Timeline for Is the fundamental group functor a left-adjoint?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2012 at 10:47 | comment | added | Ronnie Brown | See also my answer to math.stackexchange.com/questions/198348 | |
| Nov 3, 2012 at 23:14 | comment | added | Ronnie Brown | @Chris: are you asking about the higher Seifert van Kampen theorems? If so, a simple example is to give a precise description of $\pi_2(X \cup_f CA,X,x)$ in terms of the morphism of fundamental groups of $A$ and $X$ induced by the attaching map $f$ of the cone. This generalises a classic theorem of JHC Whitehead on free crossed modules. The proof uses a homotopy double groupoid of a pair. | |
| Nov 3, 2012 at 20:58 | comment | added | Ronnie Brown | @Chris: In calculating $\pi_1 S^1$ it is good to use 2 base points. In a van Kampen situation $A$ needs to meet each path component of each $1$-, $2$-, $3$-fold intersrsection of the sets of the cover. See my paper with Razak, [41] in my publication list. See also "Topology and Groupoids" Section 9.1. See also arXiv:math/0111073 for a wider application. In higher dimensions, the theory that really works well is for filtered spaces, which is against tradition; the approach yields a new exposition of basic algebraic topology without using singular homology! See EMS Tract vol 15. | |
| Oct 16, 2012 at 20:24 | comment | added | Chris Schommer-Pries | What is the easiest illustrative example of the phenomena described in the last paragraph that goes beyond the fundamental groupoid? | |
| Oct 16, 2012 at 20:09 | history | edited | Ronnie Brown | CC BY-SA 3.0 |
added some points about adjointness
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| Oct 16, 2012 at 16:44 | history | answered | Ronnie Brown | CC BY-SA 3.0 |