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Differential Topology (Definition)

The study of properties of smooth manifolds that are preserved by diffeomorphisms.
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The statement of the theorem

Let MM be a topological space. MM is a \text{C}^\\infty manifold of dimension nn if there exists an atlas \text{A} = \{\text{(U}_i, \phi_i\)}_{i \in I} such that Ui\text{U}_i are open subsets of MM, \phi_i: \text{U}_i \to \text{\{(x_1, \dots, x_n)\}}_{\mathbb{R}^n} are homeomorphisms, and for all i,j∈Ii, j \in I, the transition map ψij=ϕj∘ϕi−1:ϕi(Ui∩Uj)→ϕj(Ui∩Uj)\psi_{ij} = \phi_j \circ \phi_i^{-1}: \phi_i(\text{U}_i \cap \text{U}_j) \to \phi_j(\text{U}_i \cap \text{U}_j) is a smooth map (i.e., \text{C}^\\infty).\n\nDifferential Topology is the study of properties of such manifolds MM that are invariant under the action of the group of diffeomorphisms Diff(M)\text{Diff}(M).\n\nFormally, let Diff(M)\text{Diff}(M) be the group of all ψ:M→M\psi: M \to M such that ψ\psi and ψ−1\psi^{-1} are smooth maps. A property P\mathcal{P} of a manifold MM is a differential topological invariant if, for any diffeomorphism ψ∈Diff(M)\psi \in \text{Diff}(M), the property P\mathcal{P} holds for MM if and only if it holds for ψ(M)\psi(M) (which is MM itself, but the invariance is key). \n\nSpecifically, the theory investigates structures defined by local coordinate charts and transition maps, such as: \n\nInv(M)={P∣P is a property of M such that ∀ψ∈Diff(M),P(M)  ⟺  P(ψ(M))}\text{Inv}(M) = \{\mathcal{P} \mid \mathcal{P} \text{ is a property of } M \text{ such that } \forall \psi \in \text{Diff}(M), \mathcal{P}(M) \iff \mathcal{P}(\psi(M))\}