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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 6th 2011
• (edited Feb 6th 2011)

A manifold has

• a set of orientations;

• an xyz of topological spin structures

• a 3-groupoid of topological string structures;

• a 7-groupoid of topological fivebrane stuctures, etc.

and for some reason it is common in the literature (which of course is small in the last cases) to speak of these $n$-groupoids, but not so common to speak of the xyz here:

• A manifold has a groupoid of spin structures.

Namely the homotopy fiber of the second Stiefel-Whitney class

$Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.$

I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.

Do you have further references?

• CommentRowNumber2.
• CommentAuthorTim_Porter
• CommentTimeFeb 7th 2011

@Urs This looks worth pursuing. It may relate to HQFTs. We can discuss this when you arrive (in Lisbon).

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 7th 2011

How is it going in Lisbon? I hope it’s sunny. I need some sun. Though today it seems winter is taking a break even here.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeFeb 7th 2011

Urs, you skipped Lisboa ?

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 7th 2011

Urs, you skipped Lisboa ?

No, I’ll go. My flight is day after tomorrow, Feb 9.

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