Area preserving pl homeomorphisms and relations in K 2
Greenberg, Peter
Annales de l'Institut Fourier, Tome 48 (1998), p. 133-148 / Harvested from Numdam
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À tout homéomorphisme linéaire par morceaux à support compact du plan qui préserve l’aire est associée une relation dans le K 2 du corps de définition.

À l’aide d’une formule de J. Morita, on montre comment calculer la relation dans des cas simples. En appplication, une formule de réciprocité pour des paires de triangles dans le plan est démontrée, et des éléments de torsion sont construits dans le K 2 de certains corps de fonctions.

To any compactly supported, area preserving, piecewise linear homeomorphism of the plane is associated a relation in K 2 of the smallest field whose elements are needed to write the homeomorphism.

Using a formula of J. Morita, we show how to calculate the relation, in some simple cases. As applications, a “reciprocity” formula for a pair of triangles in the plane, and some explicit elements of torsion in K 2 of certain function fields are found.

@article{AIF_1998__48_1_133_0,
     author = {Greenberg, Peter},
     title = {Area preserving pl homeomorphisms and relations in $K\_2$},
     journal = {Annales de l'Institut Fourier},
     volume = {48},
     year = {1998},
     pages = {133-148},
     doi = {10.5802/aif.1613},
     mrnumber = {99d:19001},
     zbl = {0904.19001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1998__48_1_133_0}
}

Greenberg, Peter. Area preserving pl homeomorphisms and relations in $K_2$. Annales de l'Institut Fourier, Tome 48 (1998) pp. 133-148. doi : 10.5802/aif.1613. http://gdmltest.u-ga.fr/item/AIF_1998__48_1_133_0/

[BGSV] A. Beilinson, A. Goncharov, V. Schechtman, A. Varchenko, Aomoto dilogarithms, mixed Hodge structures, and motivic cohomology of pairs of triangles in the plane, The Grothendieck Festschrift, Birkäuser, Boston, I (1990), 135-172. | MR 92h:19007 | Zbl 0737.14003

[Du] J. Dupont, The dilogarithm as a characteristic class for flat bundles, J. Pure and Applied Alg., 44 (1987), 137-164. | MR 88k:57032 | Zbl 0624.57024

[G] A. Goncharov, Geometry of configurations, polygarithms and motivic cohomology, Adv. in Math., 114 (1995), 197-318. | MR 96g:19005 | Zbl 0863.19004

[G1] P. Greenberg, Pseudogroups of C1, piecewise projective homeomorphisms, Pacific J. Math., 129 (1987), 67-75. | Zbl 0592.58055

[G2] P. Greenberg, Piecewise SL2Z geometry, Trans. AMS, 335, 2 (1993), 705-720. | MR 93d:52015 | Zbl 0768.52009

[GS] P. Greenberg, V. Sergiescu, C1 piecewise projective homeomorphisms and a noncommutative Steinberg extension, J. K-Theory, 9 (1995), 529-544. | MR 97c:57037 | Zbl 0969.57027

[Li] S. Lictenbaum, Groups related to scissors congruence groups, Contemp. Math., 83 (1989), 151-157. | MR 90e:20030 | Zbl 0674.55012

[Mil] J. Milnor, Introduction to Algebraic K-Theory Annals of Math. Studies 72, Princeton Univ. Press, N.J., 1971. | Zbl 0237.18005

[Mor] J. Morita, K2SL2 of Euclidean domains, generalized Dennis-Stein symbols and a certain three-unit formula, J. Pure and Applied Alg., 79 (1992), 51-61. | Zbl 0759.19003

[PS] W. Parry and C.H. Sah, Third homology of SL2R made discrete, J. Pure and Applied Alg., 30 (1983), 181-209. | MR 85f:20043 | Zbl 0527.18006