##### Department of Mathematics,

University of California San Diego

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### Math 298 - Topology

## Shelly Harvey

#### UCSD

## Higher-order 3-manifold invariants and their applications; Parts I

##### Abstract:

We define an infinite sequence of new invariants, $delta_n$, of agroup G that measure the size of the successive quotients of the derivedseries of G. In the case that G is the fundamental group of a 3-manifold,we obtain new 3-manifold invariants. These invariants are closely relatedto the topology of the 3-manifold. They give lower bounds for theThurston norm which provide better estimates than the bound establishedby McMullen using the Alexander norm. We also show that the $delta_n$ giveobstructions to a 3-manifold fibering over $S^1$ and to a 3-manifold beingSeifert fibered. Moreover, we show that the $delta_n$ give computablealgebraic obstructions to a 4-manifold of the form $X x S^1$ admitting asymplectic structure even when the obstructions given by theSeiberg-Witten invariants fail. There are also applications to theminimal ropelength of knots and links in $S^3$. In addition, wediscuss the applications to the cut number of a 3-manifold (this is alsoknown as the corank of the fundamental group of the 3-manifold)

Host:

### September 27, 2002

### 4:00 PM

### AP&M 7218

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