Show that a continuously differentiable function is not 1-1

[mathwonk]

@WWGD, do you refer to the closed unit ball in R^3, i.e. the set of x such that |x| ≤ 1 as the 3ball? If so, surely this is a contractible space, hence homotopy equivalent to a point, hence has zero homology everywhere except at degree zero. What am I missing? Are you referring to the fact that a compact orientable 3 manifold (without boundary) has non zero 3rd homology? the problem here seems to be that the 3 ball has a boundary. Since it is homeomorphic to its image, the image also has no 3rd homology. What do you think?

@StoneTemplePython. Thanks for the relatively elementary reference to Chiin and Steenrod. I am somewhat reluctant to read it there however since I feel any elementary argument should suggest itself to me just by reflecting on it. And I consider winding numbers relatively elementary. The difficulty for me is how to prove that if we have an injective continuous map from the circle to the plane, that there is some point of the plane that the image curve does actually wind around in a non zero way, i.e. why does the complement of the image curve have a bounded component? I don't quite get it from your remark on projections.

Here is a simple statement I would like a simple proof of: a smooth, or even just continuous, injection from R^2 to R^2, is an open mapping, hence a homeomorphism onto an open subset of R^2. This seems to follow from Jordan curve theorem, but since the hypothesis is stronger, might be easier to prove. Notice in dimension one this is just the intermediate value theorem hence pretty elementary, although admittedly as deep as any foundational result in one variable calc.

 

[mathwonk]

A quick course in winding numbers:

Consider a continuous map f from the unit interval [0,1] to the unit circle in the plane such that f(0) = f(1). This is essentially a continuous map from the circle to the circle. We want to measure how many times f “winds around” the circle.

method 1) We subdivide the interval into small subintervals so that the image of each subinterval under f is short (say shorter than 1, just to make sure it does not contain any two antipodal points, so that the image is contained in an angle of less than 180 degrees). Then we note the angle difference between the images of the two ends of each subinterval, and we add up those angle differences. The answer must be 2πn for some n, and n is the number of times f winds around the circle counterclockwise. This must be shown to be independent of the subdivision into small subintervals.

method 2) Consider the exponential map taking t of R to the point e^it in the (complex) plane. (Thus t is the angle measure of the corresponding arc on the circle from e^i.0 to e^it.) Then given a continuous map f:[0,1]—> unit circle, there is a unique continuous map g from [0,1]—>R taking t to 0, and such that for all t in [0,1], we have e^ig(t) = f(t). I.e. g is the log of f. Then the image point g(1) is some number of form 2πn, and we define n as the winding number of f around the unit circle.

These methods are the same, since g is defined essentially the way we measured the angle differences in the images of f, using a subdivision of [0,1].

Now if f is any map from [0,1] to the plane with f(0) = f(1), i.e. a closed curve (not necessarily simple), but missing the origin, we define the winding number of f about the origin to be the winding number of the map taking t to f(t)/|f(t)|, i.e. the map to the unit circle obtained by projecting f(t) radially out to the unit circle.

If p is any point missed by f, we define the winding number of f about p, to be the winding number of the map taking t to (f(t)-p)/|f(t)-p|, i.e. the same but after translating by p to make p into the origin.This concept of winding number has the following basic properties:

1) If f, considered as a map from the circle to the plane, extends to a continuous map F of the closed disc to the plane, still missing the point p, then the winding number of f about p is zero.
2) If there is a continuous map F of a cylinder to the plane, still missing p, which restricts to f on one boundary circle of the cylinder, then the winding number of the restriction of F to the other boundary circle is the same as that of f. I.e. winding number is constant under continuous deformation.
3) If f misses both p and q, and p and q are in the same connected component of the complement of the image of f, then f has the same winding number about p and q. (At least this seems true.)Cor: (contrapositive of 1): If f is a continuous map of the circle to the plane, missing p, but with winding number non zero about p, and if F is any continuous extension of f to the closed disc, then F does not miss p, i.e. for some point q inside the unit disc we must have F(q) = p.

This corollary is the 2 dimensional analog of the intermediate value theorem.Now somehow we want to use these ideas to prove that if F is any continuous map of the 2 sphere into the plane, there are 2 points of the sphere that map to the same point. (weak Borsuk - Ulam).

E.g. if we could prove that for some disc embedded in the sphere, centered say at q, that the boundary of the disc misses F(q) but does wind around F(q) a non zero number of times, then the restriction of F to the other disc in the sphere with the same boundary circle, would have the same (or negative of) winding number number, so would have to also hit F(q). This would give us two points with the same image.

