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Topological Problems in Voting

Back in college, I developed an interest in unexpected impossibility proofs applied to real world systems. The fact that certain abstract mathematical structures inherently have limitations which profoundly impact actual applications is both captivating and sobering. Here’s a cute instance of topological properties applied to voting systems a friend shared with me.

For our second path, let’s take \(A = \{\alpha\} \cup S^1\) and its symmetric counterpart \(B = S^1 \cup \{\alpha\}\) for some \(\alpha\in S^1\) which form an orthogonal figure-eight, with \(A,B\) representing circles rotating directly on the outer and inner loop respectively. So if we take a small \(\epsilon\) sized smooth deformation at \(D\cap (A\cup B)\) to shift away from the intersection at \((\alpha,\alpha)\) to form a non self-intersecting loop \(L\), then \(\deg \phi \vert_L = 0 \ \text{mod}\ 2\) at all of its regular points as well. Here, we had to rely on smoothness for our path homotopy invariants to apply, but some heavy machinery homological approaches also extend this result to continuous settings.

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