User william - MathOverflow

Eversion of the 6-sphere in 7-space

2012-12-01T20:49:40Z

Say that $S^n$ "admits eversion" if the inclusion $S^n \rightarrow \mathbb{R}^{n+1}$ is regularly homotopic to the antipodal map (where a "regular" homotopy is a continuous path through immersions).

Smale proved that $S^2$ admits eversion by defining an appropriate algebraic invariant corresponding uniquely to regular homotopy classes, and noted that the group this invariant lives in is trivial. Many people didn't believe it until someone made a movie illustrating an explicit eversion.

It can be shown that $S^n$ admits eversion if and only if the tangent bundle of $S^{n+1}$ is trivial. That is, the only spheres which admit eversion are $S^0$, $S^2$, and $S^6$.

My question is: does anyone know of an explicit eversion of $S^6$ in $\mathbb{R}^7$?

Answer by William for What are some examples of ingenious, unexpected constructions?

2012-03-12T00:03:31Z

I notice no one mentions Godel's Incompleteness Theorems, or the basic non-computability results of Turing. Are they too obvious?

If these are not constructive enough, one that is fairly elementary is a space-filling curve. If you want something non-intuitive, how do you like a continuous surjection from a 1-dimensional space to a 2-dimensional space?

Treating the Connected Sum (and other constructions) as a Push-out

2012-03-05T01:38:45Z

It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined http://mathoverflow.net/questions/87857/characteristic-classes-of-a-fibered-sum) can be expressed as a push-out in the Smooth category, so long as the construction includes collars. The connected sum follows as a special case where the submanifold is a point. Furthermore, Thom's "shperical modifications" can also be treated as a connected sum. Basic facts about these operations (well-defined under isotopy of the embeddings, associative, etc.) can be proven as corollaries of theorems about push-outs.

I've done some work on my own developing this idea, but I'm wondering if there's some theory already in place that could help me save some time or point me in better directions. I haven't come across anything in my searches. What I'm looking for specifically are references which treat these operations as pushouts, and develop them as such. (What I am NOT looking for are modifications of the smooth category which make it closed under pushouts)

Who defined the Inertia Group $I(M^n)\subset\Theta_n$ of a smooth manifold?

2012-02-25T21:07:41Z

If you're unfamiliar with the definition, for an oriented smooth manifold $M^n$ we define the inertia group $I(M)$ to be the set of (h-coboridsm classes of) homotopy spheres $\Sigma^n$ such that $M\#\Sigma$ is orientation-preserving diffeomorphic to $M$.

I'm trying to compile results into an expository Master's thesis on the subject, and it seems silly to not know the origin. Digging through old papers about the Inertia Group, I'm having a hard time finding the start of the trail. Many early papers refer to Tamura's "Sur les sommes connexes de certaines variétés différentiables" so I expect it to be close to the beginning, but I have been unable to find a copy of this paper.

I am aware of a few members here who are familiar with this theory, and maybe were even around when it started. Does anyone happen to know in which paper/book the Inertia Group originated?

When is a Topological pushout also a Smooth pushout?

2012-02-23T18:15:09Z

I feel like this problem has not been solved, but I'm interested in knowing any results on it. More specifically, I mean:

Let $B\stackrel{f}{\leftarrow} A \stackrel{g}{\rightarrow} C$ be a diagram of Smooth manifolds and Smooth functions, and let $B\stackrel{k}{\rightarrow} D \stackrel{l}{\leftarrow} C$ be the Topological pushout (that is, $D$ obtains a Set structure as the quotient of $B \coprod C$ by the images of $f$ and $g$, and receives the weak Topology induced by $k$ and $l$). GOAL: Determine necessary and sufficient conditions for the Topological manifold $D$ to receive a natural Smooth manifold structure making the diagram a Smooth pushout.

To make it seem less daunting, break it into steps:

1) When is $D$ a Topological manifold (that is, inherits a Topological atlas from the diagram)?

2) When does $D$ inherit Smooth transition functions? (or whatever your favorite definition of "smooth structure" is)

2') If $D$ does not inherit a Smooth structure from the diagram, can we give it one? (I believe this is just Hirsch-Mazur)

3) Can Smooth functions be defined piecewise?