But how to prove that the image of some circle in the sphere, does wind around some point of the plane, a non zero number of times? I.e. how do we know that it cannot happen that the image of every circle fails to wind around any point at all?

help? (Chinn and Steenrod spend 3 pages on this theorem, so some cleverness may be used somewhere.)

Rmk: The fundamental theorem of algebra is relatively easy using these tools, since one can directly calculate that the polynomial map z^n has winding number n about the origin, and that any polynomial of degree n has the same winding number as z^n on some large circle.

But here we don't have a concrete map like z^n to compute with, just the continuity hypothesis.

heres an idea: if you think of a simple closed curve in the plane and one point inside it and another point ouside it, then an arc going from one point to the other should have to cross the closed curve once, or an odd number of times. maybe one can prove that if an arc intersects a simple closed curve once (transversely?) then its endpoints are on opposite sides of the closed curve, which would enable us to know? they have different winding numbers wrt the curve?

So if we have an injection from a 2 sphere to the plane, then a circle close to the south pole does not wind around the image of the north pole, since the constant map at the spouth pole does not. now an arc running from the north pole to the south pole crosses that little circle near the south pole exactly once, and its image joins the images of the south and north poles. So its winding number should be different for the two poles, hence non zero for the south pole. Then by the argument above, the map should actually hit the south pole again at some point on the north pole side of the little circle, a contradiction.

but this does not seem airtight at all to me, since I have not proved the key assumption nor made precise the hypothesis of transversality. But I believe it anyway. Maybe Chinn and Steenrod do something like this.

reactions?

 

[WWGD]

Habout this? We have a smooth map by assumption
.and ythe map restricts to a homeo on the ball. Don't we then have a local diffeomorphism?

 

[mathwonk]

i think the subtlety is the difference between a homeo from the ball to its image, which could be awful, and a homeo from the ball to nice subset. I.e. we have no idea what structure the image has. Unbelievably, even though the image is homeomorphic to a ball, it "could", until proven otherwise, be a strange subset of R^3, say with no interior.

As crazy as it sounds, it is not at all clear theoretically, even though obvious intuitively, that under a continuous or even smooth injective map from the disc to the plane, the image of a disc has a non empty interior. I.e. using the jordan curve theorem we know that the image of the circular boundary of the disc, is a simple closed curve, which separates the plane into two disjoint open components, and that the interior of the disc maps homeomorphically onto the bounded component, an open set in the plane homeomorphic to the disc.

But we do not know without the jordan curve theorem, that the disc might not map homeomorphically onto some strange subset of the plane with no interior at all.

At the moment I cannot even prove that, under an injective continuous map from the disc to the plane, taking 0 to 0, that the image of the boundary circle does actually wind once around the origin.

Well I have an idea, based perhaps on old memories of Chinn and Steenrod, that it may follow from some vector field results. Assume that we have a smooth map from the sphere to the plane such that no two antipodal p[oints map to the same image point. Use that to define a non zero vector field on the disc as follows: to each point of the upper hemisphere of the sphere, you have two distinct points of the plane, namely the image of the given point and the image of its antipodal point. Since thiose are by assumption different, by joining them you get a non zero oriented segment in the plane hence a non zero vector, which you can attach to the original given point of the uppr hemisphere, which is a disc. This gives you then a nerver zero vector field on the disc, which takes oppositely directed vectors at opposite points of the equator.

Now I claim that a non zero vector field on the equator, which sends opposite points to opposite vectors, must define a map from the equator to the circle with non zero winding number. hence it cannot extend to a map from the disc to the circle. i.e. by results above, any extension to the disc must map some point to zero, which does not happen if the points are all projectible onto the circle.I think this works, but it is a bit sophisticated to have to factor the argument through a vector field argument. I don't understand why it has to be so sophisticated.
almost bet this is the kind of proof in Chinn and Steenrod, indeed a quick look at their contents shows the 3 previous sections are about vector fields. So sorry, this is the kind of magic that some beautiful topology results often seem to involve.

the best I can say is that somehow this vector argument let's us use the winding number theory without picking a specific point to wind around, i.e. somehow the vector field includes winding around all points at once in terms of the twisting of the vector field...

you may recall the fixed point arguments for maps of a disc to itself also use a retraction argument from the disc to its boundary defined by the vector field going from each point to its image, assumibng there is no fixed point.

 

[WWGD]

I think the image of the ball would have non-empty interior. Homeos map interior to interior . Or just take an open subset of the ball. Since homeos are open maps, it's image would be open in R^2.

