Timeline for Classification of bundles with fixed total space
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Feb 17 at 23:57 | comment | added | Matthew Kvalheim | @IgorBelegradek: thank you for another very helpful comment. You answered multiple questions I had been wondering about. Very interesting! | |
| Feb 17 at 20:52 | comment | added | Igor Belegradek | @MatthewKvalheim: if $n\neq 4$, then any manifold homeomorphic to $\mathbb R^n$ is also diffeomorphic to it. As far as I know there is no single example of closed manifold whose universal cover is an exotic $\mathbb R^4$. On the other hand, some exotic $\mathbb R^4$s do admit group actions, see arxiv.org/pdf/1705.06644.pdf; I am not sure if any of these actions are free and properly discontinuous. For many manifolds we know their universal cover is diffeomorphic to $\mathbb R^n$, e.g. nilmanifold, nonpositively curved manifold. | |
| Feb 17 at 20:39 | comment | added | Matthew Kvalheim | @IgorBelegradek Many thanks for your helpful comments. For principal G-bundles, the equivalence relation I had in mind is indeed given by existence of a G-bundle isomorphism. Your comments about homotopy tori are quite interesting to me, since one of the main questions I hope to understand is: for which smooth n-manifolds $M$ is the universal covering space of $M$ diffeomorphic (not just homeomorphic) to $\mathbb{R}^n$? | |
| Feb 5 at 23:08 | comment | added | Igor Belegradek | Another examples where there is a sort of a classification is homotopy tori: $\mathbb Z^n$-actions on $\mathbb R^n$ by deck-transformations of a covering space. But already if $G$ is the fundamental group of a closed hyperbolic $n$-manifold, then there is no $G$-diffeomorphic classification of free properly discontinuous $G$-actions on $\mathbb R^n$ even though all of them are $G$-homeomorphic if $n\neq 4$. So there are answers in some cases, albeit complicated ones. | |
| Feb 5 at 22:57 | comment | added | Igor Belegradek | @MatthewKvalheim: for fake projective spaces a summary of what's knows is at map.mpim-bonn.mpg.de/Fake_real_projective_spaces. | |
| Feb 5 at 22:54 | comment | added | Igor Belegradek | @MatthewKvalheim: I don't have a good answer (of why there isn't a classification). I thought a lot about vector bundles with the same total space and I don't have any ideas what the classification might look like in that case. Is the equivalence relation a $G$-bundle isomorphism? People in transformation groups certainly study (not necessarily free) $G$-actions on specific manifolds such as spheres, tori, low-dimensional manifolds. Even in these cases fixing the manifold on which $G$ acts does not lead very far. Even the study of quotients of $S^n$ by a free involution is quite complex. | |
| Feb 5 at 19:28 | comment | added | Matthew Kvalheim | @IgorBelegradek thank you for your comment. As you point out, there can be many different possible base spaces of bundles having the same total space. But on the other hand, there are also many possible total spaces of bundles having the same base space, and yet there is a useful classification of these. So I remain confused: why is there a useful classification for bundles with a fixed base space but not with a fixed total space? | |
| Feb 1 at 0:01 | comment | added | Igor Belegradek | There are many examples of non-equivalent free $G$-actions on the same total space, especially when $G$ is a circle or a discrete group. For example, if two closed manifolds are h-cobordant, their products with a circle are diffeomorphic (except maybe in low dimensions), which gives free circle actions on the same $E$ with different orbit spaces. While having the same total space ties your hands somewhat, there is enough room to do topology and get different quotients of free $G$-actions. | |
| Jan 31 at 22:54 | comment | added | Matthew Kvalheim | What if we consider principal $G$-bundles with fixed total space $E$ and also fixed structure group $G$? Does that eliminate taking products as you mentioned? (I do not know if you are referring to products of groups or of spaces, and I do not know what you mean by "shove any number of actions into a...space".) | |
| Jan 31 at 20:51 | comment | added | Denis T | On the other hand, there are plenty of results of the form "groups of X type cannot act freely on a space of type Y". Finite groups with a noncyclic Sylow subgroup cannot act without fixed points on a sphere of any dimension; circle does not act freely on a space with nonzero Euler characteristic (there are a lot of papers about free circle actions). | |
| Jan 31 at 20:46 | comment | added | Denis T | It's pretty clear that any naive version of such "classification" is entirely hopeless: topological groups having decent homotopy type (at least all discrete groups) act freely on a contractible space; taking products allows you to shove any number of actions into a single contractible space. // There's a more reasonable version of your question: identifying a subset of bundles over fixed base B, such that total space is some E. This task is usually very hard and seldom has any tractable answer. | |
| Jan 31 at 19:12 | history | asked | Matthew Kvalheim | CC BY-SA 4.0 |