I'm certain that #1 is most difficult. Using $A\subset\mathbb{R}=B=C$ and $f,g$ inclusion maps, $A=0$ gives a point with 4 arms (not locally Euclidean) and $A=\mathbb{R}\setminus0$ gives the line with two origins (not Hausdorff). I find it difficult to find concise conditions that rule out those two situations. My only idea is to assume that everything is also cellular and try to come up with an obstruction class somehow (hence this question: http://mathoverflow.net/questions/88842/when-is-a-finite-cw-complex-a-compact-topological-manifold)

I have a few ideas and results for #2. For example, attaching manifolds along a submanifold (as defined in Kosinski's "Differential Manifolds") can be expressed with a smooth pushout diagram, which I proved by making a more general sufficient condition for "passing" a Smooth structure from two (or $n$) Smooth manifolds onto a Topological manifold. My intuition says that the criteria for #3 will be almost the same as for #2

I'm interested in seeing what else is out there right now, and Google/arXiv searches aren't turning anything up. I'm half-expecting to see the usual "This is basically Surgery Theory" response....

When is a finite cw-complex a compact topological manifold?

2012-02-18T17:01:06Z

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-dimensional topological manifold (possibly with boundary)? Most of the questions I have found are about the converse, "When is a topological manifold a CW-complex?" so I thought it would be useful to consider the other side of the picture.

First of all, we already have 2nd countable and Hausdorff so we only need to determine when it is locally Euclidean. One necessary condition is that every point must be contained in the closure of at least one $n$-cell. One the other hand, a point can't be in too many $n$-cells, as the wedge of spheres is not a manifold. But these criteria are clearly not sufficient (or very precise: what does "too many" mean exactly?).

Another necessary condition that I think of right away is that the space must satisfy Poincare duality (with $\mathbb{Z}/2$ coefficients). I'm not sure if this is practical at all, but maybe it is useful if you are working with an explicit cell-structure and a concrete description of the cellular chain complex.

After that, I don't know how to proceed. I am assuming this is a difficult problem, since Google searches haven't answered my question yet. One idea that I have is to try and come up with an "obstruction" to this, where cell-by-cell we determine if every point in that cell admits an $n$-dimensional Euclidean nhd, and we have that the CW-complex is a manifold iff this "obstruction," computed at the homology/cohomology level, vanishes.

Any suggestions (or references to a solution) will be appreciated.

Answer by William for Connected sum of surfaces

2012-02-09T00:22:01Z

A relatively clean and intuitive proof is given in Kosinski's "Differential Manifolds," which works in the topological setting and essentially boils down to the following:

If $M$ is path-connected and $i_1,i_2:D\rightarrow M$ are isotopic embeddings (smooth or topological), then by the so-called "Cerf-Palais disk theorem" (a consequence of the Isotopy Extension Property) there is an ambient isotopy $\Phi:M\times I\rightarrow M$ (smooth or topological) such that for all $t$, $\Phi_t$ is identity outside a contractible compact set, and $\Phi(i_1(x),1)=i_2(x)$. Intuitiely, $\Phi$ translates the image of $i_1$ to the image of $i_2$, and tries hard to not effect anything else.

So if $M, i_1, i_2$ are as above, $N$ is another topological (or smooth) manifold and $i:D\rightarrow N$ is another embedding, then let $M\#_1N$ be formed by attaching $N\setminus i(0)$ to $M\setminus i_1(0)$, and form $M\#_2N$ using $M\setminus i_2(0)$. Since these objects are actually pushouts, we can define a homeo(diffeo)morphism in pieces: If $y\in N\setminus i(0)$, send $y$ to itself; if $y\in M\setminus i_1(0)$, send $y$ to $\Phi(y,1)$. These will assemble to give the required equivalence from $M\#_1N$ to $M\#_2N$. (then you could repeat the argument on the $N$ side, or just say it follows from commutativity)

The fact that the connected sum is associative and commutative follows naturally from the fact that it is actually a pushout (if you're careful, it does make a pushout in the smooth category). Then to show that it doesn't depend on the attaching disk, I think you need something equivalent to the "Cerf-Palais" theorem I mentioned.

