PlanetPhysics/Anabelian Geometry and Algebraic Topology

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This is a new topic in which the [[../IsomorphismClass/|Anabelian Geometry]] approach will be defined and compared with other appoaches that are disticnt from it such as

[[../ModuleAlgebraic/|non-Abelian algebraic topology]] and [[../NoncommutativeGeometry4/|noncommutative geometry]]{ http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}. The latter two [[../CosmologicalConstant2/|fields]] have already made an impact on [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]] that seek a new setting beyond SUSY--the Standard Model of modern physics. Moreover, it is also possible to consider in this topic novel, possible approaches to relativity theories, especially to [[../SR/|general relativity]] on [[../SR/|spacetimes]] that are more general than pseudo- or quasi- Riemannian `spaces'. Furthermore, other [[../PhysicalMathematics2/|theoretical physics]] developments may expand specific Anabelian Geometry applications to [[../TriangulationMethodsForQuantizedSpacetimes2/|quantum geometry]] and [[../TriangulationMethodsForQuantizedSpacetimes2/|Quantum Algebraic Topology]].

Anabelian Geometry

The area of mathematics called Anabelian Geometry (AAG) began with Alexander Grothendieck's introduction of the term in his seminal and influential [[../Work/|work]] "Esquisse d'un Programme" $[1]$ produced in 1980. The basic setting of his anabelian geometry is that of the [[../CoIntersections/|algebraic]] [[../SingularComplexOfASpace/|fundamental group]] $𝒢$ of an [[../CoIntersections/|algebraic]] variety $X$ (which is a basic [[../PreciseIdea/|concept]] in Algebraic Geometry), and also possibly a more generally defined, but related, geometric [[../TrivialGroupoid/|object]]. The algebraic fundamental group , $𝒢$, in this case determines how the [[../IsomorphismClass/|algebraic variety]] $X$ can be mapped into, or linked to, another geometric [[../TrivialGroupoid/|object]] $Y$, assuming that $𝒢$ is [[../AbelianCategory3/|non-Abelian]] or [[../AbelianCategory3/|noncommutative]]. This specific approach differs significantly, of course, from that of Noncommutative Geometry introduced by Alain Connes. It also differs from the main-stream [[../NAQAT2/|nonabelian algebraic topology]] (NAAT)'s generalized approach to topology in terms of [[../GroupoidHomomorphism2/|groupoids]] and [[../CubicalHigherHomotopyGroupoid/|fundamental groupoids]] of a [[../CoIntersections/|topological]] space (that generalize the [[../PreciseIdea/|concept]] of fundamental space), as well as from that of [[../InfinityGroupoid/|higher dimensional algebra]] ([[../InfinityGroupoid/|HDA]]). Thus, the fundamental [[../IsomorphismClass/|anabelian]] question posed by Grothendieck was, and is: "how much information about the isomorphism class of the variety $X$ is contained in the knowledge of the etale fundamental group?" (on p. 2 in $http://www.math.jussieu.fr/leila/SchnepsLM.pdf$ ).

At this point, stepping down from the general, abstract setting of the Anabelian Geometry it would be useful to consider a specific, concrete example.

A Concrete Example

In the case of curves, $C$ , these could be either affine (as in [[../AlbertEinstein/|Einstein's]] or Weyl's approaches to General Relativity), or projective , as in a variety $V$. Consider here a specific hyperbolic curve $H$ , that is defined as the complement of $n$ points in a projective algebriac curve of genus $g$ , which is assumed to be both smooth and irreducible, and also defined over a [[../CosmologicalConstant/|field]] $K$ (that is finitely generated over its prime field ) such that: $2-2g-n<0$. Grothendieck conjectured in 1979 that the [[../SingularComplexOfASpace/|fundamental group]] $𝒢$ of $C$ , which is a profinite [[../TrivialGroupoid/|group]], determines the curve $C$ itself, or that the \htmladdnormallink{isomorphism {http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} class of $𝒢$ determines the [[../IsomorphismClass/|isomorphism class]] of $C$ ; this also points towards a conjecture regarding the [[../IsomorphismClass/|natural equivalence]] $\eta$ of their corresponding [[../Cod/|categories]].

Generalizations

Much more elaborate, generalizations of Grothendieck's Anabelian Geometry are posible by considering higher-dimensional, $pro-l$, $Hom$-- versions, and so on, involving for example [[../CubicalHigherHomotopyGroupoid/|fundamental groupoids]] and fundamental [[../ThinEquivalence/|double groupoids]] $[2]$.

References

1. [[../AlexanderGrothendieck/|Alexander Grothendieck]], 1984. "Esquisse d'un Programme", (1984 manuscript), published in "Geometric Galois Actions", L. Schneps, P. Lochak, eds., London Math. Soc. Lecture Notes 242, Cambridge University Press, 1997, pp. 5--48; English transl., ibid., pp. 243--283

2. Jochen Koenigsmann.2001. Anabelian geometry over almost arbitrary fields $http://www.uni-math.gwdg.de/tschinkel/SS04/koenigsmann2.pdf$

3. S. Mochizuki, H. Nakamura, A. Tamagawa. The Grothendieck Conjecture on the fundamental groups of algebraic curves, Sugaku Expositions {\mathbf 14}(1), (2001), 31--53.

{\mathbf[[../Work/|...work]] in progress}