Stable homology of automorphism groups of free groups

Abstract

Homology of the group $\operatorname{Aut}(F_n)$ of automorphisms of a free group on $n$ generators is known to be independent of $n$ in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular we confirm the conjecture that stable rational homology of $\operatorname{Aut}(F_n)$ vanishes.

  • [MR0436137] Go to document J. C. Becker and D. H. Gottlieb, "Transfer maps for fibrations and duality," Compositio Math., vol. 33, iss. 2, pp. 107-133, 1976.
    @article {MR0436137,
      author = {Becker, J. C. and Gottlieb, D. H.},
      TITLE = {Transfer maps for fibrations and duality},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {33},
      YEAR = {1976},
      NUMBER = {2},
      PAGES = {107--133},
      ISSN = {0010-437X},
      MRCLASS = {55E10 (55F05)},
      MRNUMBER = {0436137},
      MRREVIEWER = {S. Gitler},
      URL = {http://www.numdam.org/item?id=CM_1976__33_2_107_0},
      ZBLNUMBER = {0354.55009},
      }
  • [MR1621939] Go to document J. C. Becker and R. E. Schultz, "Axioms for bundle transfers and traces," Math. Z., vol. 227, iss. 4, pp. 583-605, 1998.
    @article {MR1621939,
      author = {Becker, James C. and Schultz, Reinhard E.},
      TITLE = {Axioms for bundle transfers and traces},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {227},
      YEAR = {1998},
      NUMBER = {4},
      PAGES = {583--605},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {55R12 (55M20)},
      MRNUMBER = {1621939},
      MRREVIEWER = {Donald W. Kahn},
      DOI = {10.1007/PL00004392},
      ZBLNUMBER = {0907.55015},
      }
  • [MR0365573] Go to document A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, New York: Springer-Verlag, 1972, vol. 304.
    @book {MR0365573,
      author = {Bousfield, A. K. and Kan, D. M.},
      TITLE = {Homotopy Limits, Completions and Localizations},
      SERIES = {Lecture Notes in Math.},
      VOLUME={304},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1972},
      PAGES = {287--324},
      MRCLASS = {55J05},
      MRNUMBER = {0365573},
      MRREVIEWER = {Harold Hastings},
      ZBLNUMBER = {0259.55004},
      DOI = {10.1007/978-3-540-38117-4_11},
      }
  • [MR0155330] Go to document A. Dold, "Partitions of unity in the theory of fibrations," Ann. of Math., vol. 78, pp. 223-255, 1963.
    @article {MR0155330,
      author = {Dold, Albrecht},
      TITLE = {Partitions of unity in the theory of fibrations},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {78},
      YEAR = {1963},
      PAGES = {223--255},
      ISSN = {0003-486X},
      MRCLASS = {55.50},
      MRNUMBER = {0155330},
      MRREVIEWER = {E. Fadell},
      DOI = {10.2307/1970341},
      ZBLNUMBER = {0203.25402},
      }
  • [MR0353298] Go to document G. Segal, "Categories and cohomology theories," Topology, vol. 13, pp. 293-312, 1974.
    @article {MR0353298,
      author = {Segal, Graeme},
      TITLE = {Categories and cohomology theories},
      JOURNAL = {Topology},
      VOLUME = {13},
      YEAR = {1974},
      PAGES = {293--312},
      MRCLASS = {55B20},
      MRNUMBER = {0353298},
      MRREVIEWER = {J. P. May},
      DOI = {10.1016/0040-9383(74)90022-6},
      ZBLNUMBER = {0284.55016},
      }
  • [MR0402733] Go to document D. McDuff and G. Segal, "Homology fibrations and the “group-completion” theorem," Invent. Math., vol. 31, iss. 3, pp. 279-284, 1975/76.
    @article {MR0402733,
      author = {McDuff, D. and Segal, G.},
      TITLE = {Homology fibrations and the ``group-completion'' theorem},
      JOURNAL = {Invent. Math.},
      VOLUME = {31},
      YEAR = {1975/76},
      NUMBER = {3},
      PAGES = {279--284},
      MRCLASS = {55D45},
      MRNUMBER = {0402733},
      MRREVIEWER = {Harold Hastings},
      ZBLNUMBER={0306.55020},
      DOI = {10.1007/BF01403148},
      }
  • [MR1678155] Go to document A. Hatcher and K. Vogtmann, "Cerf theory for graphs," J. London Math. Soc., vol. 58, iss. 3, pp. 633-655, 1998.
    @article {MR1678155,
      author = {Hatcher, Allen and Vogtmann, Karen},
      TITLE = {Cerf theory for graphs},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {58},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {633--655},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {20E05 (20F28 57M07)},
      MRNUMBER = {1678155},
      MRREVIEWER = {Michel Coornaert},
      DOI = {10.1112/S0024610798006644},
      ZBLNUMBER = {0922.57001},
      }
  • [MR0112134] Go to document M. Nakaoka, "Decomposition theorem for homology groups of symmetric groups," Ann. of Math., vol. 71, pp. 16-42, 1960.
    @article {MR0112134,
      author = {Nakaoka, Minoru},
      TITLE = {Decomposition theorem for homology groups of symmetric groups},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {71},
      YEAR = {1960},
      PAGES = {16--42},
      ISSN = {0003-486X},
      MRCLASS = {55.00 (18.00)},
      MRNUMBER = {0112134},
      MRREVIEWER = {A. Dold},
      DOI = {10.2307/1969878},
      ZBLNUMBER = {0090.39002},
      }
  • [MR0131874] Go to document M. Nakaoka, "Homology of the infinite symmetric group," Ann. of Math., vol. 73, pp. 229-257, 1961.
    @article {MR0131874,
      author = {Nakaoka, Minoru},
      TITLE = {Homology of the infinite symmetric group},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {73},
      YEAR = {1961},
      PAGES = {229--257},
      ISSN = {0003-486X},
      MRCLASS = {55.30},
      MRNUMBER = {0131874},
      MRREVIEWER = {A. Dold},
      DOI = {10.2307/1970333},
      ZBLNUMBER = {0099.25301},
      }
  • [MR0097062] Go to document A. Dold and R. Thom, "Quasifaserungen und unendliche symmetrische Produkte," Ann. of Math., vol. 67, pp. 239-281, 1958.
    @article {MR0097062,
      author = {Dold, Albrecht and Thom, Ren{é}},
      TITLE = {Quasifaserungen und unendliche symmetrische {P}rodukte},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {67},
      YEAR = {1958},
      PAGES = {239--281},
      ISSN = {0003-486X},
      MRCLASS = {55.00},
      MRNUMBER = {0097062},
      MRREVIEWER = {Sze-tsen Hu},
      DOI = {10.2307/1970005},
      ZBLNUMBER = {0091.37102},
      }
  • [MR0649409] J. A. Berrick, An Approach to Algebraic $K$-Theory, Boston, MA: Pitman (Advanced Publishing Program), 1982, vol. 56.
    @book {MR0649409,
      author = {Berrick, A. Jon},
      TITLE = {An Approach to Algebraic {$K$}-Theory},
      SERIES = {Res. Notes Math.},
      VOLUME= {56},
      PUBLISHER = {Pitman (Advanced Publishing Program)},
      ADDRESS = {Boston, MA},
      YEAR = {1982},
      PAGES = {iii+108},
      ISBN = {0-273-08529-8},
      MRCLASS = {18F25 (55N15)},
      MRNUMBER = {0649409},
      MRREVIEWER = {K. Vogtmann},
      ZBLNUMBER = {0479.18006},
      }
  • [MR2058353] A. Beilinson and V. Drinfeld, Chiral Algebras, Providence, RI: Amer. Math. Soc., 2004, vol. 51.
    @book {MR2058353,
      author = {Beilinson, Alexander and Drinfeld, Vladimir},
      TITLE = {Chiral Algebras},
      SERIES = {Amer. Math. Soc. Colloq. Publ.},
      VOLUME = {51},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2004},
      PAGES = {vi+375},
      ISBN = {0-8218-3528-9},
      MRCLASS = {17Bxx (14F43)},
      MRNUMBER = {2058353},
      MRREVIEWER = {Francisco J. Plaza Mart{\'ı}n},
      ZBLNUMBER = {1138.17300},
      }
  • [MR0794488] Go to document D. Curtis and N. To Nhu, "Hyperspaces of finite subsets which are homeomorphic to $\aleph_0$-dimensional linear metric spaces," Topology Appl., vol. 19, iss. 3, pp. 251-260, 1985.
    @article {MR0794488,
      author = {Curtis, Doug and To Nhu, Nguyen},
      TITLE = {Hyperspaces of finite subsets which are homeomorphic to {$\aleph\sb 0$}-dimensional linear metric spaces},
      JOURNAL = {Topology Appl.},
      FJOURNAL = {Topology and its Applications},
      VOLUME = {19},
      YEAR = {1985},
      NUMBER = {3},
      PAGES = {251--260},
      ISSN = {0166-8641},
      CODEN = {TIAPD9},
      MRCLASS = {54B20 (54F65)},
      MRNUMBER = {0794488},
      MRREVIEWER = {T. Ma{ć}kowiak},
      DOI = {10.1016/0166-8641(85)90005-7},
      ZBLNUMBER = {0587.54015},
      }
  • [MR2220689] Go to document A. Hatcher, K. Vogtmann, and N. Wahl, "Erratum to: Homology stability for outer automorphism groups of free groups," Algebr. Geom. Topol., vol. 6, pp. 573-579, 2006.
    @article {MR2220689,
      author = {Hatcher, Allan and Vogtmann, Karen and Wahl, Natalie},
      TITLE = {Erratum to: {H}omology stability for outer automorphism groups of free groups},
      JOURNAL = {Algebr. Geom. Topol.},
      FJOURNAL = {Algebraic \& Geometric Topology},
      VOLUME = {6},
      YEAR = {2006},
      PAGES = {573--579},
      ISSN = {1472-2747},
      MRCLASS = {20F28 (20E05 20F65 20J06 57M07)},
      MRNUMBER = {2220689},
      DOI = {10.2140/agt.2006.6.573},
      ZBLNUMBER = {1093.20501},
      }
  • [MR0510404] Go to document R. W. Thomason, "Homotopy colimits in the category of small categories," Math. Proc. Cambridge Philos. Soc., vol. 85, iss. 1, pp. 91-109, 1979.
    @article {MR0510404,
      author = {Thomason, R. W.},
      TITLE = {Homotopy colimits in the category of small categories},
      JOURNAL = {Math. Proc. Cambridge Philos. Soc.},
      FJOURNAL = {Mathematical Proceedings of the Cambridge Philosophical Society},
      VOLUME = {85},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {91--109},
      ISSN = {0305-0041},
      CODEN = {MPCPCO},
      MRCLASS = {18F25},
      MRNUMBER = {0510404},
      MRREVIEWER = {Daniel R. Grayson},
      DOI = {10.1017/S0305004100055535},
      ZBLNUMBER = {0392.18001},
      }
  • [MR0533892] Go to document G. Segal, "The topology of spaces of rational functions," Acta Math., vol. 143, iss. 1-2, pp. 39-72, 1979.
    @article {MR0533892,
      author = {Segal, Graeme},
      TITLE = {The topology of spaces of rational functions},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {143},
      YEAR = {1979},
      NUMBER = {1-2},
      PAGES = {39--72},
      ISSN = {0001-5962},
      CODEN = {ACMAA8},
      MRCLASS = {55P10 (32C42 81E10)},
      MRNUMBER = {0533892},
      MRREVIEWER = {D. B. Fuks},
      DOI = {10.1007/BF02392088},
      ZBLNUMBER = {0427.55006},
      }
  • [MR0516216] Go to document G. Segal, "Classifying spaces related to foliations," Topology, vol. 17, iss. 4, pp. 367-382, 1978.
    @article {MR0516216,
      author = {Segal, Graeme},
      TITLE = {Classifying spaces related to foliations},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {17},
      YEAR = {1978},
      NUMBER = {4},
      PAGES = {367--382},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {57R30},
      MRNUMBER = {0516216},
      MRREVIEWER = {D. B. Fuks},
      DOI = {10.1016/0040-9383(78)90004-6},
      ZBLNUMBER = {0398.57018},
      }
  • [MR0830040] Go to document M. Culler and K. Vogtmann, "Moduli of graphs and automorphisms of free groups," Invent. Math., vol. 84, iss. 1, pp. 91-119, 1986.
    @article {MR0830040,
      author = {Culler, Marc and Vogtmann, Karen},
      TITLE = {Moduli of graphs and automorphisms of free groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {84},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {91--119},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20F28 (20J05 57S30)},
      MRNUMBER = {0830040},
      MRREVIEWER = {Christopher W. Stark},
      DOI = {10.1007/BF01388734},
      ZBLNUMBER = {0589.20022},
      }
  • [MR0786348] Go to document J. L. Harer, "Stability of the homology of the mapping class groups of orientable surfaces," Ann. of Math., vol. 121, iss. 2, pp. 215-249, 1985.
    @article {MR0786348,
      author = {Harer, John L.},
      TITLE = {Stability of the homology of the mapping class groups of orientable surfaces},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {121},
      YEAR = {1985},
      NUMBER = {2},
      PAGES = {215--249},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {57M99 (20F34)},
      MRNUMBER = {0786348},
      MRREVIEWER = {K. Vogtmann},
      DOI = {10.2307/1971172},
      ZBLNUMBER = {0579.57005},
      }
  • [MR1015128] N. V. Ivanov, "Stabilization of the homology of Teichmüller modular groups," Algebra i Analiz, vol. 1, iss. 3, pp. 110-126, 1989.
