Holomorphic factorization of mappings into $\mathrm{SL}_n(\mathbb{C})$

Björn Ivarsson; Frank Kutzschebauch

doi:https://doi.org/10.4007/annals.2012.175.1.3

Abstract

We solve Gromov’s Vaserstein problem. Namely, we show that a null-homotopic holomorphic mapping from a finite dimensional reduced Stein space into $\mathrm{SL}_n(\mathbb{C})$ can be factored into a finite product of unipotent matrices with holomorphic entries.

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[CohnSGL2R] $Go to document$ P. M. Cohn, "On the structure of the ${ GL}_{2}$ of a ring," Inst. Hautes Études Sci. Publ. Math., vol. 30, pp. 5-53, 1966.

Show bibtex

@article {CohnSGL2R, MRKEY = {0207856}, AUTHOR = {Cohn, P. M.}, TITLE = {On the structure of the {${\rm GL}\sb{2}$} of a ring}, JOURNAL = {Inst. Hautes Études Sci. Publ. Math.}, FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques}, VOLUME = {30}, YEAR = {1966}, PAGES = {5--53}, ISSN = {0073-8301}, MRCLASS = {20.70 (16.48)}, MRNUMBER = {0207856}, MRREVIEWER = {I. Reiner}, DOI={10.1007/BF02684355}, ZBLNUMBER = {0144.26301}, }
[EliashbergESMAS] $Go to document$ Y. Eliashberg and M. Gromov, "Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $3n/2+1$," Ann. of Math., vol. 136, iss. 1, pp. 123-135, 1992.

Show bibtex

@article {EliashbergESMAS, MRKEY = {1173927}, AUTHOR = {Eliashberg, Yakov and Gromov, Mikhael}, TITLE = {Embeddings of {S}tein manifolds of dimension {$n$} into the affine space of dimension {$3n/2+1$}}, JOURNAL = {Ann. of Math.}, FJOURNAL = {Annals of Mathematics. Second Series}, VOLUME = {136}, YEAR = {1992}, NUMBER = {1}, PAGES = {123--135}, ISSN = {0003-486X}, CODEN = {ANMAAH}, MRCLASS = {32H35 (32E10)}, MRNUMBER = {1173927}, MRREVIEWER = {Zong Yuan Yao}, DOI = {10.2307/2946547}, ZBLNUMBER = {0758.32012}, }
[ForsterTMSR] O. Forster, "Topologische Methoden in der Theorie Steinscher Räume," in Actes du Congrès International des Mathématiciens, Tome 2, Paris: Gauthier-Villars, 1971, pp. 613-618.

Show bibtex

@incollection {ForsterTMSR, MRKEY = {0419834}, AUTHOR = {Forster, Otto}, TITLE = {Topologische {M}ethoden in der {T}heorie {S}teinscher {R}äume}, BOOKTITLE = {Actes du {C}ongrès {I}nternational des {M}athématiciens, {T}ome 2}, VENUE={{N}ice, 1970}, PAGES = {613--618}, PUBLISHER = {Gauthier-Villars}, ADDRESS = {Paris}, YEAR = {1971}, MRCLASS = {32E10 (32C15)}, MRNUMBER = {0419834}, MRREVIEWER = {W. Barth}, ZBLNUMBER={0245.32006}, }
[ForsterAME] $Go to document$ O. Forster and K. J. Ramspott, "Analytische Modulgarben und Endromisbündel," Invent. Math., vol. 2, pp. 145-170, 1966.

Show bibtex

@article {ForsterAME, MRKEY = {0218618}, AUTHOR = {Forster, Otto and Ramspott, Karl Josef}, TITLE = {Analytische {M}odulgarben und {E}ndromisbündel}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {2}, YEAR = {1966}, PAGES = {145--170}, ISSN = {0020-9910}, MRCLASS = {32.49}, MRNUMBER = {0218618}, MRREVIEWER = {E. Brieskorn}, DOI = {10.1007/BF01404550}, ZBLNUMBER={0154.33401}, }
[ForsterOPGG] $Go to document$ O. Forster and K. J. Ramspott, "Okasche Paare von Garben nicht-abelscher Gruppen," Invent. Math., vol. 1, pp. 260-286, 1966.

