Timeline for The fundamental group of space which has both an H and a co-H structure
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Mar 6, 2011 at 14:48 | comment | added | Mark Grant | @Jino: This is Exercise 1.21 in the book "Lusternik-Schnirelmann Category" by Cornea, Lupton, Oprea and Tanré. You'll find some handy hints there. | |
| Feb 28, 2011 at 14:18 | comment | added | Jeff Strom | @Jino: $X$ is a co-H-space, so the diagonal factors through $X\vee X$ (up to homotopy), and $\pi_1(X\vee X) = \pi_1(X) * \pi_1(X)$ by Van Kampen. | |
| Feb 28, 2011 at 12:18 | comment | added | Jino | @Jeff. I don't know how show that $\Delta_\ast$ induces above factorization. Please explain to me. | |
| Feb 27, 2011 at 16:38 | comment | added | John Klein | @Mark: I was asking about H and co-H spaces, not (co)-groups, so $S^7$ is an example. | |
| Feb 27, 2011 at 16:37 | comment | added | John Klein | @Mark: of course. I wonder if that's the whole list. | |
| Feb 27, 2011 at 16:36 | comment | added | Mark Grant | Correction: $S^3$ is an example, but not $S^7$. Only $S^1$ and $S^3$ admit a homotopy associative multiplication (James, "Multiplications on Spheres. II") | |
| Feb 27, 2011 at 11:57 | history | edited | Jeff Strom | CC BY-SA 2.5 |
fixed typos
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| Feb 27, 2011 at 11:19 | comment | added | Mark Grant | I guess $S^3$ and $S^7$ also qualify? But I can't think of any other examples. | |
| Feb 27, 2011 at 11:04 | comment | added | Jino | @Jeff. I don't know that your first answer : when $X$ is co-H-group, then $\pi_1(X)$ is free. How can proof this? | |
| Feb 27, 2011 at 2:04 | comment | added | Jeff Strom | @John: Not that I can think of. I guess its homology with field coefficients would be a Hopf algebra with trivial diagonal, so that narrows it down considerably. | |
| Feb 27, 2011 at 0:48 | comment | added | John Klein | @Jeff: Do you know of any examples of connected spaces which have both H and co-H structures, other than $S^1$? | |
| Feb 26, 2011 at 21:55 | comment | added | Mariano Suárez-Álvarez | That makes for a fun computation of the fundamentak group of $S^1$, provided one can show it is not zero. The least illuminating computation, I guess! :) | |
| Feb 26, 2011 at 21:22 | history | answered | Jeff Strom | CC BY-SA 2.5 |