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Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ are connected. Given two intersection points $x,y$ of opposite sign, I am looking for a concrete obstruction to finding a paths $\gamma_P$ in $P$ and $\gamma_Q$ in $Q$ connecting $x$ and $y$ such that the concatenation is a null homotopic loop in $M.$ In other words, an obstruction to finding a Whitney disk. This is possible when $M$ is simply connected but I am looking for a more refined obstruction.

Is there such a thing in the literature?

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  • $\begingroup$ The homotopy-class of the concatenation? $\endgroup$
    – Ryan Budney
    Commented Apr 6 at 20:35
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    $\begingroup$ @RyanBudney The homotopy classes of the paths in P and Q are not fixed, so the homotopy class of the concatenation is not well defined. The problem is whether one can find paths such that its concatenation is trivial. $\endgroup$
    – João Lobo Fernandes
    Commented Apr 6 at 22:13
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    $\begingroup$ For M not simply-connected, you need to look at the $\mathbb{Z}[\pi_1(M)]$-valued (self-)intersection numbers that are explained in surgery theory books. Is (the proof) 7.27 in Ranicki's "Algebraic and geometric surgery" helpful? For p=q=2, the discussion in Section 11.3 of "The disc embedding theorem" book or in 2.29 of arxiv.org/pdf/2201.03961.pdf might also be helpful? $\endgroup$
    – Anthony Conway
    Commented Apr 6 at 23:55
  • $\begingroup$ @AnthonyConway Thank you, that helps a lot. $\endgroup$
    – João Lobo Fernandes
    Commented Apr 7 at 9:00

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