 

[mathwonk]

i think you are assuming that "homeo" means homeo to an open set, but all you know is homeo to some set with the induced topology. i.e. your homeo only maps an open set to the intersection of an open set in R^2 with the image set. that is what an open set of the image means. this is very unintuitive i agree, but i think it is unavoidable.

in fact it is not true that a homeo maps interior to interior. take a homeo from an open disc in the plane to an open disc in R^3, which has no interior. it is still a homeo onto its image but the image has empty interior in R^3. you have to prove that cannot happen when the image is in R^2. it is not at all obvious. well it depends what you mean by obvious. it seems obvious but it is not easy to prove.

 

[mathwonk]

note in the vector field argument above how sneaky, i.e. brilliant, it was of Borsuk Ulam to assume that no two antipodal points mapped to the same image, rather than just that no two points at all did so. That gave us the ability to construct a map from the unit circle to itself with opposite values on opposite points, which forces the winding number to be non zero. I.e. when you go half way around, the image point must be half way around from the original image point, hence the total angle change has form 2πn + π. And when you go the rest of the way around you must repeat the same values so you must complete the tour around the circle, giving an odd winding number, i.e. angle change of form 4πn + 2π = (2n+1).2π, hence non zero winding number 2n+1. I know this is vague but it seems clear to my visual mind that this trick allows you to actually compute a non zero winding number, something I lamented not being able to do with merely a hypothesis of continuity. Thus it was actually easier to prove the more precise statement that some pair of antipodal points mapped to the same image than to prove that some arbitrary two points did so. this is cleverness personified.

A quick perusal of one of my most sophisticated algebraic topology books, e.g. the lectures of Albrecht Dold, reveal that the theorem that no continuous map from an n sphere to euclidean n space can be injective, is rather advanced, using the full power of homology and neighborhood retracts, etc...occurring on about page 78 of his somewhat condensed and elegant book, in the section on jordan curve theorem and invariance of domain.

by the way, in the differential geometry and differential topology books i have, the discussion of smooth jordan curve theorems seem rather weak, since they only prove that a submanifold of R^n+1 which is diffeomorphic to S^n separates the ambient space into two pieces, each open, one bounded and one not. The weakness is in the assumption that the space is a submanifold, rather than just assuming we have the image of a smooth injective map from S^n into R^n+1. I.e. the hard part seems to be to prove that the image is indeed a submanifold, which these books just assume. Or maybe I have skipped some earlier preliminaries where they prove this but i did not notice it. And these are fine books, by authors far more expert than me, so I could well be wrong, indeed I hope so.

but i think we are learning from this discussion that assumptions on the rank of a smooth map are quite crucial in making arguments on the structure of the image easier, since they are needed for applying the implicit function theorem. i.e. without assuming maximal rank, it is hard to draw conclusions. i suspect that's why the theory of singularities of maps is so difficult.

Yes, I think now one gets very little benefit from assuming just smoothness, and to get a real bump one needs to assume smooth of maximal rank. Maybe I should revise my estimate of the cleverness or usefulness of the OP's original problem, since his prof has found a way to get mileage just out of, no wait a minute, for a map from R^2 to R^1, rank 1 is maximal. So again we have a maximal rank smooth map, which then is goosed up to a rank 2 map from R^2 to R^2 to use the inverse function theorem.

So smoothness does not help much on understanding the structure of the image of a smooth injective map from a circle to the plane unless we also assume the map is an immersion, i.e. rank 1 everywhere. Then one can prove the image curve is a submanifold, hence each point has a nice local "collar", so one can understand the two "sides" of the curve, and what iot means to cross the curve transversally, etc...

well this has helped me a lot to understand why authors make the hypotheses they do. I recall when I taught calc of one variable, I always said (exaggerating somewhat) that "when the derivative is zero, i.e. not of maximal rank, it tells you zero". I.e. you cannot conclude you have a max or min or anything at all really at that point. (Zero derivative at one point tells you little that is, since of course one important case where you do get something useful is when the derivative is zero everywhere.) The same seems to hold in diff top, you need to know your space is a manifold, or your subspace is a submanifold. After all you want to use smoothness to approximate your situation well by a linear situation, and for that you need maximal rank, so that the linear approximating object has the same dimension as the manifold.

If anyone figures out how to use these elementary, i.e. winding number arguments, to prove that an injective continuous map from an open disc to the plane is an open map, i.e. that every injective contin. map R^2-->R^2 is a homeo onto not just some subset, but onto an open subset of R^2, I think that would be interesting. By way of encouragement, this is presumably somewhat easier than the jordan curve theorem, since we are assuming given an injective continuous map on the whole disc. The Jordan curve theorem and Schoenflies theorem together say, if I recall correctly, that every continuous injection from the circle to the plane extends to a continuous injection of the disc to the plane, and the extension is a homeomorphism from the open disc to an open subset of the plane bounded by the image of the circle; but we are giving ourselves this injective extension.

