On the classification of topological field theories.(English)Zbl 1180.81122
Jerison, David (ed.) et al., Current developments in mathematics, 2008. Somerville, MA: International Press (ISBN 978-1-57146-139-1/pbk). 129-280 (2009).
Many people are excited by this publication, and the reviewer is one of them.
The expository paper under review outlines a large programme, developed by the author jointly with Mike Hopkins, for reformulating and proving the Baez-Dolan cobordism hypothesis. It is exciting to read and difficult to put down – a rare virtue for a mathematics paper. In particular, the first chapter imparts a great deal of insight into why we are interested in extended TQFT’s and into why the cobordism hypothesis itself is so beautiful and fundamental.
The author stresses that this paper is a gentle exposition rather than a full-fledged proof. Most crucially, the key notion of an \((\infty,n)\)-category, which underlies the whole paper, is never actually defined, and its key properties, listed at the top of page 161, are never proven. An additional warning is that there are many different definitions for an \(n\)-category, and a proof for one definition does not imply a proof for any other definition.
The key new idea which this paper brings to the table is the reformulation of the cobordism hypothesis in terms of \((\infty,n)\)-categories (rather than \(n\)-categories), and fully dualizable objects (an original idea of Lurie), allowing homotopy theory to be brought to bear. The programme sounds original and plausible, and we look forward to reading the details, which Lurie promises to publish soon.
For the entire collection see [Zbl 1173.00021].
The expository paper under review outlines a large programme, developed by the author jointly with Mike Hopkins, for reformulating and proving the Baez-Dolan cobordism hypothesis. It is exciting to read and difficult to put down – a rare virtue for a mathematics paper. In particular, the first chapter imparts a great deal of insight into why we are interested in extended TQFT’s and into why the cobordism hypothesis itself is so beautiful and fundamental.
The author stresses that this paper is a gentle exposition rather than a full-fledged proof. Most crucially, the key notion of an \((\infty,n)\)-category, which underlies the whole paper, is never actually defined, and its key properties, listed at the top of page 161, are never proven. An additional warning is that there are many different definitions for an \(n\)-category, and a proof for one definition does not imply a proof for any other definition.
The key new idea which this paper brings to the table is the reformulation of the cobordism hypothesis in terms of \((\infty,n)\)-categories (rather than \(n\)-categories), and fully dualizable objects (an original idea of Lurie), allowing homotopy theory to be brought to bear. The programme sounds original and plausible, and we look forward to reading the details, which Lurie promises to publish soon.
For the entire collection see [Zbl 1173.00021].
Reviewer: Daniel Moskovich (Kyoto)
MSC:
| 81T45 | Topological field theories in quantum mechanics |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
| 55P42 | Stable homotopy theory, spectra |
| 18D10 | Monoidal, symmetric monoidal and braided categories (MSC2010) |
| 18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |
| 57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |
| 57R75 | \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism |
