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Derived Differential Geometry Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations Derived Differential Geometry Lecture 9 of 14: Differential Geometry of derived manifolds and orbifolds Dominic Joyce, Oxford University Summer 2015 These slides, and references, etc., available at http://people.maths.ox.ac.uk/∼joyce/DDG2015 1 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations Plan of talk: 9 Differential Geometry of derived manifolds and orbifolds 9.1 Orbifold groups, tangent and obstruction spaces 9.2 Immersions, embeddings and d-submanifolds 9.3 Embedding derived manifolds into manifolds 9.4 Submersions 9.5 Orientations 2 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations 9. Differential Geometry of derived manifolds and orbifolds Here are some important topics in ordinary differential geometry: Immersions, embeddings, and submanifolds. The Whitney Embedding Theorem. Submersions. Orientations. Transverse fibre products. Manifolds with boundary and corners. (Oriented) bordism groups. Fundamental classes of compact oriented manifolds in homology. The next four lectures will explain how all these extend to derived manifolds and orbifolds. 3 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations 9.1. Orbifold groups, tangent and obstruction spaces for d-orbifolds and Kuranishi spaces In x5.4 and x6.3 we explained that a d-manifold / M-Kuranishi space X has functorial tangent spaces Tx X and obstruction spaces Ox X for x 2 X. We now discuss the orbifold versions of these, which are not quite functorial. Let X be a d-orbifold or Kuranishi space, and x 2 X. Then we can define the orbifold group Gx X, a finite group, and the tangent space Tx X and obstruction space Ox X, both finite-dimensional real representations of Gx X. If (V ; E; Γ; s; ) is a Kuranishi neighbourhood on X with x 2 Im , and v 2 s−1(0) ⊆ V with (vΓ) = x then we may write Gx X = StabΓ(v) = γ 2 Γ: γ · v = v ; Tx X = Ker dsjv : Tv V ! Ejv ; (9.1) Ox X = Coker dsjv : Tv V ! Ejv : 4 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations For d-manifolds, Tx X; Ox X are unique up to canonical isomorphism, so we can treat them as unique. For d-orbifolds (and classical orbifolds), things are more subtle: Gx X; Tx X; Ox X are unique up to isomorphism, but not up to canonical isomorphism. That is, to define Gx X; Tx X; Ox X in (9.1) we had to choose v 2 s−1(0) with (v) = x. 0 0 0 0 If v is an alternative choice yielding Gx X ; Tx X ; Ox X , then v 0 = δ · v for some δ 2 Γ, and we have isomorphisms 0 −1 Gx X −! Gx X ; γ 7−! δγδ ; 0 Tx X −! Tx X ; t 7−! Tv δ(t); 0 Ox X −! Ox X ; o 7−! T^v δ(o); where Tv δ : Tv V ! Tδ·v V , T^v δ : Ejv ! Ejδ·v are induced by the Γ-actions on V ; E. But these isomorphisms are not unique, as we could replace δ by δ for any 2 Gx X. 5 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations Because of this, Gx X is canonical up to conjugation, i.e. up to −1 automorphisms Gx X ! Gx X of the form γ 7! γ for 2 Gx X, and similarly Tx X; Ox X are canonical up to the action of elements of Gx X, i.e. up to automorphisms Tx X ! Tx X mapping t 7! · t. Our solution is to use the Axiom of Choice to choose an allowed triple (Gx X; Tx X; Ox X) for all derived orbifolds X and x 2 X. Similarly, for any 1-morphism f : X ! Y of derived orbifolds and x 2 X with f(x) = y 2 Y, we can define a group morphism Gx f : Gx X ! Gy Y, and Gx f-equivariant linear maps Tx f : Tx X ! Ty Y, Ox f : Ox X ! Oy Y. These Gx f; Tx f; Ty f are only unique up to the action of an element of Gy Y, so again we use the Axiom of Choice. We may not have Gx (g ◦ f) = Gx g ◦ Gx f, etc. If η : f ) g is a 2-morphism of derived orbifolds, there is a canonical element Gx η 2 Gy Y such that −1 Gx g(γ) = (Gx η)(Gx f(γ))(Gx η) ; Tx g(t) = Gx η · Tx f(t); Ox g(t) = Gx η · Ox f(t): 6 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations You can mostly ignore this issue about Gx X; Tx X; Ox X only being unique up to conjugation by an element of Gx X, it is not very important in practice. What is important is that Tx X; Ox X are representations of the orbifold group Gx X, so we can think about them using representation theory. For example, we have natural splittings tr nt tr nt Tx X = (Tx X) ⊕ (Tx X) ; Ox X = (Ox X) ⊕ (Ox X) into trivial (··· )tr and nontrivial (··· )nt subrepresentations. Then it is easy to prove: Lemma 9.1 tr Let X be a derived orbifold and x 2 X; and suppose (Ox X) = 0 nt and (Ox X) is not isomorphic to a Gx X-subrepresentation of nt (Tx X) . Then it is not possible to make a small deformation of X near x so that it becomes a classical orbifold. In contrast, derived manifolds can always be perturbed to manifolds. 7 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations 9.2. Immersions, embeddings and d-submanifolds A smooth map of manifolds f : X ! Y is an immersion if Tx f : Tx X ! Ty Y is injective for all x 2 X . It is an embedding if also f is a homeomorphism with its image f (X ). Definition Let f : X ! Y be a 1-morphism of derived manifolds. We call f a weak immersion, or w-immersion, if Tx f : Tx X ! Tf(x)Y is injective for all x 2 X. We call f an immersion if Tx f : Tx X ! Tf(x)Y is injective and Ox f : Ox X ! Of(x)Y is surjective for all x 2 X. We call f a (w-)embedding if it is a (w-)immersion and f : X ! f (X ) is a homeomorphism. If instead X; Y are derived orbifolds, we also require that Gx f : Gx X ! Gf(x)Y is injective for all x 2 X for (w-)immersions, and Gx f : Gx X ! Gf(x)Y is an isomorphism for (w-)embeddings. 8 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations Proposition 9.2 Let SV 0;f ;f^ : SV ;E;s ! SW ;F ;t be a standard model 1-morphism in dMan. Then SV 0;f ;f^ is a w-immersion (or an immersion) if and only if for all x 2 s−1(0) ⊆ V with f (x) = y 2 t−1(0) ⊆ W ; the following sequence is exact at the second term (or the second and fourth terms, respectively): dsjx ⊕ Tx f f^jx ⊕ −dtjy 0 / Tx V / Ejx ⊕ Ty W / F jy / 0: Proof. This follows from the diagram with exact rows, (5.4) in x5: 0 / Tx SV ;E;s / Tx V / Ejx / Ox SV ;E;s / 0 dsjx Tx SV 0;f ;f^ Tx f f^jx Ox SV 0;f ;f^ dtjy 0 / Ty SW ;F ;t / Ty W / F jy / Oy SW ;F ;t / 0: 9 / 48 Dominic Joyce, Oxford University Lecture 9: Differential geometry of derived manifolds Orbifold groups, tangent and obstruction spaces Immersions, embeddings and d-submanifolds Differential Geometry of derived manifolds and orbifolds Embedding derived manifolds into manifolds Fibre products of derived manifolds and orbifolds Submersions Orientations Local models for (w-)immersions Locally, (w-)immersions are modelled on standard model SV ;f ;f^ : SV ;E;s ! SW ;F ;t with f : V ! W an immersion.
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