no. 1
Area preserving pl homeomorphisms and relations in K 2
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 133-148.

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.

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     author = {Greenberg, Peter},
     title = {Area preserving pl homeomorphisms and relations in $K_2$},
     journal = {Annales de l'Institut Fourier},
     pages = {133--148},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {1},
     year = {1998},
     doi = {10.5802/aif.1613},
     mrnumber = {99d:19001},
     zbl = {0904.19001},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1613/}
}
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Greenberg, Peter. Area preserving pl homeomorphisms and relations in $K_2$. Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 133-148. doi : 10.5802/aif.1613. http://www.numdam.org/articles/10.5802/aif.1613/

[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 | Zbl

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

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

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

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

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

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

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

[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

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

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