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Peter Franek
Peter Franek
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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: $n>0$.

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?

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: $n>0$.

Source Link
Peter Franek
Peter Franek
  • 972
  • 5
  • 23

Connected representant of a framed cobordism class (reference needed)

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?