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Let me try to give a more elementary argument. Assume there is a compression disk. One can assume it is transversal to $A\times\left\{\frac{1}{2}\right\}$, thus it intersects $A\times\left\{\frac{1}{ …

Perhaps I'm wrong but I thought that historically the notions were considered first in the work of Bennequin who proved the existence of contact structures not equivalent to the standard contact struc …

In fact, for manifolds with torus boundary you have $$\parallel M,\partial M\parallel_0=\parallel M,\partial M\parallel$$ so that the two inequalities are plainly equivalent. (More generally, this eq …

There is also an explicit argument which shows $$\parallel M_1\cup_{A}M_2\parallel \le \parallel M_1,\partial M_1\parallel + \parallel M_2,\partial M_2\parallel$$ whenever $A=\partial M_1=\partial M_2 …

For $M_0=M_1=S^2$, this follows from the 3-dimensional Poincaré conjecture: glueing in 3-balls, you get a simply connected $3$-manifold, that has to be $S^3$, so $W=S^2\times\left[0,1\right]$. For sur …

(Edit: Everything what follows is about closed and orientable $3$-manifolds.) Non-spherical geometric $3$-manifolds are determined by their fundamental group. This is proved in Peter Scott's paper "T …

See Bowditch: link text Systems of bands in hyperbolic 3-manifolds with an approach to the Brock-Canary-Minsky Theorem (though not through their model manifold) that is, in principle, effective. Thou …

Dehn's Lemma (= Papakyriakopoulos' Theorem) asserts: if C represents 0 in $\pi_1M$ and if C is a simple closed curve, then C bounds an embedded disk. The assumption on C being a simple closed curve i …

No, the braid index of a Montesinos link can be as large as one wishes. One way to see this is as follows. There is a lower bound for the braid index in terms of the Homfly polynomial due to Morton …

Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K …

Ad (2): the mapping class group of surfaces is isomorphic to the outer automorphism group of its fundamental group - this is a theorem of Baer-Dehn-Nielsen. In particular there is even a finite set of …

Which hyperbolic $3$-manifolds are known to have quadratic cusp shape? Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, wh …

The natural definition of a higher Dehn twist at a surface $F$ is to identify a neighborhood of $F$ homeomorphic to $F\times\left[0,1\right]$, take a loop $\phi_t$ in $Homeo(F)$ based at $id$, and con …

The statement is only true if you restrict to geometrically finite hyperbolic metrics (possibly of infinite volume) and ignore parabolic elements, which basically means that you ignore the boundary …

McMullen and Taubes 4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations constructs 3-manifolds $N$ with different fibrations, whose Euler classes do not lie in …