Timeline for "structure group" for fibration
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2016 at 2:15 | answer | added | Jeff Strom | timeline score: 1 | |
| Apr 25, 2016 at 16:33 | comment | added | Peter May | Mike, thanks for the reference. For others, his comment about larger X is that there is no need to restrict to finite CW complexes. That ensures that Aut(X) has the homotopy type of a CW complex, but the proofs in ``Classifying spaces and fibrations'' don't require that. Denis, if memory serves, I think that Milnor was the first to see that you can replace M by a topological group. | |
| Apr 25, 2016 at 14:52 | comment | added | mme | Sorry, the above comment should say "even if you change $X$ to a homotopy equivalent complex"; of course you cannot change the homotopy type. Is it worth collecting these comments as one answer? | |
| Apr 25, 2016 at 3:36 | comment | added | Igor Belegradek | In particular, if $G$ is a closed subgroup of the homeomorphism group of $X$, then one has the map $BG\to BAut(X)$ and reduction of the structure group of the fibration to $G$ is a lift of the classifying map to $BG$. | |
| Apr 24, 2016 at 22:27 | comment | added | Denis Nardin | I'll add that you can always find, for every group-like topological monoid $M$, a model for $M$ that is an actual topological group (the usefulness of this fact is not very high, but it's cool to know). This is done, for example, in Quillen's On the group-completion of a simplicial monoid | |
| Apr 24, 2016 at 22:24 | comment | added | mme | If $X$ is a finite CW complex, the space of self-homotopy equivalences $\text{Aut}(X)$ is a grouplike topological monoid. It's the appropriate notion of "structure group"; it admits a classifying space $B\text{Aut}(X)$, well-defined up to homotopy equivalence (even if you change the homotopy type of $X$). Fibrations with fiber $X$ are classified by maps into the above classifying space. This was all done, IIRC, by Stasheff, and then generalized by May to a larger class of $X$ - see May, "Classifying spaces and fibrations". | |
| Apr 24, 2016 at 22:02 | review | First posts | |||
| Apr 24, 2016 at 23:03 | |||||
| Apr 24, 2016 at 21:58 | history | asked | user20165678 | CC BY-SA 3.0 |