This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit a taut foliation. The essential part of this paper focuses on Seifert fibered homology $3$-spheres. The result is quite different if they are integral or rational but non-integral homology $3$-spheres. Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the Poincaré $3$-sphere admit a taut foliation. Concerning non-integral homology $3$-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology $3$-spheres.

This paper concerns the problem of existence of taut foliations among $3$-manifolds. Since the contribution of David Gabai, we know that closed $3$-manifolds with non-trivial second homology group admit a taut foliation. The essential part of this paper focuses on Seifert fibered homology $3$-spheres. The result is quite different if they are integral or rational but non-integral homology $3$-spheres. Concerning integral homology $3$-spheres, we can see that all but the $3$-sphere and the Poincaré $3$-sphere admit a taut foliation. Concerning non-integral homology $3$-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology $3$-spheres.

We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

Let $K$ be a knot in $S^3$. This paper is devoted to Dehn surgeries which create $3$-manifolds containing a closed non-orientable surface $\ch S$. We look at the slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the surface and the intersection number $s$ between $\ch S$ and the core of the Dehn surgery. We prove that if $\chi(\hat S) \geq 15 - 3q$, then $s=1$. Furthermore, if $s=1$ then $q\leq 4-3\chi(\ch S)$ or $K$ is cabled and $q\leq 8-5\chi(\ch S)$. As consequence, if $K$ is hyperbolic and $\chi(\ch S)=-1$, then $q\leq 7$.

Let $\tilde M$ be a regular branched cover of a homology 3-sphere $M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph $\Gamma$; such a cover is determined by a coloring of the edges of $\Gamma$ with elements of $G$. For each index-2 subgroup $H$ of $G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma has proved that $H_1(\tilde M)$ is isomorphic, modulo 2-torsion, to $\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain cases; specifically, when $d=2$ and the coloring is such that the branch set of each cover $M_H\to M$ is connected, and when $d=3$ and $\Gamma$ is the complete graph $K_4$. We prove this for a larger class of coverings: when $d=2$, for any coloring of a connected graph; when $d=3$ or $4$, for an infinite class of colored graphs; and when $d=5$, for a single coloring of the Petersen graph.

We prove an integration formula involving Jack polynomials conjectured by I.~P.~Goulden and D.~M.~Jackson in connection with enumeration of maps in surfaces.