There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian multiplication on $\mathrm{CP}(\infty)$. What is the formula and where is it written?
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1$\begingroup$ I guess you want this with respect to a particular set of coordinates on $\mathbb{CP}^{\infty}$? There are other ways of presenting this multiplication, e.g. in terms of the infinite symmetric product $\text{SP}(S^2)$. $\endgroup$– Qiaochu YuanNov 22, 2013 at 1:05
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$\begingroup$ We essentially had this question a while ago: mathoverflow.net/questions/11117/… $\endgroup$– Mariano Suárez-ÁlvarezNov 22, 2013 at 3:20
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$\begingroup$ Thanks. None of those tags occurred to me. It is indeed the Segre map I was trying to recover. Can one use them to begin an explicit coordiante construction of K(Z,3)? $\endgroup$– Jim StasheffNov 22, 2013 at 13:49
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$\begingroup$ @Jim: again, depending on what kind of explicit coordinates you want, you can think of $K(\mathbb{Z}, 3)$ as $\text{SP}(S^3)$ and present the multiplication that way. Maybe this has some interpretation in terms of quaternionic polynomials. $\endgroup$– Qiaochu YuanNov 22, 2013 at 18:48
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1 Answer
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NOTE: I am not sure if this answers the OP, certainly it gives an expression for the map in homogeneous coordinates which should be the equivalent of Plücker coordinates in the case of the projective space.
If you mean an H-space structure giving the tensor product on line bundles on $\mathbb{CP}^\infty$ the easiest way I know to get it is to see $\mathbb{CP}^\infty$ as the projective space corresponding to the vector space $\mathbb{C}[T]$ of polynomials.
In fact the multiplication map $\mathbb{C}[T]\times\mathbb{C}[T]\to \mathbb{C}[T]$ passes to the quotient (this follows from $\mathbb{C}[T]$ being an integral domain).
You can check that it represents tensor product of line bundles by computing it in the case of the universal example (that is $\mathbb{CP}^\infty\times\mathbb{CP}^\infty$ with the two line bundles obtained by the tautological under pullback of either projection).
EDIT: while writing my answer Tom Goodwillie posted what amount to the content of it as a comment. Should I delete this answer? (Sorry I'm new on MO and I'm not sure what's the correct etiquette).
answered Nov 22, 2013 at 1:28
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4$\begingroup$ I deleted my comment. A detailed answer is better than a comment. $\endgroup$ Nov 22, 2013 at 1:32