Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold $M\times \{\ast\}\times\cdots\times \{\ast\}$?
For some particularly nice manifold,i tend to believe,the answer is positive.Say torus $T^n$. In general,I guess,this is not true.
(Following the idea from the classification of knot theory,I think,a good understanding of the diagonal complement would help here.Say, for the $k=2$ case,the diagonal complement is a fiber bundle over $M$ with fiber $M-\{\ast\}$.But $M\times M-M\times\{\ast\}$ is always a trivial bundle.)
1.Are there some more examples for $k\geq 3?$
2.Have people already established some criterion for the existence of such an isotropy or any obstruction theory in terms of the algebraic topology of $M$? Are there some nice references on this topic?