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Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$.

In general, there are a variety of obstructions to the existence of a cobordism between two representatives of a homology class. However, I have a strong memory of reading a paper that states that in this particular setting one can say something quite strong. The result I remember reading is that: there exists a sequence $C_0,C_1,\ldots, C_k$ of embedded surfaces in $M$ and submanifolds $N_1,\ldots N_k$ such that:

  • $A = C_0$
  • $B = C_k$
  • $N_i$ is a cobordism from $C_{i-1}$ to $C_i$

Unfortunately, I have been unable to locate this paper or recall the author. I would appreciate it very much if someone recalls the reference (or can tell me my memory is wrong and this is false).

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This “sort of thing” is proven by Kakimizu [1992, 2005] and by others in many later papers. If you want more references then Google “The Kakimizu complex is connected” (or simply-connected, or contractible, or has various fixed point properties).

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