_1(X),SU(2)) modulo conjugation by SU(2). In particular, we compute the (rational) equivariant cohomology ring of Hom(π_1(X),SU(2)) and use this to compute the ordinary cohomology groups of the quotient Hom(π_1(X),SU(2))/SU(2). A key property is that the conjugation action is equivariantly formal.
We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.
For a simply connected, compact, simple Lie group G, the moduli space of flat G-bundles over a closed surface is known to be pre-quantizable at integer levels. For non-simply connected G, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction -- namely a certain cohomology class in H^3(G^2;Z) -- that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G.
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.
We prove that group homology of the diffeomorphism group of $\#^g S^n \times S^n$ as a discrete group is independent of $g$ in a range, provided that $n>2$. This answers the high dimensional version of a question posed by Morita about surface diffeomorphism groups made discrete. The stable homology is isomorphic to the homology of a certain infinite loop space related to the Haefliger's classifying space of foliations. One geometric consequence of this description of the stable homology is a splitting theorem that implies certain classes called generalized Mumford-Morita-Miller classes can be …
We continue our study of symplectically flat bundles. We broaden the notion of symplectically flat connections on symplectic manifolds to $ζ$-flat connections on smooth manifolds. These connections on principal bundles can be represented by maps from an extension of the base's fundamental group to the structure group. We introduce functionals with zeroes being symplectically flat connections and study their critical points. Such functionals lead to novel geometric flows. We also describe some characteristic classes of the $ζ$-flat bundles.
Remarks on flat $S^1$-bundles, $C^\infty$ vs $C^ω$
本文利用Thurston定理刻画微分同胚群的低维同调，并探讨实解析平坦圆丛有理Euler类幂消失的条件，指出该情形将导致特定同调群中出现两类挠类。
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