##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry

## Mario Micallef

#### Warwick Univ

## Surgeries on manifolds with almost positive isotropic curvature

##### Abstract:

It is known that a simply connected manifold with positive isotropic curvature (PIC) is homeomorphic to a sphere. The observation that the product metric on $S^1 imes S^n$ has PIC and the fact that the class of manifolds with PIC is closed under connected sums led to the conjecture that the fundamental group of manifolds with PIC is almost free. Significant progress on this conjecture has recently been made by Ailana Fraser. In this talk I will describe the notion of almost PIC (which still implies positive scalar curvature) and I will indicate why this class of manifolds is closed under surgery over a circle. In particular, there is no restriction on the fundamental group for manifolds with almost PIC. By considering higher dimensional surgeries, there is even reason to believe that the class of simply connected manifolds with almost PIC coincides with that of positive scalar curvature. This is joint work in progress with Ingi Petursson.

Host: Lei Ni

### April 9, 2003

### 4:00 PM

### AP&M 5829

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