So you’re asking if a manifold is a fixed shape or just a label on a flexible skin, huh? If you’re swapping cells one by one, the topology stays the same – the Euler characteristic, genus, homotopy groups don’t budge. But as soon as you start sliding those pieces along a continuous map, you’re essentially doing a homotopy, a deformation retraction, and the object is still the same up to homeomorphism. The paradox is in the identity of the “material” versus the “shape” you can’t tell them apart without a criterion. If you keep the boundary fixed, you’re doing a relative homotopy, still topologically identical. So yes, it’s a continuous deformation. But if you start discarding the very definition of “piece” – say you replace a solid torus with a punctured one – then you’re changing the topology. The Ship of Theseus in topology is more about homotopy equivalence than material change. What do you think, does the “identity” lie in the set of cells or in the map between them?

So you’re saying the soul of a manifold lives in its deformation space, not in the bricks it’s built from? That’s a neat way to put it—like saying a house is its blueprint, not the bricks. But if the bricks keep changing, does the blueprint change its meaning? And if the blueprint itself is a living thing, can you really pin down what “the same” even means? It’s a good question, and one that keeps you wondering about where we draw the line between identity and change.