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I looked on the site but I did not find anything matching my problem.

I am working in 4D with cubical complexes, also known as Khalimsky grids; I mean that I work with "cubical" manifolds.

Do you know if it is possible that, starting from two topological 4-manifolds A and B, the union of A and B is a 4-manifold too, even if the intersection of A and B is the union of two 3D cubes which share a vertex, that is, even if the shared space between the two manifolds is not a 3-manifold ?

I heard about "connected sums" of manifolds which preserve manifoldness, however the intersection of my two manifolds here is not a manifold itself.

Thank you in advance guys, I would really appreciate your help.

Nicolas Boutry
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  • You have to be a bit more specific : Are you assuming that your manifolds are compact? Do you allow manifolds to have boundary? – Moishe Kohan Mar 22 '17 at 00:22
  • Effectively, the two 4-manifolds A and B are made of a finite number of 4D unitary cubes, and then they are compact 4-manifolds with a compact boundary. – Nicolas Boutry Mar 22 '17 at 21:17
  • The answer is that $A\cup_C B$ is a manifold (possibly with boundary) (in your setting) if and only if $C$ is a submanifold (possibly with boundary) of both $\partial A$ and of $\partial B$, or if $C$ equals a connected component of either $A$ or of $B$. In particular, $C$ has to be a manifold (possibly with boundary). The proof of necessity is based on some local homology computations; I do not sure how comfortable you are with homology groups. – Moishe Kohan Mar 23 '17 at 03:14
  • Thank you for your answer, It helps me a lot ! Do you have any reference or any book with proves this theorem ? And can you tell me more about this proof, I'm not very confortable with homology groups but I'm learning homology with the book of Computational homology of Kaczynski et al. :) – Nicolas Boutry Mar 23 '17 at 09:49
  • There is no textbook reference I am aware of; I will write a proof when I have more time. – Moishe Kohan Mar 23 '17 at 12:25
  • Or perhaps could you advise me a reference in matter of homology related to manifolds that you know? I can start to (try to) think about the proof by myself. – Nicolas Boutry Mar 24 '17 at 00:00
  • Since you never worked with homology, I doubt it, but just in case: The homological condition for a space $M$ to be an $n$-dimensional manifold (possibly with boundary) is that for every $x\in M$ (not on the potential boundary) $H_k(M,M-{x})$ is isomorphic to ${\mathbb Z}$ for $k=n$ and zero otherwise. In your case, you would be computing homology using the Mayer-Vietoris long exact sequence (with respect to the decomposition of $M$ as the union of two manifolds with boundary $M_1, M_2$). – Moishe Kohan Mar 24 '17 at 02:44
  • Thank you so much Moishe :) – Nicolas Boutry Mar 24 '17 at 16:09

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