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The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
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Novikov-Wall non-additivity theorem with twisted coefficients
Let $Y$ be a compact manifold and let $\pi_1(Y) \to \mathbb{Z}^n= \langle t_1,\ldots,t_n\rangle$ be a homomorphism. Extend it to the group rings $\mathbb{Z}[\pi_1(Y)] \to \mathbb{Z}[ t_1,\ldots,t_n]$ …