Non injectivity of the "hair" map

特征流形上迹映射的非单射性

Non-injectivity of the trace map for character varieties

本文研究了特征流形上迹映射的非单射性问题，通过构造显式的非零元素证明了该映射在任意秩情况下都不可逆。

An Injectivity Theorem for Casson-Gordon Type Representations relating to the Concordance of Knots and Links

In the study of homology cobordisms, knot concordance and link concordance, the following technical problem arises frequently: let $π$ be a group and let $M \to N$ be a homomorphism between projective $\Z[π]$-modules such that $\Z_p \otimes_{\Z[π]} M\to \Z_p \otimes_{\Z[π]} N$ is injective; for which other right $\Z[π]$-modules $V$ is the induced map $V \otimes_{\Z[π]} M\to V\otimes_{\Z[π]}N$ also injective? Our main theorem gives a new criterion which combines and generalizes many previous results.

Harmonic maps between surfaces homotopic to a (branched) covering map

In the paper, we consider the harmonic maps between surfaces $Σ$ and $S$ in the homotopy class of a (branched) covering map $u_0$. We prove the uniqueness of critical points of energy function and the injectivity of Hopf differential if $u_0$ is a covering map. On the other hand, if $u_0$ is a branched covering, we show that the uniqueness of critical points fails if $u_0$ is a non-simple branched covering, and prove the injectivity of Hopf differential $Φ:\mc{T}(S)\to \op{QD}(Σ,g)$ when $g=[u_0^* h]$ for some hyperbolic metric $h$ on $S$.

Assembly maps for topological cyclic homology of group algebras

We use assembly maps to study $\mathbf{TC}(\mathbb{A}[G];p)$, the topological cyclic homology at a prime $p$ of the group algebra of a discrete group $G$ with coefficients in a connective ring spectrum $\mathbb{A}$. For any finite group, we prove that the assembly map for the family of cyclic subgroups is an isomorphism on homotopy groups. For infinite groups, we establish pro-isomorphism, (split) injectivity, and rational injectivity results, as well as counterexamples to injectivity and surjectivity. In particular, for hyperbolic groups and for virtually finitely generated abelian groups, we…

Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree…

On the welded Tube map

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quotient under classical Reidemeister moves and reformulates these moves and the known sources of non-injectivity of the Tube map in terms of filling changes.