# References for the connectivity of complements of smooth submanifolds

Suppose $$M$$ is a smooth $$m$$-manifold and $$Z$$ a codimension-$$d$$ closed submanifold. Neither $$M$$ nor $$Z$$ has a boundary and neither is assumed to be compact. I believe that the inclusion $$M \setminus Z \to M$$ is $$d-1$$-connected, in that one can construct an argument based on the argument of Proposition 4.1 of https://people.math.osu.edu/anderson.2804/eilenberg/appA.pdf

The idea is that to show $$i_*: \pi_n(M\setminus Z, m_0) \to \pi_n(M, m_0)$$ is surjective when $$n \le d-1$$, you can take any class in the target and find a representative $$f: S^n \to M$$ that is smooth and meets $$Z$$ transversely. Dimension counting then shows that $$f(S^n) \cap Z = \emptyset$$, so the class is in the image of $$i_*$$.

To show it is injective when $$n < d-1$$, you argue in a similar way, but this time with smooth maps $$F: D^{n+1} \to M$$ that restrict to $$f: S^n \to M \setminus Z$$ on the boundary, where $$f$$ represents a class in the kernel of $$i_*$$. Again, you can choose a $$D^{n+1}$$ that is smooth and transverse to $$Z$$, and again conclude by counting dimensions.

Does anyone know of a reference for this fact in the (published) literature?

• This is Theorem 1.1.4 in the notes ivv5hpp.uni-muenster.de/u/jeber_02/skripten/bordism-skript.pdf . It seems hard to track down a published reference. Many authors would just say By a standard transversality argument...". – Mark Grant Apr 1 at 12:16
• Thanks! I'm glad it appears somewhere online. The treatment of transversality in Guillemin & Polalck's Differential Topology (ams.org/books/chel/370) is sufficiently strong to prove the result without too much extra work, since it handles manifolds with boundary. Other treatments I found restrict to manifolds without boundary for transversality, so aren't quite as convenient. – Ben Williams Apr 5 at 23:07