Homotopy Theory

3:14 AM

is it true that S^oo (loops S^5) smash C(eta) = X(2)?

1:39 PM

@skd I don't think that's true, the two have different Steenrod operations on their mod 2 homology (Sq^2 and Sq^1 act the same but the higher ones don't)

Arrow

3:28 PM

Hello,
Can a local homeomorphism with homeomorphic fibers fail to be a covering map?

Tyler Lawson

3:40 PM

@Arrow In fact, it can. Let R be the real line, and let Y be a disjoint union of the intervals (q,q+1), one for each rational number q. Then the map Y -> R which sends these intervals to the corresponding intervals in R is a local homeomorphism and the fibers are all homeomorphic to countable discrete sets, but the map isn't a covering map.

Denis Nardin

It is true if you assume that the fibers are finite, but the proof uses that the intersection of finitely many open subsets is open and so it doesn't extend

Arrow

@TylerLawson ah, thank you. @DenisNardin how to prove the claim if the fibers are assumed finite? From a little googling it seems properness is a key property for questions of this kind.

Denis Nardin

So, I might be wrong but it should go this way: Let n be the cardinality of the fiber. Then I need to prove that for every point b∊B of the base, there is an open neighbourhood U of b such that the preimage is homeomorphic to U×n over U. Let b_1,...,b_n be the fiber over b. Then there are U_1,..,U_n neighbourhoods of b_1,...,b_n such that p|_{U_i} is a homeomorphism onto p(U_i) open neighbourhood of b (cont.)

I am assuming the total space to be Hausdorff, so we can assume U_1,...,U_n to be disjoint. Let U be the intersection of p(U_1),.., p(U_n). Then I claim that p^{-1}U is equal to the disjoint union of U_i ∩ p^{-1}U for each U. In fact U contains ∐ (U_i ∩ p^{-1}U). Moreover for each b' in U the preimage consists of exactly n points, so it must be contained in the disjoint union.

I'm actually not quite sure of what happens if the space is not Hausdorff, but a lot of covering space theory breaks down in that case, so I won't lose my sleep over it

Tyler Lawson

oh, that Hausdorff requirement is sneaky. you can take the disjoint union of R \ {0} and the line with the doubled origin, projecting down to R, and get a counterexample there too

Denis Nardin

Aha! Good point.

skd

4:04 PM

@TylerLawson i see, thanks. supposedly there's a cell complex Y with one cell in each dimension divisible by 4 such that Y smash C(eta) = X(2)

i thought that Y = loops S^5, but that's not right, as you pointed out

how do you construct this Y?

Tyler Lawson

my guess is that it's something like HP^infty

or maybe a Thom spectrum on same

skd

i'll try to check that

i guess i should look in mike hopkins' thesis

Arrow

That example of the line with double origin is why I'd like to avoid Hausdorff assumptions. I'll try to understand the importance of properness here. Thanks again!

Arrow

5:46 PM

@TylerLawson I have a question about your example of the coproduct of intervals of length one starting at each rational: "what" is the problem which prevents this map from being covering?

The example of the line with a double origin is an instance of gluing spaces along an open subset, which results in discrete fibers whose elements need increasingly smaller neighborhoods to be separated as we tend towards the boundary. On the other hand, this same problem of "fibers collapsing in the limit" can happen when we tend towards infinity, which (I think) is the role of properness in related ques…

Denis Nardin

6:39 PM

@Arrow Imagine that the intervals are shaped like the graph of a tangent function, then it should be clearer in which sense they are "running off at infinity"

Arrow

7:01 PM

@DenisNardin I don't get it :( In my mind the "problem" is that points in a fiber can be arbitrarily close, and the solution is to ask that distance of points in the same fiber has a lower bound. I'm thinking of this condition as the missing property making a local homeo into a covering map. But this doesn't seem to be a problem with Tyler Lawson's example since different elements of every fiber live in different coproduct components.

Drawing the intervals as graph of the tangent function seems to pretend the tip of some intervals is close to the tip of another, but isn't that wrong? Sorry for being slow by the way.

