Manifolds |

**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* × *M* → *M*.

* __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*): Δ*f*(*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 *X*^{n} and
*Y*^{ p}, their direct product
is *Z*^{n+p}
= *X*^{n} × *Y*^{ p} as
a set, with the product topology and the product
charts: *U*_{Z}
= *U*_{X} × *U*_{Y},
*φ*_{Z}(*x*,*y*) =
(*φ*_{X}(*x*),
*φ*_{Y}(*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 *N* ⊂ *M* such that for all *p* in *N*,
*f*_{*}(T_{p}*N*)
= *S*_{p}, with *f* : *N* → *M* the embedding map.

$ __Connected sum__: In sloppy
notation, *X* # *Y*:= (*X* \
D^{n}) ∪ (*Y* \
D^{n}), where *n* is the dimension of *X* and *Y*;
It is associative and commutative, and has S^{n} 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

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send feedback and suggestions to bombelli at olemiss.edu – modified 4 jun 2020