arXiv subjects
Publications and source records attributed to Ricard Riba.
In this note we show that the image of the third Johnson homomorphism $τ_3$ coincides with the kernel of Morita's trace map for the case of a surface of genus $g$ with one boundary component $Σ_{g,1}$, when $g \geq 6$. Moreover, as a consequence, we get that the image of $τ_3$ on the handlebody subgroups coincides with the intersection of kernels of the Morita trace map and the Lagrangian trace maps.
By results of Morita, Pitsch and, more recently, Faes, it is known that any integral homology 3-sphere can be constructed as a Heegaard splitting with a gluing map an element of the fourth Johnson subgroup. In this work we prove that the equivalence relation on the fourth Johnson subgroup induced by this construction admits an intrinsic description in terms of the fourth Johnson handlebody subgroups. In addition, we give an ``antisymmetic'' Lagrangian trace map inspired in the Lagrangian trace map introduced by Faes and compute the image of the third Johnson handlebody subgroups by the third Johnson homomorphism.
We study the behaviour of the Casson invariant $λ$, its square, and Othsuki's second invariant $λ_2$ as functions on the Johnson subgroup of the mapping class group. We show that since $λ$ and $d_2 = λ_2 - 18 λ^2$ are invariants that are morphisms on respectively the second and the third level of the Johnson filtration they never vanish on any level of this filtration. In contrast we prove that the invariant $λ_2-18λ^2 +3λ$ vanishes on the fifth level of the Johnson filtration, $\mathcal{M}_{g,1}(5)$, and as a consequence we prove that, for instance, the Poincaré homology sphere does not admit any Heegaard splitting with gluing map an element in $\mathcal{M}_{g,1}(5)$. Finally we determine a surgery formula for Othsuki's second invariant $λ_2$.
We study the relation between the set of oriented $\mathbb{Z}/d$-homology $3$-spheres and the level-$d$ mapping class groups, the kernels of the canonical maps from the mapping class group of an oriented surface to the symplectic group with coefficients in $\mathbb{Z}/d\mathbb{Z}$. We formulate a criterion to decide whenever a $\mathbb{Z}/d$-homology $3$-sphere can be constructed from a Heegaard splitting with gluing map an element of the level-$d$ mapping class group. Then we give a tool to construct invariants of $\mathbb{Z}/d$-homology $3$-spheres from families of trivial $2$-cocycles on the level-$d$ mapping class groups. We apply this tool to find all the invariants of $\mathbb{Z}/p$-homology $3$-spheres constructed from families of $2$-cocycles on the abelianization of the level-$p$ mapping class group with $p$ prime and to disprove the conjectured extension of the Casson invariant modulo a prime $p$ to rational homology $3$-spheres due B. Perron.
In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology $3$-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with $2$-torsion.
The main target of this thesis is to solve the Perron's conjecture. This conjecture affirms that some function on the mod p Torelli group, with values in Z/p, is an invariant of mod p homology 3-spheres. In order to solve this conjecture, in this thesis we first study the mod p homology 3-spheres, the rational homology 3-spheres and those that can be realized as a Heegaard splitting with gluing map an element of the mod p Torelli group. In particular we give a criterion to determine whenever a rational homology 3-sphere has a Heegaard splitting with gluing map an element of the Torelli group mod p, and using this criterion we prove that not all mod p homology 3-spheres can be realized in such way. Next, we extend the results of the article ''Trivial cocycles and invariants of homology 3-spheres'' obtaining a construction of invariants with values to an abelian group without restrictions, from a suitable family of 2-cocycles on the Torelli group. In particular, we explain the influence of the invariant of Rohlin in the lost of uniqueness in such construction. Later, using the same tools, we obtain a construction of invariants of rational homology spheres that have a Heegaard splitting with gluing map an element of the mod p Torelli group, from a suitable family of 2-cocycles on the mod p Torelli group where appears an invariant of mod p homology spheres which does not appear in the literature, who plays the same role that Rohlin invariant in the lost of uniqueness of our construction. Finally, we prove that Perrron's conjecture is false providing a cohomological obstruction that is given by the fact that the first characteristic class of surface bundles reduced modulo p does not vanish.
Given a free group $Γ$ of finite rank $n$ and a prime number $p,$ denote by $Γ_k^\bullet$ the $k^\text{th}$ layer of the Stallings ($\bullet=S$) or Zassenhaus ($\bullet=Z$) $p$-central series, by $\mathcal{N}_{k}^\bullet$ the quotient $Γ/Γ_{k+1}^\bullet$ and by $\mathcal{L}_{k}^\bullet$ the quotient $Γ_k^\bullet /Γ_{k+1}^\bullet.$ In this paper we prove that there is a non-central extension of groups $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1, \mathcal{L}^\bullet_{k+1}) \longrightarrow Aut\;\mathcal{N}^\bullet_{k+1} \longrightarrow Aut \;\mathcal{N}^\bullet_k \longrightarrow 1, $ which splits if and only if $k=1$ and $p$ is odd if $\bullet=Z$ or, $k=1$ and $(p,n)= (3,2), (2,2)$ if $\bullet=S$. Moreover, if we denote by $IA^p(\mathcal{N}^\bullet_k )$ the subgroup of $Aut \;\mathcal{N}^\bullet_k$ formed by the automorphisms that acts trivially on $\mathcal{N}_1^\bullet,$ then the restriction of this extension to $IA^p(\mathcal{N}^\bullet_{k+1})$ give us a non-split central extension of groups $ 0 \longrightarrow Hom(\mathcal{N}^\bullet_1,\mathcal{L}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_{k+1}) \longrightarrow IA^p(\mathcal{N}^\bullet_k ) \longrightarrow 1. $