For references, there is actually a presentation, in the wonderfully impressive analysis book by Dieudonne'. Foundations of Modern Analysis, of the Jordan curve theorem, but I think not the Schoenflies theorem, using as a tool only the winding number concept and some complex analysis, in an appendix to his chapter IX, which treats elementary complex analysis. I have never read it, but always meant to, since valid elementary proofs of the Jordan theorem are famously hard to find. I sat through one in grad school many years ago but learned essentially nothing fom it, except that I did not want to specialize in that sort of topology. It may have been taken from the classic book of Newman.

https://www.amazon.com/dp/0313249563/?tag=pfamazon01-20

 

[mathwonk]

By the way, in reference to the attempted homology arguments above, I cannot find a reference that shows all subsets of R^2 have zero degree 2 homology. It is true however for neighborhood retracts, and the property of being a neighborhood retract is a homeomorphism invariant. Thus since an n sphere is a neighborhood retract in R^(n+1), then also any subspace of any euclidean space that is homeomorphic to a sphere is also a neighborhood retract. It is also possible to give a direct proof using the Tietze extension lemma, for the case of a subspace homeomorphic to a sphere. Thus indeed an injective continuous map from R^3 to R^2 would induce a homeomorphism from the 2 sphere to some neighborhood retract in R^2. This is then a contradiction since the 2 sphere has non zero 2nd homology while the neighborhood retract in R^2 does not.

These arguments depend on rather sophisticated and lengthy arguments at least in the book of Dold, using local homology, retracts, and numerous exact sequences. The vanishing result for retracts has something like 6 steps, spread over some 4 1/2 pages. And this is not a wordy book. This is far more sophisticated than the winding number argument above that no continuous map from the 2 sphere to the plane can be injective, via Borsuk Ulam. Of course you do get more from the homology version, since homology let's you count connected components. Thus Dold also concludes that the complement of a subset of the plane homeomorphic to a disc is connected. and the complement of a subset homeo to a circle has two components. Thus given an injective continuous map of a disc to the plane, the interior of the disc must map onto the unique bounded connected component of the complement of the image of the boundary circle, which must be open. We can then conclude that any injective continuous map R^2 to R^2 is a homeo onto an open subset. But this uses a lot of machinery.

 

[mathwonk]

I just reviewed the presentation in the little book of Marvin Greenberg, who benefits from borrowing heavily from the book of Emil Artin, and the argument for these results looks much easier. Apparently that is because he argues only for the case at hand (sets homeo to a sphere or a ball) rather than proving vanishing of homology for all neighborhood retracts as Dold does. So Dold's argument is more difficult and longer but yields a much more general result.

Indeed that special case seems to be covered by just one of Dold's 6 steps, (step 2), which he deals with even more briefly than does Greenberg.

 

[mathwonk]

The question of when an injective map is an open embedding is quite interesting and has different answers in different settings. We have been exploring the theorem that an injective continuous map of n dimensional topological manifolds is indeed a homeomorphism onto an open set, in particular the inverse function not only exists but is guaranteed to be continuous.

A more elementary example is the theorem that a linear injection of vector spaces of the same dimension is an isomorphism, i.e.is also surjective and the inverse function is linear.

On the category of smooth real manifolds, the result is false, since we have the smooth bijective map of R to R taking t to t^3, which is infinitely differentiable and bijective but whose inverse function is not differentiable at t=0.

If we pass to the realm of complex holomorphic manifolds we have a different result. A smooth map of complex manifolds, whose derivatives are everywhere complex linear, i.e. a holomorphic map, is holomorphically invertible if and only if it is bijective.

The same result applies to algebraic varieties over the complex numbers, or any algebraically closed field of characteristic zero; namely an injective regular map of non singular algebraic varieties is an open immersion.

The result is not true for singular varieties however, since the map taking t to (t^2,t^3) is a bijection from C to the curve y^2=x^3 which is not invertible; i.e. the inverse map taking (t^2,t^3) = (x,y) to t = y/x, is not regular at (0,0).

So in general we need both a non singular (or at least "normal") target space and an algebraically closed field of definition, but in the case of continuous maps over R we are good. Notice that similar results are much easier in algebra, e.g. bijective homomorphisms are also isomorphisms for groups and rings. This is also true for Banach spaces but there the requirement the inverse linear map be continuous makes the proof non trivial in infinite dimensions where linear does not automatically imply continuous.