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Finger Tip Topologies
Σ
Hans Tilgner

 
Abstract

 We sugguest to develop a se­ries of finger tip to­po­lo­gies, ge­ne­ra­li­sing the con­cept of met­ric, to area, vo­lume, ... — Examp­les are ta­ken from short exact se­quen­ces in some ca­te­go­ries of al­ge­bra­ic struc­tures like groups, al­ge­bras, ... and from sym­plec­tic vec­tor spa­ces. — If it is pos­sib­le to find axi­oms for are­as, like those for me­trics, it should be pos­sib­le to find axioms for vo­lu­me such that the­re are vec­tor spa­ces with vo­lu­me. — We ex­pect examp­les from de­ter­min­ants and from pseu­do‌-‌or­tho­go­nal vec­tor spa­ces, where the top­most one-di­men­sio­nal vec­tor spa­ce in the ex­te­rior and Clif­ford-al­ge­bra has such a vo­lume-struc­ture. The spe­cial li­near groups are their auto­mor­phism groups and the trace­less li­ne­ar Lie al­ge­bras are their der­iva­tion Lie al­ge­bras.
 

Incepta Physica Mathematica
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first published
24. May 2002

revised upload
 Look at the concept of a metric in elementary topology: For a set  𝕄  a metric is a real-valu­ed map
𝕄 × 𝕄  –→  ℝ   ,
fulfilling the three axioms reflexivity, sym­met­ry and the triangle un­equality. But, there is mis­sing some­thing, some­thing has been over­looked by the mathematician's com­munity: Stop here and remem­ber, how a metric leads to a topology on  𝕄, using systems of neigh­bor­hoods. Still no idea? ...Rethink... there is a con­cept in which metric is only the first of a se­ries of to­po­lo­gi­cal con­cepts, the best notion of which would be finger tip to­po­lo­gies. Still no idea? ....

are the
real numbers
Metric-
Area-
Volume-

Topologies
Well – an area is a map
𝕄 × 𝕄 × 𝕄  –→  ℝ   ,
fulfilling the four (?) axioms: Skew symmetry against permutation of adjacent entries, Ja­co­bi iden­ti­ty, the te­tra­eder in­equality and one (?) more axiom.  To de­fine a neigh­bor­hood one needs to con­struct the concept of a point of gravity, around which one can whirl a triangle of given area. The con­se­cu­tive  finger tip to­po­lo­gy  on  𝕄  in this line of argu­men­ta­tion is gi­ven by a vo­lu­me‌-‌map
𝕄 × 𝕄 × 𝕄 × 𝕄  –→  ℝ   ,
subject to five (?) axioms. Thus the category of topolo­gical spaces contains also area-, vo­lu­me‌-, ... spa­ces be­sides the metric ones.
 
Short
Exact
Sequences


Lie Groups
Lie Algebras
Examples of this area-concept should be given by symplectic manifolds, especially symplectic vector spa­ces, i.‌e. pairs of a set  𝕄  and a skew map resp. bi­li­near form
𝕄 × 𝕄  –→  ℝ   ,
the area being con­structed in terms of the symplec­tic form. This topologises  𝕄  en­tire­ly in terms of the sym­plec­tic form and the area.
Other examples should come from the 2nd coho­mo­lo­gy of groups, i.e. short exact se­quen­ces
𝔽  >—→  𝔾  —→𝕊   
of groups, where there exists a pair of maps Σ and Δ, ful­filling two axioms. There are two well‌-‌known spe­cial cases, the retract case and the split case, i.e. in group‌-‌theo­re­ti­cal lan­guage, al­most‌- and se­mi‌-‌di­rect pro­ducts ( which when both are ful­fil­led result in a di­rect pro­duct). In the split case there is a skew map
𝕊 × 𝕊  –→  𝔽    ,
usually called σ, which in the case of sym­plectic vector spaces is the sym­plec­tic bi­linear form. In this case the resulting group  𝔾  is the Heisen­berg group of quantum mecha­nics and the se­cond ex­amp­le of an area map co­in­cides with the first one.
𝔽 𝔾 𝕊
are groups

and even

symmetric spaces

if the latter have a

2nd cohomology
and
short exact sequences
2nd
Cohomology
It would be nice to show that for all short exact sequences the third group  𝕊  carries an area map, con­struc­ted in terms of Σ and Δ . In case the group is a Lie group, there is a 2‌nd co­ho­mology of its Lie al­ge­bra, and there should be an induced area map. 
  
 seeany book on general and metric topology.
scroll up:start / topmetric spacesfinger tip topolshort exact sequeLie theory2nd Cohomology
 
