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Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.

I believe it could be proved by directly constructing the ambient framed cobordism (using some tubular neighborhoods and surgery..) but is there not a reference for the claim?

EDIT: $n>0$.

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The claim doesn't seem to be true when $n=0$ and $M^m$ is orientable.

In that case framed cobordism classes correspond to homotopy classes of maps $f:M^m\to S^m$, which by the Hopf degree theorem are classified by their degree. Any map with $|\operatorname{deg}(f)|>1$ gives a counter-example.

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  • $\begingroup$ Thanks, I edited the question; I'm mainly interested in the $n>0$ case. $\endgroup$
    – Peter Franek
    Commented Jul 31, 2014 at 15:44
  • $\begingroup$ Ah, OK. Where does your proposed proof for $n>0$ break down when $n=0$? $\endgroup$
    – Mark Grant
    Commented Jul 31, 2014 at 16:12
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    $\begingroup$ $(n>1)$-manifolds can be surgered to make them connected (connected sum). Not true for 0-manifolds. $\endgroup$
    – Kevin Walker
    Commented Jul 31, 2014 at 18:55
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    $\begingroup$ Oops -- that should have been $(n\ge 1)$-manifolds. (Too late to edit the original comment now.) $\endgroup$
    – Kevin Walker
    Commented Jul 31, 2014 at 21:02

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