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starting something, on the kind of theorems originating with
Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)
added statement of the analogous theorem (here) for maps between projective spaces, from
Jacob Mostovoy, Spaces of rational maps and the Stone–Weierstrass theorem, Topology Volume 45, Issue 2, March 2006, Pages 281-293 (doi:10.1016/j.top.2005.08.003)
Jacob Mostovoy, Truncated Simplicial Resolutions and Spaces of Rational Maps, The Quarterly Journal of Mathematics, Volume 63, Issue 1, March 2012, Pages 181–187 (doi:10.1093/qmath/haq031)
I have added a remark (here) that the space of rational maps $\mathbb{C}P^1 \to \mathbb{C}P^n$ that is considered in Segal’s theorem is also considered in Gromov-Witten theory (after compactification and quotienting), as is nicely explicit in Bertram 02, p. 9.
This confluence looks like it ought to have drawn attention, but I don’t find literature in this direction.
re #8:
I see that it is this connection which the preprint
was after, before the author discovered the mistake highlighted in v2.
Mistake or not, that’s the right question to ask. But it looks like it wasn’t followed up.
added pointer to:
added (here) brief statement of the main theorem from
identifying the full homotopy type of the space of pointed rational maps $\mathbb{C}P^1 \to \mathbb{C}P^n$ with that of a configuration space of points. (Am making a similar addition now to this latter entry, too.)
Fixed the statement of Segal’s theorem (here), now pointing to n-equivalence (following Segal’s note on terminology on p. 44)
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