Literature with commentswith [ links to publications] and [ conference books]
[Bou]N Bourbaki  Algebra I  Hermann, Paris [1970] ISBN 2 7056 5675 8  remains the standard in­for­ma­tion on universality in al­ge­bra and chap. III §6 in that on sym­me­tric algebras.
[Col]G P Collins  Die Lösung eines Jahrhundertproblems  Spektrum der Wissenschaft Sep [2004] p 86-94  translated from the  Sci­en­ti­fic Ame­ri­can  on re­sults of Pe­rel­man for 3-di­mensional spaces.
[Fmk]A F Fomenko  Differential Geometry and Topology  Consultants Bureau, NY [1987]  translated from Russian with more re­fe­ren­ces.
[God]C Godbillon  Géometrie Différentielle et Mécanique Analytique  Hermann, Paris [1969]  de­fi­nes Pois­son brackets on p 125.
[How]R E Howe  On the Role of the Heisenberg Group in Harmonic Analy­sis  Bull. Am. Math. Soc 3 no 2 [1981] p 821-843
[How]R E Howe  Remarks on Classical Invariant Theory  Trans. Am. Math. Soc 313 no 2 [1989] p 539-570
[Kch]M Koecher  The Minnesota Notes on Jordan Algebras and Their Applications  Springer Lecture Notes in Mathematics 1710 [1999] His con­cept of mutations IV 2 still has to be included here.
[Kow]O Kowalski  Partial Curvature Structures and Conformal Transformations  J. Differential Geometry 8 [1973] p 53-70
[Kul]R S Kulkarni  Curvature and Metric  Annals of Mathematics 91 [1970] p 311-331   follows, together with Katsumi Nomizu and Oldrich Kowalski, the cur­va­ture axioms Singer & Thorpe's.
[K70]R S Kulkarni  Curvature Structures and Conformal Transformations  J. Differential Geometry 4 [1970] p 425-451
[Lic]A Lichnérowics  Ondes et Radiations Électromagnétique et Gravitationelles en Relativité Générale  Annali di Mate­ma­ti­ca Pu­ra & Appl. 4 [1960] p 1-95  is a very elegant representation of the concept, although written in the old‌-­fashioned in­dex no­ta­tion. We rate this work as  t h e  major contribution to gravitational radiation end even to general relativity after Einatein !
[Loo]O Loos  Symmetric Spaces I + II  Benjamin N.Y. [1969] no ISBN  We only use the first volume. We generalize his symme­tric mul­ti­pli­ca­tion from hyperboloids (spheres) to arbitrary elements outside null-cones, this being well-known to the Artin school of ma­the­ma­tics, and this in turn to the complex case of pseudo-Hilbert spaces.
 One of the main points, following from Loos' definition of a symmetric space, is that instead of studying  ℂℂ*-algebras -in quan­tum me­cha­nics - one should study algebras with an in­volu­tion ( in­stead of the *-anti-involution).
[L&N]O Loos, E Neher  Reflection Systems and Partial Root Systems  Forum Math 23 [2011] p 349-411  see also
O Loos, E Neher  Locally Finite Root Sys­tems   Mem Am Math Soc 171 n°811 [2004] p 1-214
[Nom]K Nomizu  The Decomposition of Generalized Curvature Tensor Fields  p 335-345 in  Differential Geometry, Papers in Ho­nor of Ken­ta­ro Yano  Kinukuniya, Tokyo [1972] no ISBN
[Poi]H Poincaré  Sur la Dynamique de l'Électron  Comptes Rendus de l'Académie de France  [1905] p 1504-1508  first launched the con­cept of  gravitational waves  in the framework of special relativity only, however.
[Pos]E J Post  Formal Structure of Electromagnetics  North Holland, Amsterdam [1962] no ISBN is an inspiring book, though writ­ten in an old‌-‌fashio­ned way with indices: The electromagnetic constituency 4-tensor between the 2-tensors (4×4 matrices) of field-strengths and -in­ten­si­ties has the same symmetries as a curvature tensor.
[Seg]I Segal  A variant of Special Relativity  Pergamon, N.Y. [1969] no ISBN  did first have this idea on a  aariant  between special and general relativity. Together with Loos' concept of  loacally symmetric spaces  this should lead to a quantization of general re­la­tivity.
[ST]M Singer, J A Thorpe  The Curvature of 4-Dimensional Einstein Spaces  in  Global Analysis, Papers in Ho­nor of K Kodaira  Prin­ce­ton University Press, Princeton N.J. [1968] no ISBN  is the basic work to formulate  general relativity  in a mo­dern ma­the­ma­ti­cal lan­gua­ge.
[Sou]J-M Souriau  Structure des Systèmes Dynamiques  Dunod, Paris [1970] no ISBN  remains the classic book uniting theo­re­ti­cal phy­sics and mo­dern ma­the­ma­tics. There is an eng­lish trans­la­tion:
[Sou]J-M Souriau  Structure of Dynamical Systems  Birkhäuser, Basel [1997] ISBN 0 8176 3695 1  de­fines the basis-free Pois­son bracket on p 88.
[T78]H Tilgner  The Group Structure of Pseudo-Riemannian Curvature Spaces  J.Math.Phys. 19 [1978] p 1118-1125  shows how ele­gant­ly in- resp. co­variance groups can be chased around Singer & Thorpe's curvature diagrams.
[T84]H Tilgner  Conformal Orbits of Electromagnetic Rie­man­nian Cur­va­ture Ten­sors – Elec­tro­mag­ne­tic Implies Gra­vi­ta­tio­nal Ra­dia­tion  p 317-339 in Springer Lecture Notes in Mathematics 1156 [1984]  ISBN 0 387 15994 0  has more details and re­fe­ren­ces on the ma­the­ma­ti­cal description of electromagnetic and gravitational radiation.

 

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