In General
$ Def: A topological space in which every point has an open neighborhood homeomorphic to an open n-ball.
* Properties: It is finite-dimensional, locally contractible, satisfies the Alexander-Lefschetz duality relationships.
> Types of manifolds: see cell complex; differential geometry; differentiable manifolds; types of manifolds [combinatorial, PL, topological, etc].
> Online resources: see the Manifold Atlas Project site.

Algebraic Characterization
* Idea: The structure of a manifold can be recovered from the C*-algebra generated by appropriate functions (abstractly, in several ways, e.g., as ideals or as propagators corresponding to point sources).
$ Gel'fand-Naimark theorem: A C*-algebra \({\cal A}\) with identity is isomorphic to the C*-algebra of continuous bounded functions on a compact Hausdorff space, the spectrum of \({\cal A}\); The spectrum can be constructed directly, as the set of maximal ideals, or *-homomorphisms \({\cal A}\) → \({\mathbb C}\).
$ Gel'fand-Kolmogorov theorem: (1939) A compact Hausdorff topological space X can be canonically embedded into the infinite-dimensional vector space C(X)*, the dual space of the algebra of continuous functions C(X) as an "algebraic variety" specified by an infinite system of quadratic equations.
@ General references: Fell & Doran 88; Khudaverdian & Voronov AIP(07)-a0709 [generalization of Gel'fand-Kolmogorov]; Izzo AM(11).
@ Related topics: Connes a0810 [spectral triples]; > s.a. non-commutative geometry.

Additional Structures of Manifolds > s.a. fiber bundles.
* Possibilities: Different levels of structure are manifold, triangulable manifold, PL manifold, differentiable manifold; For a while, the first two or three of these structures were conjectured to be equivalent, but now this has been shown to be false.
* Multiplication structure: A map *: M × MM.
* Comultiplication: A mapping F: C(M) → C(M) ⊗ C(M) on the algebra of functions on M.
* Relationship: If M has a multiplication, then it gets a (diagonal) comultiplication Δ defined by f \(\mapsto\) Δ(f): Δf(x,y) = F(x * y).
@ References: Kankaanrinta T&A(05) [G-manifolds and Riemannian metrics].

Constructions and Operations on Manifolds > s.a. category; embeddings; foliations; tensor [product].
$ Direct product: Given two manifolds Xn and Y p, their direct product is Zn+p = Xn × Y p as a set, with the product topology and the product charts: UZ = UX × UY, φZ(x,y) = (φX(x), φY(y)).
$ Submanifold: N is a submanifold of M if it is a topological subspace of M and the inclusion map is an embedding (if it is an immersion we have an immersed submanifold).
$ Integral submanifold: A submanifold NM such that for all p in N, f*(TpN) = Sp, with f : NM the embedding map.
$ Connected sum: In sloppy notation, X # Y:= (X \ Dn) ∪ (Y \ Dn), where n is the dimension of X and Y; It is associative and commutative, and has Sn as identity; Examples: X # \(\mathbb R\)n = X \ {p}; > s.a. laplacian; 3D manifolds.
@ Submanifolds: Carter JGP(92) [outer curvature]; Giachetta et al mp/06 [Lagrangian and Hamiltonian dynamics].

Superspace, Supermanifolds > s.a. complex structures; Gegenbauer, Hermite and Jack Polynomials; geometric quantization.
* Idea: A manifold with the bundle-like addition of a vector space of Grassmann numbers at each point.
* Q-manifolds: A Q-manifold is a supermanifold equipped with an odd vector field satisfying {Q, Q} = 0.
@ General references: Rabin & Crane CMP(86); Bagger pr(87); Nelson IJMPA(88); Bandyopadhyay & Ghosh IJMPA(89); De Bie & Sommen AP(07)a0707 [Clifford analysis approach]; Santi a0905 [homogeneous supermanifolds]; Hübsch a0906 [size and algebro-geometric structure]; Cattaneo & Schätz RVMP(11)-a1011-ln; Voronov a1409 [extending the category]; Castellani et al NPB(15)-a1503, NPB(15)-a1507 [geometry, Hodge dual].
@ Texts, reviews: Hermann 77; Chow IJTP(78); DeWitt 92; Howe & Hartwell CQG(95); Howe & Rogers 01; Rogers 07; Sardanashvily a0910-ln; Fioresi & Lledó 15.
@ Quantum superspace: Brink & Schwarz PLB(81).
@ With metric structure: Sardanashvily IJGMP(08) [supermetrics]; Asorey & Lavrov JMP(09) [symplectic and metric structures]; Dumitrescu et al JHEP(12)-a1205 [curved]; > s.a. killing tensors.
@ Integration: Gates ht/97-conf; Cartier et al mp/02-in; De Bie & Sommen JPA(07)-a0705 [and spherical harmonics]; > s.a. grassmann structures.
@ Fedosov supermanifolds: Geyer & Lavrov IJMPA(04)ht/03 [symplectic]; Lavrov & Radchenko TMP(06); Asorey et al a0809-conf [and Riemannian]; Monterde et al JGP(09).
@ Q-manifolds: Schwarz LMP(00)ht [and gauge theory]; > s.a. characteristic classes.
@ Physics: Kochan JGP(04)m.DG/03 [supergeometry, electromagnetism and gravity]; Cirilo-Lombardo EPJC(12)-a1205 [geometrical properties]; Hack et al CMP-a1501 [and locally covariant quantum field theory]; Nicolis a1606-conf [and the partition function, fluctuations of a non-relativistic particle]; > s.a. Double Field Theory; FLRW spacetimes; scalar fields [on supersphere].
@ Related topics: Bruzzo & Pestov JGP(99); Philbin ht/03-wd [topology]; Constantinescu JPA(05)ht/04, MPLA(05)ht/04 [inner product, Hilbert-Krein structure]; Desrosiers et al m.CO/04 [symmetric functions]; Bruce ArchM-a1401 [curve and jet of a curve]; Randall a1412-th [closed forms in superspace]; > s.a. brownian motion; spherical functions.
> Other generalizations: see differential geometry [including fuzzy]; differentiable manifolds [exotic structures]; types of manifolds [including pseudomanifolds]; Wilson Loop.

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