[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]

This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global equations?

Suppose $M$ is a smooth compact $d$-dimensional submanifold of $\mathbb{R}^n$ given as the transversal zero locus of $k=n-d$ functions. The normal bundle of $M$ in $\mathbb{R}^n$ is framed and moreover, it turns out that $M$ is framed cobordant to 0 bounds a submanifold of $\mathbb{R}^n$, unless $d=0$. (At first I thought this was a consequence of Sard's lemma; now I think this is not quite so obvious but true nonetheless.)

Q1. I would like to ask: is there a submanifold $M$ of $\mathbb{R}^n$ with trivial normal bundle such that no framing of this bundle makes $M$ framed cobordant to 0? A positive answer to this would mean that the answer to the above-mentioned question is negative. [This question still stands, but I don't think it is directly related to the above mentioned question in the other thread.]

Q2. Is there a manifold $M\subset\mathbb{R}^n$ with trivial normal bundle such that no $N$ with boundary $M$ can be embedded in $\mathbb{R}^n$? Presumably this is more difficult than Q1. [But if the answer is positive, this would mean that there are submanifolds of $\mathbb{R}^n$ with trivial normal bundles that can't be given by global equations.]

In general, if $M$ is embeddable in $\mathbb{R}^n$, there seems to be no reason any of the manifolds bounded by $M$ should be. However I do not know of any obstructions or counter-examples.

Q3. What if we drop the condition that the normal bundle is trivial in Q2 and replace it with the weaker condition that $M$ is cobordant to 0, i.e., that all Stiefel-Whitney numbers of $M$ are 0?

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    $\begingroup$ Regarding your 2nd paragraph, consider the map $\mathbb R^4 \to \mathbb R^3$ which you obtain from $S^3 \to S^2$ by "coning off" to get a map $D^4 \to D^3$, then removing the boundary to give a map $\mathbb R^4 \to \mathbb R^3$. The pre-image of any point other than the origin is a Hopf fibre, but stabilized to be considered as a submanifold of $\mathbb R^4$. This represents $2$-torsion in the framed cobordism, since suspension $\mathbb Z \simeq \pi_3 S^2 \to \pi_4 S^3 \simeq \mathbb Z_2$ is reduction mod 2. $\endgroup$ – Ryan Budney Oct 26 '11 at 18:19
  • $\begingroup$ Ryan -- true, the "Hopf" framed circle represents 2-torsion, but after changing the framing we get 0 $\endgroup$ – algori Oct 26 '11 at 18:28
  • $\begingroup$ Perhaps you meant to include the change-of-framing statement in the 2nd paragraph ? The way I'm reading the statement, it appears false for the reasons given in my first comment. $\endgroup$ – Ryan Budney Oct 26 '11 at 20:09
  • $\begingroup$ Ryan -- I think you are right and the statement is false. Moreover, I'm not sure that even a change of framing would save it (nor that it won't). $\endgroup$ – algori Oct 26 '11 at 23:40

The Lie group $SU(2)\cong S^3$, with its left-invariant framing represents a generator of the cobordinsm group of stably-framed 3-manifolds: $$[(S^3,\text{left-invariant framing})]=\bar 1\in \mathbb Z/24\cong\Omega^\text{fr}_3.$$

If you change the framing on $S^3$, you can reach anything in the set $$ \{\bar 1,\bar 3, \bar 5, \bar 7,\ldots,\bar {23}\}\subset\mathbb Z/24, $$ but not the other elements.

My answer deals with tangential framings, whereas the question was about normal framings.
I guess I'll leave it here as it might be of independent interest...

  • $\begingroup$ Andre -- thanks, that is interesting, but yes, this is not exactly what I was asking. $\endgroup$ – algori Oct 26 '11 at 23:47

To Q3. If you ask whether an $M^d$ embedded into $R^n$ bounds a manifold in $R^{n+1}_+$, then here is a partial answer. Let $Emb(d,n)$ denote the cobordism group of embeddings of $d$-dimensional manifolds into $R^n$. Zvagelski has computed that $Emb(3,5) = Z_2$ (See: MR1756714 (2001e:57032) 57R40 57R90) Take an embedding $f: M^3\to R^5$ representing the non-trivial element of this group. This $M^3$ , like any other $3$-manifold, bounds a $4$-manifold, but that $4$-manifold can not be embedded into $R^6_+$ if the boundary must be mapped by $f.$


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