Show bibtex

@article {ForsterOPGG, MRKEY = {0212211}, AUTHOR = {Forster, Otto and Ramspott, Karl Josef}, TITLE = {Okasche {P}aare von {G}arben nicht-abelscher {G}ruppen}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {1}, YEAR = {1966}, PAGES = {260--286}, ISSN = {0020-9910}, MRCLASS = {32.25 (32.50)}, MRNUMBER = {0212211}, MRREVIEWER = {H. R{ö}hrl}, DOI = {10.1007/BF01452245}, }
[ForsterHISA] $Go to document$ O. Forster and K. J. Ramspott, "Homotopieklassen von Idealbasen in Steinschen Algebren," Invent. Math., vol. 5, pp. 255-276, 1968.

Show bibtex

@article {ForsterHISA, MRKEY = {0232965}, AUTHOR = {Forster, Otto and Ramspott, Karl Josef}, TITLE = {Homotopieklassen von {I}dealbasen in {S}teinschen {A}lgebren}, JOURNAL = {Invent. Math.}, FJOURNAL = {Inventiones Mathematicae}, VOLUME = {5}, YEAR = {1968}, PAGES = {255--276}, ISSN = {0020-9910}, MRCLASS = {32.49}, MRNUMBER = {0232965}, MRREVIEWER = {F. Gherardelli}, DOI = {10.1007/BF01389776}, ZBLNUMBER = {0157.13302}, }
[ForsterAEPSM] $Go to document$ O. Forster and K. J. Ramspott, "Über die Anzahl der Erzeugenden von projektiven Steinschen Moduln," Arch. Math. $($Basel$)$, vol. 19, pp. 417-422, 1968.

Show bibtex

@article {ForsterAEPSM, MRKEY = {0236959}, AUTHOR = {Forster, Otto and Ramspott, Karl Josef}, TITLE = {Über die {A}nzahl der {E}rzeugenden von projektiven {S}teinschen {M}oduln}, JOURNAL = {Arch. Math. $($Basel$)$}, FJOURNAL = {Archiv der Mathematik}, VOLUME = {19}, YEAR = {1968}, PAGES = {417--422}, ISSN = {0003-889X}, MRCLASS = {57.60 (32.00)}, MRNUMBER = {0236959}, MRREVIEWER = {O. F. G. Schilling}, DOI = {10.1007/BF01898424}, ZBLNUMBER = {0162.38502}, }
[ForstnericOPSSFB] F. Forstnerivc, "The Oka principle for sections of stratified fiber bundles," Pure Appl. Math. Q., vol. 6, iss. 3, Special Issue: In honor of Joseph J. Kohn. Part 1, pp. 843-874, 2010.

Show bibtex

@article {ForstnericOPSSFB, MRKEY = {2677316}, AUTHOR = {Forstneri{\v{c}}, Franc}, TITLE = {The {O}ka principle for sections of stratified fiber bundles}, JOURNAL = {Pure Appl. Math. Q.}, FJOURNAL = {Pure and Applied Mathematics Quarterly}, VOLUME = {6}, YEAR = {2010}, NUMBER = {3, Special Issue: In honor of Joseph J. Kohn. Part 1}, PAGES = {843--874}, ISSN = {1558-8599}, MRCLASS = {32E30 (32E10 32Q55)}, MRNUMBER = {2677316}, MRREVIEWER = {Erlend Forn{æ}ss Wold}, ZBLNUMBER = {1216.32005}, }
[ForstnericEHSCS] $Go to document$ F. Forstnerivc and J. Prezelj, "Extending holomorphic sections from complex subvarieties," Math. Z., vol. 236, iss. 1, pp. 43-68, 2001.

Show bibtex

@article {ForstnericEHSCS, MRKEY = {1812449}, AUTHOR = {Forstneri{\v{c}}, Franc and Prezelj, Jasna}, TITLE = {Extending holomorphic sections from complex subvarieties}, JOURNAL = {Math. Z.}, FJOURNAL = {Mathematische Zeitschrift}, VOLUME = {236}, YEAR = {2001}, NUMBER = {1}, PAGES = {43--68}, ISSN = {0025-5874}, CODEN = {MAZEAX}, MRCLASS = {32D15 (32E10)}, MRNUMBER = {1812449}, MRREVIEWER = {Bert Fischer}, DOI = {10.1007/PL00004826}, ZBLNUMBER = {0968.32005}, }
[ForstnericOPHSWS] $Go to document$ F. Forstnerivc and J. Prezelj, "Oka’s principle for holomorphic submersions with sprays," Math. Ann., vol. 322, iss. 4, pp. 633-666, 2002.