Edit: because of what was mentioned in the comments above, it was necessary for me to assume that the embeddings were isotopic to begin with

Characteristic Classes of a Fibered Sum

2012-02-07T23:54:14Z

I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.

Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for simplicity), $N$ a closed $k$-manifold, $D$ a closed $(n-k)$-disk bundle over $N$ (so that $D$ is an $n$-dimensional manifold whose boundary is the associated $(n-k-1)$-sphere bundle), and suppose we have embeddings $f_i:D\rightarrow M_i$.

Now say $\tilde{M_i}=M_i\setminus f(N)$, let $E_0$ be the punctured-open-disk bundle associated with $D$, and let $\alpha:E_0\rightarrow E_0$ be defined by sending $v_x$ in the fiber of $x\in N$ to the point $(1-|v_x|)\frac{v_x}{|v_x|}$ in the same fiber (intuitively, $\alpha$ turns $E_0$ inside-out so that we can attach the manifolds with a "collar").

Then if we form the Topological pushout of $\tilde{M_1}$ and $\tilde{M_2}$ using the smooth embeddings $f_1|_{E_0}$ and $f_2|_{E_0}\circ \alpha$, this in fact produces a pushout in the smooth category. The resulting manifold $M$ then has a tangent bundle, and in fact this tangent bundle can be formed by attaching the tangent bundles of $M_1$ and $M_2$ using the same recipe.

So, finally, here is the question: is there a formula for characteristic classes of $M$ (maybe just restrict attention to Stiefel-Whitney, Chern, Pontryagin classes) in terms of the characteristic classes of $M_1$, $M_2$, and $N$ (and the embeddings $f_1$, $f_2$)? More generally, is there a similar formula for attaching arbitrary bundles over arbitrary topological spaces (i.e. we would form a topological pushout on the base spaces and indicate how the fibers would be identified over points that are glued together)?

The inertia subgroup of `$\Theta_n$` for Lie groups

2011-12-29T18:42:17Z

See http://mathoverflow.net/questions/79991/smooth-manifold-with-non-trivial-inertia-group-wrt-homotopy-spheres for the definition of $\Theta_n$ and inertia subgroups.

I'm wondering what can be said about Lie groups. If $M^n$ is an n-dimensional manifold with Lie group structure and $\Sigma^n$ is a homotopy n-sphere, is there a Lie group structure on $M\#\Sigma$ that is in some sense compatible with the original structure? If this new group structure is isomorphic to the old structure, this implies Lie group isomorphism, correct?

What I would like to see is that there is a canonically induced Lie group structure on $M\#\Sigma$ and that this structure is isomorphic to that of $M$ , and hence the inertia group for Lie groups is the full $\Theta^n$

Higher dimensional Heegaard splittings?

2011-11-16T06:28:16Z

Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing homeomorphism $f$ of their boundaries so that $M=V\ \cup_f W$. Such a decomposition is called a Heegaard splitting.

I want to know: Does this kind of symmetric handlebody decomposition extend into higher dimensions? More specifically, given an $n=2k+1$ manifold $M$, can we construct a $V$ and $W$ by attaching handles $D^i \times \small{D^{n-i}}\ (i\leq k)$ to an $n$-disk, and find an orientation reversing homeomorphism $f:\partial V\rightarrow\partial W$ so that $M=V\ \cup_f W$? Since $f$ is only continuous here, it might not provide a unique smooth structure for $M$; could we remedy this by requiring $f$ to be a diffeomorphism instead? After all, any exotic sphere $\Sigma\in\Theta_n$ can be constructed by gluing two copies of $D^n$ together with an orientation reversing diffeomorphism of the boundary (except possibly $n=4$?).

Smooth manifold with non-trivial inertia group? (wrt homotopy spheres)

2011-11-04T01:03:17Z

Let $\Theta_n$ be the set of orientation-preserving diffeomorphism classes of homotopy spheres, with abelian group structure given by #. Then for any smooth manifold $M^n$ one defines the "inertia subgroup" as

$I(M)=${$\Sigma\in\Theta_n$ | $M$#$\Sigma\cong M$}

In other words, the inertia group is the set of homotopy spheres which fix the smooth structure on $M$.

