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Is there some natural notion of a fractional degree of a map? The degree of a map is a generalization of the winding number, and fractional winding numbers appear in the (mathematical physics) literature1,2 (literature that I have not penetrated). I am interesting in capturing some notion of, for example, geometric "engulfing $2\frac{1}{2}$ times." I can concoct an ad hoc definition, but I'd prefer to connect to the literature on the topic. Thanks!

1Rothe, Swieca, "Fractional winding numbers and the $U(1)$ problem", Nuclear Physics B, Volume 168, Issues 3–4, 16–23 June 1980, Pages 454–464. (Journal link)

2"Gauge transformations with fractional winding numbers." Phys. Rev. D, 54, 2889–2898 (1996). (Journal link)

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  • $\begingroup$ Could you elaborate a bit more about what you need? One of the basic things that makes degree theory interesting is the fact that the analytic definition (which comes from a signed count of inverse images) is equivalent to the topological definition (which comes from the multiplicative factor of the induced map on local orientation). Surely, no fractional analog will produce such an equivalence. $\endgroup$
    – Vidit Nanda
    Commented Oct 19, 2013 at 0:19
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    $\begingroup$ I was trying to point out that the cost of looking beyond integers is that you'd immediately lose homotopy invariance, and at that point it isn't quite clear what such a degree would be "good for". $\endgroup$
    – Vidit Nanda
    Commented Oct 19, 2013 at 0:30
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    $\begingroup$ I think that in some situations losing some features would actually be consistent with the theory. For instance, one can extend intersection theory (at least in AG) to some singular varieties and get fractional intersection numbers. For instance two lines intersecting at the vertex of a quadric cone should have an intersection number $\frac 12$. Indeed, twice one of the lines is just the intersection of a plane tangent to that line and the intersection of the other line with this plane should definitely be $1$, so to have the intersection theory of the ambient $3$-space consistent... $\endgroup$
    – Sándor Kovács
    Commented Oct 19, 2013 at 1:18
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    $\begingroup$ If $f:X\to Y$ is a map of degree $a$ and $p:Z\to Y$ is a covering space of degree $b$, you coud say that "$p^{-1}\circ f$ has degree $a/b$, modulo inverting finite coverings. Similarly, you could attach degrees to correspondences $\Gamma\subseteq X\times Y$ instead of just to maps $X\to Y$, with sensible hypothesis like, say, that the projections to both factors aree coverings, or at least one of them. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 19, 2013 at 23:37
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    $\begingroup$ (The category of manifolds localized at finite coverings must have been considered already...) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 19, 2013 at 23:45

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