Show bibtex

@article {ForstnericOPHSWS, MRKEY = {1905108}, AUTHOR = {Forstneri{\v{c}}, Franc and Prezelj, Jasna}, TITLE = {Oka's principle for holomorphic submersions with sprays}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {322}, YEAR = {2002}, NUMBER = {4}, PAGES = {633--666}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {32Q28 (32L05 32Q55)}, MRNUMBER = {1905108}, MRREVIEWER = {Finnur L{á}russon}, DOI = {10.1007/s002080100249}, ZBLNUMBER = {1011.32006}, }
[GrauertAHFWKR] $Go to document$ H. Grauert, "Approximationssätze für holomorphe Funktionen mit Werten in komplexen Räumen," Math. Ann., vol. 133, pp. 139-159, 1957.

Show bibtex

@article {GrauertAHFWKR, MRKEY = {0098197}, AUTHOR = {Grauert, Hans}, TITLE = {Approximationssätze für holomorphe {F}unktionen mit {W}erten in komplexen {R}äumen}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {133}, YEAR = {1957}, PAGES = {139--159}, ISSN = {0025-5831}, MRCLASS = {32.00}, MRNUMBER = {0098197}, MRREVIEWER = {H. J. Bremermann}, DOI = {10.1007/BF01343296}, ZBLNUMBER={0080.29201}, }
[GrauertHFWLG] $Go to document$ H. Grauert, "Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen," Math. Ann., vol. 133, pp. 450-472, 1957.

Show bibtex

@article {GrauertHFWLG, MRKEY = {0098198}, AUTHOR = {Grauert, Hans}, TITLE = {Holomorphe {F}unktionen mit {W}erten in komplexen {L}ieschen {G}ruppen}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {133}, YEAR = {1957}, PAGES = {450--472}, ISSN = {0025-5831}, MRCLASS = {32.00}, MRNUMBER = {0098198}, MRREVIEWER = {H. J. Bremermann}, DOI = {10.1007/BF01343758}, ZBLNUMBER = {0080.29202}, }
[GrauertAFHR] $Go to document$ H. Grauert, "Analytische Faserungen über holomorph-vollständigen Räumen," Math. Ann., vol. 135, pp. 263-273, 1958.

Show bibtex

@article {GrauertAFHR, MRKEY = {0098199}, AUTHOR = {Grauert, Hans}, TITLE = {Analytische {F}aserungen über holomorph-vollständigen {R}äumen}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {135}, YEAR = {1958}, PAGES = {263--273}, ISSN = {0025-5831}, MRCLASS = {32.00}, MRNUMBER = {0098199}, MRREVIEWER = {H. Cartan}, DOI = {10.1007/BF01351803}, ZBLNUMBER={0081.07401}, }
[GromovOPHSEB] $Go to document$ M. Gromov, "Oka’s principle for holomorphic sections of elliptic bundles," J. Amer. Math. Soc., vol. 2, iss. 4, pp. 851-897, 1989.

Show bibtex

@article {GromovOPHSEB, MRKEY = {1001851}, AUTHOR = {Gromov, M.}, TITLE = {Oka's principle for holomorphic sections of elliptic bundles}, JOURNAL = {J. Amer. Math. Soc.}, FJOURNAL = {Journal of the American Mathematical Society}, VOLUME = {2}, YEAR = {1989}, NUMBER = {4}, PAGES = {851--897}, ISSN = {0894-0347}, MRCLASS = {32E10 (32E15 32L05 58G30)}, MRNUMBER = {1001851}, MRREVIEWER = {David L. Johnson}, DOI = {10.2307/1990897}, ZBLNUMBER = {0686.32012}, }
[GrunewaldGSL2] $Go to document$ F. Grunewald, J. Mennicke, and L. Vaserstein, "On the groups ${ SL}_2({\bf Z}[x])$ and ${ SL}_2(k[x,y])$," Israel J. Math., vol. 86, iss. 1-3, pp. 157-193, 1994.

Show bibtex

@article {GrunewaldGSL2, MRKEY = {1276133}, AUTHOR = {Grunewald, Fritz and Mennicke, Jens and Vaserstein, Leonid}, TITLE = {On the groups {${\rm SL}\sb 2({\bf Z}[x])$} and {${\rm SL}\sb 2(k[x,y])$}}, JOURNAL = {Israel J. Math.}, FJOURNAL = {Israel Journal of Mathematics}, VOLUME = {86}, YEAR = {1994}, NUMBER = {1-3}, PAGES = {157--193}, ISSN = {0021-2172}, CODEN = {ISJMAP}, MRCLASS = {20G30 (20H05)}, MRNUMBER = {1276133}, MRREVIEWER = {Zong Zhu Lin}, DOI = {10.1007/BF02773676}, ZBLNUMBER = {0805.20042}, }
[HW] W. Hurewicz and H. Wallman, Dimension Theory, Princeton, NJ: Princeton Univ. Press, 1941, vol. 4.

