Timeline for H-space structure on infinite projective spaces
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jan 8, 2010 at 17:57 | comment | added | Allen Hatcher | The H-space structure in a $K(A,n)$ is unique up to homotopy since homotopy classes of maps $K(A,n)\times K(A,n) \to K(A,n)$ correspond bijectively with homomorphisms $A\times A \to A $, and the H-space condition says the homomorphism restricts to the identity on each factor so it is just the addition operation in the abelian group $A$. | |
| Jan 8, 2010 at 17:17 | comment | added | Reid Barton | In the complex case, you can also use the fundamental theorem of algebra to replace $\mathbb{CP}^\infty$ to the infinite symmetric power of $\mathbb{CP}^1$. | |
| Jan 8, 2010 at 17:14 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Jan 8, 2010 at 16:41 | comment | added | Jason DeVito - on hiatus | Is it somehow obvious that this H-space structure and the one Hanno was talking about are the same? | |
| Jan 8, 2010 at 10:29 | vote | accept | Hanno Becker | ||
| Jan 8, 2010 at 10:15 | vote | accept | Hanno Becker | ||
| Jan 8, 2010 at 10:27 | |||||
| Jan 8, 2010 at 10:14 | comment | added | Hanno Becker | Thank you, Mariano & Allen! This is really beautiful, and the structure you described is even strictly associative and unital. What about the (homotopy) inversion of this H-space structure - is there a nice way to describe it, too? | |
| Jan 8, 2010 at 7:48 | comment | added | Allen Hatcher | As a footnote, the construction does not extend to the quaternionic case since commutativity of multiplication of coefficients is needed in order for the multiplication of polynomials to be well defined modulo scalar multiplication. In the complex case, if you only factor out by scalar multiplication by numbers that are a real number times a p-th root of unity, you get an H-space structure on an infinite-dimensional lens space, a $K({\mathbb Z}_p,1)$. | |
| Jan 8, 2010 at 7:38 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Jan 8, 2010 at 7:31 | history | answered | Mariano Suárez-Álvarez | CC BY-SA 2.5 |