Does anyone know an example of a diffeomorphism (as explicit as possible) between an $M$ and $M$#$\Sigma$, where $\Sigma$ is as exotic sphere? Moreover, if the dimension were sufficiently large can we always find a diffeomorphism which induces the identity on $\pi_1$?

(The big question I'm trying to think about is "If $\Sigma$ is in the inertia group of $M$, how does its smooth structure get 'unraveled' in $M$?")

Comment by William

2012-03-15T19:22:15Z

I think they are non-intuitive simply because I wouldn't expect them to happen. Maybe that's just me.

Comment by William

2012-03-07T02:44:46Z

Yes, that is the "with collars" construction I am thinking about (equivalently you could take $S^{n-1} \times (0,1)$) Do you think I should edit the OP to make that more explicit?

Comment by William

2012-03-05T17:58:22Z

@Daniel No, that is not the question I am interested in. I am more interested in the cases where the topological pushout is actually a smooth pushout. Does this not happen even when the construction includes collars?

Comment by William

2012-03-05T17:29:30Z

This is not an example of the construction I am considering. I am aware that a collar is needed in order to induce a smooth structure on an open overlap. I am not asserting that every topological pushout is a Smooth pushout, I am looking for a reference in the cases where it is.

Comment by William

2012-03-05T01:20:05Z

Update: My supervisor found me a copy of Tamura's paper! In one corollary at the end he has a diffeomorphism between a $7$-manifold and its connected sum with a non-trivial Milnor sphere, but nowhere in the paper does he use "inertial" or "$I(M)$" (or French equivalents). I haven't managed to find the definition in anything by Milnor yet either (he does use "$I(M)$" in "Differentiable Manifolds which are homotopy spheres," but here it refers to the index aka signature).

Comment by William

2012-03-03T17:48:55Z

Just out of curiosity, why do we know $\pi_3(G)\cong\mathbb{Z}$? Is there a simple reason, like maybe it has $S^3$ or $S^2$ as universal cover or something?

Comment by William

2012-02-28T05:17:50Z

Not a good question. Isomorphic in what sense? Maybe this is a question better suited for math.stackexchange.

Comment by William

2012-02-24T17:31:01Z

Thank you for the references. I was aware of "cartesean closed" expansions of the Smooth category, which your second reference goes into, but I am interested in pushouts that are specifically Smooth. This "diptych" idea in Pradines looks interesting, possibly providing a good way to phrase nec./suf. conditions for a smooth pushout. I'll keep looking at it, hopefully I find something

Comment by William

2012-02-24T03:11:15Z

Great thread. I think the fundamental fallacies behind this phenomenon are misinterpretation of terms, and thinking that real-world situations are part of a formal system. Approximations are often effectively used, but to conclude that your result applies you have to demonstrate with HIGH confidence that your approximation is sufficiently accurate.

Comment by William

2012-02-24T02:42:26Z

Apparently (according to Wikipedia) Goodwin had also proven such "truths" as trisecting a given angle, and had them published in American Mathematical Monthly, with the disclaimer 'published by request of the author.' Although this happened in the late 1800s, it makes me skeptical of ALL published mathematical results...

Comment by William

2012-02-24T02:28:24Z

I agree, this "application" is extremely laughable. I was expecting a million answers about Godel's theorems, but Noether's theorem applied to public relations? grooooaaaaaan..... I guess the problem is in thinking that our definition of "symmetry" is the same as their definition of "symmetry"

Comment by William

2012-02-23T23:51:55Z

Edited accordingly

Comment by William

2012-02-21T20:58:07Z

Just to be pedantic, if the functions are continuous on a closed disk then by the extreme value theorem (more generally, the continuous image of a compact set is compact) they are already bounded

Comment by William

2012-02-21T05:22:30Z

Yeah, I think "$A^{-1} = \frac{1}{\det(A)}\adj (A)$" is the easiest way to remember, because for a 2x2 matrix computing the adjugate is trivial

Comment by William

2012-02-20T18:24:47Z

I vote to have this question closed