Denis Nardin

No, I meant that the boundary of the interval goes off to infinity. As I can see it, there are two problems for local homeomorphisms that want to be covering maps: 1) you might have a "ramification" kind of deal, where two distinct points of the fiber come crashing to only one. Properness won't help you with this, but this is a rather infrequent situation, since being Hausdorff (basically a requirement for any serious theorem) will prevent it

2) Points of the fiber "disappearing at infinity". Let us consider the projection {(x,y)∊R² | xy=1}→R sending (x,y) to x. This is a local homeomorphism (in fact it is an open embedding) but it is not a covering map, because the fiber "runs off at infinity". This is exactly what properness excludes

To see that properness won't help you with the first kind of problem, consider the projection of the line with two origins onto R. This is a proper local homeomorphism, but it is not a covering map

Arrow

I understand why properness won't help. I think properness can be characterized by taking infty to infty when extending to Stone-Čech compactifications for nice spaces

I think I understand my wrong mental image for (2) now, thanks! If I may continue to bug you, my "problem" with properness is that it forces a covering map to be finite. I would like to formulate the property possessed by all covering maps which is absent from general local homeomorphisms with isomorphic fibers.

Denis Nardin

Easier: taking infinity to infinity for the 1 point compactification

Arrow

(Sorry, the 1-point compactification is what I meant.)

Denis Nardin

Well, the unique path lifting property comes to mind

(and in all fairness, it is more being continous when extended to the 1-point compactification than taking ∞ to ∞, but I think you understand that :))

Arrow

7:15 PM

(by "extended" I meant "continuously extended :D)
Well, I was hoping for something that doesn't involve the unit interval. For (1), I thought about asking that nets moving between fibers which are not eventually equal should have no common accumulation point. (Or something like that)

Does that make sense?

Denis Nardin

I'm not sure. This feels very much like Hausdorffness... And covering space theory uses the interval so much that I'm not sure you gain much by avoiding it

Arrow

I'm originally motivated by trying to concretely justify the motto "sheaves on X=continuously variable sets over X". Since I can live with "coverings of X=continuously variable fixed set over X" (i.e only the orientation of the fiber is changing, not the fiber as a set), and sheaves are equivalent to local homeos, I'm trying to "solve" for the property P such that "covering map = local homeo with homeomorphic fibers + P". It would just be nice to understand P (whatever it may be) without [0,1].

Denis Nardin

I think what you want is "covering spaces= locally constant sheaves"

Arrow

That follows from the equivalence of sheaves and étalé spaces. I am looking to justify to myself why a local homeomorphism (therefore a sheaf) deserves to be called a continuously variable set

Since covering maps are continuous variations of a fixed set, I want to understand what properties on a mere local homeomorphism with homeomorphic fibers makes it locally trivial.

Denis Nardin

7:31 PM

I'm not sure you will find anything that's not "locally trivial" thinly disguised, or the path lifting property

Arrow

But even those two seem vastly different

Also, moving to path lifting properties eventually makes me try to pin down the difference between Hurewicz fibration and fiber bundle, which seems harder

Denis Nardin

If anything you should consider Serre fibrations, not Hurewicz ones (they are the well behaved ones), but yeah, the path lifting moves you sharply in the homotopical direction (aside: my understanding of the story there is that fiber bundles up to isomorphism are homotopy classes of maps B→BAut(F), while Serre fibrations up to fiberwise homotopy are homotopy classes of maps B→BhAut(F), where hAut(F) is the monoid of self-homotopy equivalences)

Arrow

That's completely above my head and is part of the reason why I'm hoping to avoid the homotopical viewpoint for something so (seemingly) innocent. I really feel that (1),(2) you cited are what I would like to capture without mentioning the unit interval.

Being proper does the trick for (2) for finite coverings, I'm just not sure what to do for the general case... From your experience are infinite covers important? Do they pop up often?

Denis Nardin

7:50 PM

It depends on what you want to use them for, but yeah they do show up fairly often

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