Small knots and large handle additions

We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic) which contains no essential closed surfaces, but for any even integer $g> 0$ there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$, the 3-manifold obtained by attaching 2-handle to $M$ along $r$, contains an essential separating closed surface of genus $g$ and is still hyperbolic. The result contrasts sharply with those known finiteness results for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a simple small knot in a handlebody.Comment: 25 pages, 14 figure

Handle additions producing essential surfaces

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is hyperbolic and contains an essential separating closed surface of genus $g$. The result contrasts sharply with those known finiteness results on Dehn filling, and it also contrasts sharply with the known finiteness result on handle addition for the cases $g=0,1$. Our 3-manifold $M$ is the complement of a hyperbolic, small knot in a handlebody of genus 3.Comment: 28 pages, 22 figure

Non-zero degree maps between 2n2n-manifolds

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $\phi : H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds $M$ and $L$, $n>1$. A corollary is that each $(n-1)$-connected $2n$-manifold admits selfmaps of degree larger than 1, $n>1$. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree $k$ map from a closed orientable 4-manifold $M$ to a closed simply connected 4-manifold $L$ in terms of their intersection forms, in particular there is a map $f:M\to L$ of degree 1 if and only if the intersection form of $L$ is isomorphic to a direct summand of that of $M$.Comment: 18 page