I have heard stated the following

**Theorem.** If $\Sigma$ is a (orientable) surface, then $\mathrm b_1(\Sigma)$ counts the maximum number of "circular cuts" (embedded circles $C_1,\ldots,C_m$) that you can make on $\Sigma$ without disconnecting it (i.e. with $\Sigma\smallsetminus (C_1\cup\ldots\cup C_m)$ still being connected).

Is there a similar interpretation also for higher Betti numbers $\mathrm b_k(M)$ of manifolds in general?

(I've been asked this by a non-mathematician, but I think it fits MO. If the topologists here think it's too trivial or well known, please move it to MSE).

**Edit1:** It wouldn't be bad to know if the above "theorem" is *actually* a theorem, and a reference to a rigorous proof.

**Edit2:** In the above, by a "generalization" I mean a statement of the form: $b=b_k(M)$ counts the maximum number of embedded orientable submanifolds $N_1,\ldots,N_b$ such that $M\setminus (N_1\cup\ldots\cup N_b)$ has clear topological property $\boldsymbol{\mathrm{P}}$. Maybe property $\boldsymbol{\mathrm{P}}$ could be about some $(k-1)$-connectedness condition? For the $b_1$ case, this would involve $0$-connectedness, i.e. just connectedness, and this is the case for $\dim M=2$ if the theorem I quoted above is true.

On the Cut Number of a 3-manifoldby S.L.Harvey arxiv.org/abs/math/0112193 which may (or may not) be useful for an answer. $\endgroup$ – Qfwfq May 3 '17 at 20:25codimensional phenomenon. $\endgroup$ – Kevin Casto May 3 '17 at 22:57