Kommende Vorträge

We find some restrictions on the intersection forms of smooth definite manifolds bounded by rational homology spheres which are rationally cobordant to lens spaces. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite rank cokernels. Further, we show that there is no $n$ such that every lens space smoothly embeds in $n$ copies of the complex projective plane. This is joint work with Paolo Aceto and Daniele Celoria.
 

$$\frac{a}{n}=\frac{1}{m_1}+\cdots+\frac{1}{m_k}$$ with positive integers $a,n,m_1,\ldots,m_k$. What is of interest is, given $n$, to count $A^*_k(n)=\{a:(a,n)=1, a/n=1/m_1+\cdots+1/m_k~{\text{for some}}~m_1,\ldots,m_k\}$ as well as $A_k(n)$ which is the same as $A_k^*(n)$ except that without the corporality condition on $a$ and $n$. In my talk, I will survey what is known about this problem for $k=2$ and I will show that $$ x(\log x)^3\ll \sum_{p\le x} A_3^*(p)\ll x(\log x)^5.$$

A knot is slice if it bounds an embedded disc in the 4-ball. There are (at least) two natural generalizations of sliceness: one might weaken either 'disc' to 'small genus surface' or 'the 4-ball' to 'any 4-manifold that is simple in some sense.'  In this talk, I'll discuss joint work with Jae Choon Cha and Mark Powell that gives new evidence that these two approaches measure very different things. Our tools include Casson-Gordon style representations of knot groups, L2 signatures of 3-manifolds, and the notion of a minimal generating set for a module.
 

David Gabai's smooth 4-dimensional "Light Bulb Theorem'' says that in the absence of involutions in the fundamental group of the ambient 4-manifold, homotopy implies isotopy for embedded 2-spheres which have a common geometric dual. In joint work with Peter Teichner we extend his result to orientable 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to the "homotopy implies isotopy'' question. The invariant takes values in an F2-vector space generated by the involutions in the fundamental group.