Let $K$ be a finite $d$-dimensional simplicial complex embedded in $\mathbb{R}^{d+1}$. The setting of this question is simplicial homology with coefficients over $\mathbb{Z}\_2$. By Alexander duality $K$ partitions $\mathbb{R}^{d+1}$ into $\beta\_d + 1$ connected components, where $\beta\_d$ is the $d$th Betti number of $K$. We define $G$ to be the dual graph of $K$. $G$ is the graph whose vertices are the connected components of $\mathbb{R}^{d+1}$ and whose edges are the $d$-dimensional simplices of $K$. Two vertices are adjacent if and only if their corresponding connected components share a common $d$-simplex in the intersection of their boundaries.


It is easy to see that there is a one-to-one correspondence between $d$-dimensional cycles in $K$ and minimal edge cuts in $G$. This is analogous to cycle/cut duality in planar graphs.


What does a cycle in $G$ correspond to in $K$? My intuition is that it should be something analogous to a cut in a graph. In terms of homology a graph cut is a set of edges whose removal increases the rank of the zeroth homology group of the graph (because this homology group counts the number of connected components.)


Is the dual of a cycle in $G$ a set of $d$-dimensional simplices in $K$ whose removal increases the rank of the $(d-1)$th homology group? This seems likely to me, but I do not know how to prove it.



Planar duality generalized to embedded simplicial complexes
Like you guessed, the dual of a cycle in $G$ is a set of $d$-simplices in $K$ whose removal increases the rank of the $(d-1)$-th homology group.


In the following I use that $H\_i(X;\mathbb{Z}\_2) \cong H^i(X;\mathbb{Z}\_2)$. It might have been better to keep track of the distinction...


Consider the closures of the connected components of $K$'s complement in $\mathbb{R}^{d+1}$, $\{X\_1,\ldots,X\_n\}$, where $n=\beta\_d(K)+1$. 
These are $(d+1)$-dimensional simplicial complexes,and their boundaries are $d$-dimensional. 




---


**Edit:** It is of course not true that the connected components of the complement are simplicial complexes, although if the embedding is a PL embedding in $S^{d+1}$ instead then they can be triangulated. I mistakenly answered this more combinatorial variant of the question. 


Nevertheless the method of proof works if we think of the components of $\mathbb{R}^{d+1}\setminus K$ as being topological subspaces of $\mathbb{R}^{d+1}$, with the following addition: note that no more than two connected components can contain a single $d$-dimensional simplex of $K$ in their common boundary. A reference is Daverman and Venema, "Embeddings in Manifolds," corollary 7.1.2 and the preceding proposition (the section is "Codimension-one separation properties," accessible from Google Books).




---


Let us consider the boundaries of each pair of distinct $X\_i,X\_j$ as though they were disjoint: we want to think of them as different simplicial complexes. 


Let $C$ be a cycle in $G$. We want to know what happens when the $d$-simplices $``C\cap K"$ is removed from $K$. Since $K$ is the Alexander dual of $\bigsqcup\_{i=1}^n X\_i,$ we can determine this by looking at what happens to the $\{X\_i\}$ instead. Removing a $d$-simplex from $K$ is equivalent to gluing some pair along it, say $X\_1,X\_2$, so we can use the Mayer-Vietoris sequence for reduced homology:
$$ \ldots\rightarrow H\_k(X\_1 \cap X\_2) \rightarrow H\_k(X\_1)\oplus H\_k(X\_2) \rightarrow H\_k(X\_1 \cup X\_2) \rightarrow H\_{k-1}(X\_1\cap X\_2) \rightarrow \ldots $$


Here, the intersection $X\_1 \cap X\_2$ is the intersection after the gluing. Hence it is a $(d-1)$-dimensional simplex on the boundary, and its reduced homology is zero in all dimensions. Thus the first homology of $X\_1 \cup X\_2$ is just the direct sum $H\_1(X\_1)\oplus H\_1(X\_2)$.


Now we can remove, one by one, the edges (or $d$-simplices) in a simple cycle of $G$. Let us remove the last pair of edges at once; this means we are gluing the connected component $X\_t$ dual to the last vertex in the cycle by both of its edges simultaneously. The other piece glued to $X\_t$ is the union of components corresponding to the other vertices of the cycle, let's call it $Y$. The reduced Mayer-Vietoris sequence in low degrees gives us
$$ H\_1(X\_t \cap Y) \rightarrow H\_1(X\_t)\oplus H\_1(Y) \rightarrow H\_1(X\_t \cup Y) \rightarrow \tilde{H}\_0(X\_t \cap Y) \overset{j}{\rightarrow} \tilde{H}\_0(X\_t) \oplus \tilde{H}\_0(Y).$$
Here the intersection $X\_t \cap Y$ is a disjoint union of two simplices, which has $H\_1 = 0$ but $\tilde{H}\_0 = \mathbb{Z}\_2$. However, the map $j$ in the sequence above is $0$. So the rank of $H\_1(X\_t \cup Y)$ is exactly
$$\mathrm{rk}(H\_1(X\_t)\oplus H\_1(Y)) + \mathrm{rk}(\tilde{H}\_0(X\_t \cap Y)) = \mathrm{rk}(H\_1(X\_t))+\mathrm{rk}(H\_1(Y))+1. $$


By induction on the length of the cycle, and using that the first homology of a space with several connected components is the direct sum of their first homologies, we see that the result of cutting out the dual of a cycle of $G$ from $K$ increases the first homology of $\mathbb{R}^{d+1}\setminus K$ by $1$. Alexander duality (this time applied to the complement) gives the result on $H\_{d-1}(K)$.