Show bibtex

@book {HW, MRKEY = {0006493}, AUTHOR = {Hurewicz, Witold and Wallman, Henry}, TITLE = {Dimension {T}heory}, SERIES = {Princeton Math. Series}, VOLUME={4}, PUBLISHER = {Princeton Univ. Press}, ADDRESS = {Princeton, NJ}, YEAR = {1941}, PAGES = {vii+165}, MRCLASS = {56.0X}, MRNUMBER = {0006493}, MRREVIEWER = {H. Whitney}, ZBLNUMBER = {0060.39808}, }
[IK] $Go to document$ B. Ivarsson and F. Kutzschebauch, "A solution of Gromov’s Vaserstein problem," C. R. Math. Acad. Sci. Paris, vol. 346, iss. 23-24, pp. 1239-1243, 2008.

Show bibtex

@article {IK, MRKEY = {2473300}, AUTHOR = {Ivarsson, Bj{ö}rn and Kutzschebauch, Frank}, TITLE = {A solution of {G}romov's {V}aserstein problem}, JOURNAL = {C. R. Math. Acad. Sci. Paris}, FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris}, VOLUME = {346}, YEAR = {2008}, NUMBER = {23-24}, PAGES = {1239--1243}, ISSN = {1631-073X}, MRCLASS = {32E10 (15A23 15A54)}, MRNUMBER = {2473300}, MRREVIEWER = {Sergey Ivashkovich}, DOI = {10.1016/j.crma.2008.10.017}, ZBLNUMBER = {1160.32017}, }
[IvarssonKutzschebauchProcAMS] $Go to document$ B. Ivarsson and F. Kutzschebauch, On the number of factors in the unipotent factorization of holomorphic mappings into ${{SL}}_2({\mathbb{C}})$.

Show bibtex

@misc{IvarssonKutzschebauchProcAMS, author = {Ivarsson, Bj{ö}rn and Kutzschebauch, Frank}, TITLE = {On the number of factors in the unipotent factorization of holomorphic mappings into {${\rm{SL}}_2({\mathbb{C}})$}}, NOTE={to appear in \textit{{P}roc. {A}mer. {M}ath. {S}oc.}}, URL={http://dx.doi.org/10.1090/S0002-9939-2011-11025-6}, SORTYEAR={2011}, }
[RosenbergAKTA] J. Rosenberg, Algebraic $K$-Theory and its Applications, New York: Springer-Verlag, 1994, vol. 147.

Show bibtex

@book {RosenbergAKTA, MRKEY = {1282290}, AUTHOR = {Rosenberg, Jonathan}, TITLE = {Algebraic {$K$}-Theory and its Applications}, SERIES = {Graduate Texts in Math.}, VOLUME = {147}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1994}, PAGES = {x+392}, ISBN = {0-387-94248-3}, MRCLASS = {19-01 (19-02)}, MRNUMBER = {1282290}, MRREVIEWER = {Dominique Arlettaz}, ZBLNUMBER = {0801.19001}, }
[SchurmannESSMD] $Go to document$ J. Schürmann, "Embeddings of Stein spaces into affine spaces of minimal dimension," Math. Ann., vol. 307, iss. 3, pp. 381-399, 1997.

Show bibtex

@article {SchurmannESSMD, MRKEY = {1437045}, AUTHOR = {Sch{ü}rmann, J.}, TITLE = {Embeddings of {S}tein spaces into affine spaces of minimal dimension}, JOURNAL = {Math. Ann.}, FJOURNAL = {Mathematische Annalen}, VOLUME = {307}, YEAR = {1997}, NUMBER = {3}, PAGES = {381--399}, ISSN = {0025-5831}, CODEN = {MAANA}, MRCLASS = {32E10}, MRNUMBER = {1437045}, MRREVIEWER = {Finnur L{á}russon}, DOI = {10.1007/s002080050040}, ZBLNUMBER={0881.32007}, }
[SuslinSSLGRP] A. A. Suslin, "The structure of the special linear group over rings of polynomials," Izv. Akad. Nauk SSSR Ser. Mat., vol. 41, iss. 2, pp. 235-252, 477, 1977.

