is non-zero
whenever x is non-zero. If this is the case, we call the pairing p
positive. The question arises for each dimension d=0,1,2,.... We find
p(d) positive for d=0,1, and 2 and not positive for d=4. We conjecture
that p(3) is also positive. Similar questions may be phrased for
(manifold, submanifold) pairs and manifolds with other additional
structure. The results in dimension 4 imply that unitary TQFTs cannot
distinguish homotopy equivalent simply connected 4-manifolds, nor can
they distinguish smoothly s-cobordant 4-manifolds. This may illuminate
the difficulties that have been met by several authors in their
attempts to formulate unitary TQFTs for d=3+1. There is a further
physical implication of this paper. Whereas 3-dimensional Chern-Simons
theory appears to be well-encoded within 2-dimensional quantum
physics, eg in the fractional quantum Hall effect,
Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional
quantum system. The positivity of the physical Hilbert spaces means
they cannot see null vectors of the universal pairing; such vectors
must map to zero.
**Keywords**.
Manifold pairing, unitary, positivity, TQFT, s-cobordism

**AMS subject classification**.
Primary: 57R56, 53D45.
Secondary: 57R80, 57N05, 57N10, 57N12, 57N13.

**E-print:** `arXiv:math.GT/0503054`

**DOI:** 10.2140/gt.2005.9.2305

Submitted to G&T on 25 May 2005.
(Revised 2 December 2005.)
Paper accepted 3 December 2005.
Paper published 10 December 2005.

Notes on file formats
Michael H Freedman, Alexei Kitaev, Chetan Nayak, Johannes K Slingerland, Kevin Walker and Zhenghan Wang

MHF,CN,JKS,KW: Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA

AK: California Institute of Technology, Pasadena, CA 91125, USA

CN: Department of Physics and Astronomy, UCLA, CA 90095-1547, USA

ZW: Dept of Mathematics, Indiana University, Bloomington, IN

Email: michaelf@microsoft.com, kitaev@iqi.caltech.edu, nayak@physics.ucla.edu, joost@microsoft.com, kwalker@microsoft.com, zhewang@indiana.edu

GT home page

## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://msp.warwick.ac.uk/.
**