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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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 symplectic topology
first published
24. May 2002

revised upload
 Look at the concept of a metric in elementary topology: For a set  MMa metric is a real-valu­ed map
MM × MM –––>     ,
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
MM × MM × MM  –––>  ,
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  MMin this line of argu­men­ta­tion is gi­ven by a vo­lu­me‌-‌map
MM × MM × MM × MM –––>  ,
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 the area-concept should by given by symplectic vector spaces ( not ne­ces­sari­ly ma­ni­folds ),  i.‌e. pairs of a set  MMand a skew bi­li­near form
MM × MM –––>  ,
the area being con­structed in terms of the symplec­tic form. This topologises  MMen­tire­ly 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>–––> GG –––>> SS
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
SS × SS  –––>  FF,
usually called σ, which in the case of sym­plectic vector spaces is the sym­plec­tic bi­linear form. In this case the resulting group  GGis the Heisen­berg group of quantum mecha­nics and the se­cond ex­amp­le of an area map co­in­cides with the first one.
  FF,GG,SS
are groups
2nd
Cohomology
It would be nice to show that for all short exact sequences the third group  SScarries an area map, con­struc­ted in terms of sigma and delta. 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. 
  
 Literature with comments

[...]    any book on general and metric topology.
scroll up:start / topmetric spacesfinger tip topolshort exact sequeLie theory2nd Cohomology

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