Show bibtex

@article {SuslinSSLGRP, MRKEY = {0472792}, AUTHOR = {Suslin, A. A.}, TITLE = {The structure of the special linear group over rings of polynomials}, JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.}, FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya}, VOLUME = {41}, YEAR = {1977}, NUMBER = {2}, PAGES = {235--252, 477}, ISSN = {0373-2436}, MRCLASS = {13C10 (14F05)}, MRNUMBER = {0472792}, MRREVIEWER = {Chr. U. Jensen}, ZBLNUMBER={0378.13002}, }
[ThurstonVasersteinK1ES] $Go to document$ W. Thurston and L. Vaserstein, "On $K_1$-theory of the Euclidean space," Topology Appl., vol. 23, iss. 2, pp. 145-148, 1986.

Show bibtex

@article {ThurstonVasersteinK1ES, MRKEY = {0855453}, AUTHOR = {Thurston, W. and Vaserstein, L.}, TITLE = {On {$K\sb 1$}-theory of the {E}uclidean space}, JOURNAL = {Topology Appl.}, FJOURNAL = {Topology and its Applications}, VOLUME = {23}, YEAR = {1986}, NUMBER = {2}, PAGES = {145--148}, ISSN = {0166-8641}, CODEN = {TIAPD9}, MRCLASS = {18F25 (19B14 57N12)}, MRNUMBER = {0855453}, MRREVIEWER = {Bruce A. Magurn}, DOI = {10.1016/0166-8641(86)90035-0}, ZBLNUMBER = {0611.18007}, }
[vanderKallenSL3BWL] $Go to document$ W. van der Kallen, "${ SL}_{3}({\bf C}[X])$ does not have bounded word length," in Algebraic $K$-theory, Part I, New York: Springer-Verlag, 1982, vol. 966, pp. 357-361.

Show bibtex

@incollection {vanderKallenSL3BWL, MRKEY = {0689383}, AUTHOR = {van der Kallen, Wilberd}, TITLE = {{${\rm SL}\sb{3}({\bf C}[X])$} does not have bounded word length}, BOOKTITLE = {Algebraic {$K$}-theory, {P}art {I}}, VENUE={{O}berwolfach, 1980}, SERIES = {Lecture Notes in Math.}, VOLUME = {966}, PAGES = {357--361}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1982}, MRCLASS = {20F05 (12A65 12F20 20G35)}, MRNUMBER = {0689383}, ZBLNUMBER = {0935.20501}, DOI={10.1007/BF02684355}, }
[VasersteinRMDPDFAO] $Go to document$ L. N. Vaserstein, "Reduction of a matrix depending on parameters to a diagonal form by addition operations," Proc. Amer. Math. Soc., vol. 103, iss. 3, pp. 741-746, 1988.

Show bibtex

@article {VasersteinRMDPDFAO, MRKEY = {0947649}, AUTHOR = {Vaserstein, L. N.}, TITLE = {Reduction of a matrix depending on parameters to a diagonal form by addition operations}, JOURNAL = {Proc. Amer. Math. Soc.}, FJOURNAL = {Proceedings of the American Mathematical Society}, VOLUME = {103}, YEAR = {1988}, NUMBER = {3}, PAGES = {741--746}, ISSN = {0002-9939}, CODEN = {PAMYAR}, MRCLASS = {18F25 (19B10)}, MRNUMBER = {0947649}, MRREVIEWER = {Charles Weibel}, DOI = {10.2307/2046844}, ZBLNUMBER = {0657.55005}, }
[WrightAFPS] $Go to document$ D. Wright, "The amalgamated free product structure of ${ GL}_{2}(k[X_{1},\ldots,X_{n}])$ and the weak Jacobian theorem for two variables," J. Pure Appl. Algebra, vol. 12, iss. 3, pp. 235-251, 1978.

Show bibtex

@article {WrightAFPS, MRKEY = {0501951}, AUTHOR = {Wright, David}, TITLE = {The amalgamated free product structure of {${\rm GL}\sb{2}(k[X\sb{1}, \ldots,X\sb{n}])$} and the weak {J}acobian theorem for two variables}, JOURNAL = {J. Pure Appl. Algebra}, FJOURNAL = {Journal of Pure and Applied Algebra}, VOLUME = {12}, YEAR = {1978}, NUMBER = {3}, PAGES = {235--251}, ISSN = {0022-4049}, MRCLASS = {20G15 (13C10 14K30)}, MRNUMBER = {0501951}, MRREVIEWER = {Chr. U. Jensen}, DOI = {10.1016/0022-4049(87)90004-1}, ZBLNUMBER = {0387.20039}, }

Holomorphic factorization of mappings into $\mathrm{SL}_n(\mathbb{C})$

Abstract

Authors