Hitoshi Murakami
We show an integrality of the quantum SU(2)-invariant associated with a non-trivial first cohomology class modulo two.
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On contact modulo p L-space covers
本文将 Roso（2023）关于极小 L-空间上等变接触结构的结果推广到模 p L-空间，利用 Borel Floer 上同调的 Serre 谱序列，并引入两个新的数值不变量。
Thermodynamic formalism for Quasimorphisms: Bounded Cohomology and Statistics
本文发展了用于研究紧负曲率空间基本群准同态空间的热力学框架，证明该空间与关联的Gromov测地流的Bowen函数空间在Banach空间上同构，并建立准同态的中心极限定理和不变原理。
Based in the isomorphism between Lie algebroid cohomology and piecewise smooth cohomology, it is proved that the Rham cohomology of a locally trivial Lie groupoid $G$ on a smooth manifold $M$ is isomorphic to the piecewise Rham cohomology of $G$, in which $G$ and $M$ are manifolds without boundary and $M$ is smoothly triangulated by a finite simplicial complex $K$ such that, for each simplex $Δ$ of $K$, the inverse images of $Δ$ by the source and target mappings of $G$ are transverses submanifolds in the ambient space $G$. As a consequence, it is shown that the piecewise de Rham cohomology of …
We solve some computational problems for triangulated closed three-dimensional manifolds using groups of simplicial homology and cohomology modulo 2. Two efficient algorithms for computing the intersection numbers of 1- and 2-dimensional cycles are developed. By means of these algorithms it is possible to construct a basis of cohomology group from the homology group of two cycles of complementary dimensions.
We show that in the bounded cohomology of non-abelian free groups the Massey triple product is always trivial when the second factor is represented by the coboundary of a decomposable quasi-morphism. We also show that in the bounded cohomology of a negatively curved compact Riemannian manifold the Massey triple product is always trivial when the second factor is represented by an exact differential form.
It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4. In this paper we present two results concerning the multiplicativity modulo 8: firstly we identify the obstruction to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Po…
Non-trivial and Non-tautological Cohomology of Strata of Differentials
本文通过多尺度微分模空间的边界分层几何，构建了非平凡和非琐碎的上同调类。
In this article we show that if a knot diagram admits a non-trivial coloring modulo 13 then there is an equivalent diagram which can be colored with 5 colors. Leaning on known results, this implies that the minimum number of colors modulo 13 is 5.
This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a Z_2-valued quadratic form. This result is applied to a fibration of geometric Poincare complexes with the total space 4k-dimensional. It is known that the signature is multiplicative modulo 4, and that there are examples of non-multiplicativity modulo 8. We identify the obstructi…
Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $ϕ$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hatϕ$ such that neither $\hatϕ$ nor $\hatϕ^2$ has fixed points. We give conditions under which Seidel and Smith's localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $\mathit{HF}(ϕ^2)\otimes \mathbb Z_2((θ))$ to $\mathit{HF}(ϕ)\otimes \mathbb Z_2((θ))$.
We propose a new method of computing cohomology groups of spaces of knots in $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ define at arbitrary singular knots. More generally, for any finite-order cohomology class of the space of knots we define its principal symbol, which lies in a cohomology group of a certain finite-dimensional configuration space and characterizes our class modulo the classes of smaller filtration.
We study the relationship between trivial cocycles on the Torelli group and invariants of oriented integral homology 3-spheres. We give ncecessary and sufficient conditions for a function defined on the union of the Torelli groups to be an invariant of homology spheres. We apply this study to give a new purely algebraic construction of the Casson invariant and prove in this setting its surgery properties. As a by-product we get a new 2-torsion cohomology class in the second integral cohomology of the Torelli group.
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