    @article {MR1015128,
      author = {Ivanov, N. V.},
      TITLE = {Stabilization of the homology of {T}eichmüller modular groups},
      JOURNAL = {Algebra i Analiz},
      FJOURNAL = {Algebra i Analiz},
      VOLUME = {1},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {110--126},
      ISSN = {0234-0852},
      MRCLASS = {57M99 (20F38 22E40 30F10 32G15 57N05)},
      MRNUMBER = {1015128},
      MRREVIEWER = {Darryl McCullough},
      ZBLNUMBER = {0727.30036},
      }
  • [MR0314940] Go to document M. Barratt and S. Priddy, "On the homology of non-connected monoids and their associated groups," Comment. Math. Helv., vol. 47, pp. 1-14, 1972.
    @article {MR0314940,
      author = {Barratt, Michael and Priddy, Stewart},
      TITLE = {On the homology of non-connected monoids and their associated groups},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {47},
      YEAR = {1972},
      PAGES = {1--14},
      ISSN = {0010-2571},
      MRCLASS = {18H10 (57F05)},
      MRNUMBER = {0314940},
      MRREVIEWER = {J. C. Becker},
      DOI = {10.1007/BF02566785},
      ZBLNUMBER = {0262.55015},
      }
  • [MR1314940] Go to document A. Hatcher, "Homological stability for automorphism groups of free groups," Comment. Math. Helv., vol. 70, iss. 1, pp. 39-62, 1995.
    @article {MR1314940,
      author = {Hatcher, Allen},
      TITLE = {Homological stability for automorphism groups of free groups},
      JOURNAL = {Comment. Math. Helv.},
      FJOURNAL = {Commentarii Mathematici Helvetici},
      VOLUME = {70},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {39--62},
      ISSN = {0010-2571},
      CODEN = {COMHAX},
      MRCLASS = {20E05 (20F28 20J99 57M07)},
      MRNUMBER = {1314940},
      MRREVIEWER = {Gilbert Levitt},
      DOI = {10.1007/BF02565999},
      ZBLNUMBER = {0836.57003},
      }
  • [MR2113904] Go to document A. Hatcher and K. Vogtmann, "Homology stability for outer automorphism groups of free groups," Algebr. Geom. Topol., vol. 4, pp. 1253-1272, 2004.
    @article {MR2113904,
      author = {Hatcher, Allen and Vogtmann, Karen},
      TITLE = {Homology stability for outer automorphism groups of free groups},
      JOURNAL = {Algebr. Geom. Topol.},
      FJOURNAL = {Algebraic \& Geometric Topology},
      VOLUME = {4},
      YEAR = {2004},
      PAGES = {1253--1272},
      ISSN = {1472-2747},
      MRCLASS = {20F28 (20E05 20F65 20J06 57M07)},
      MRNUMBER = {2113904},
      MRREVIEWER = {Mihalis A. Sykiotis},
      DOI = {10.2140/agt.2004.4.1253},
      ZBLNUMBER = {1093.20020},
      }
  • [MR1671188] A. Hatcher and K. Vogtmann, "Rational homology of ${ Aut}(F_n)$," Math. Res. Lett., vol. 5, iss. 6, pp. 759-780, 1998.
    @article {MR1671188,
      author = {Hatcher, Allen and Vogtmann, Karen},
      TITLE = {Rational homology of {${\rm Aut}(F\sb n)$}},
      JOURNAL = {Math. Res. Lett.},
      FJOURNAL = {Mathematical Research Letters},
      VOLUME = {5},
      YEAR = {1998},
      NUMBER = {6},
      PAGES = {759--780},
      ISSN = {1073-2780},
      MRCLASS = {20J05 (20E05 20F28)},
      MRNUMBER = {1671188},
      MRREVIEWER = {S. Andreadakis},
      ZBLNUMBER = {1040.20042},
      }
  • [MR1945530] K. Igusa, Higher Franz-Reidemeister Torsion, Providence, RI: Amer. Math. Soc., 2002, vol. 31.
    @book {MR1945530,
      author = {Igusa, Kiyoshi},
      TITLE = {Higher {F}ranz-{R}eidemeister Torsion},
      SERIES = {AMS/IP Stud. Adv. Math.},
      VOLUME = {31},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2002},
      PAGES = {xxii+370},
      ISBN = {0-8218-3170-4},
      MRCLASS = {19D10 (19F27 55R40 57Q10 57R45)},
      MRNUMBER = {1945530},
      MRREVIEWER = {Nathalie Wahl},
      ZBLNUMBER = {1016.19001},
      }
  • [MR0864505] M. Gromov, Partial Differential Relations, New York: Springer-Verlag, 1986, vol. 9.
    @book {MR0864505,
      author = {Gromov, Mikhael},
      TITLE = {Partial Differential Relations},
      SERIES = {Ergeb. Math. Grenzgeb.},
      VOLUME = {9},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1986},
      PAGES = {x+363},
      ISBN = {3-540-12177-3},
      MRCLASS = {58G99 (35A99 35B99 53C42 58-02)},
      MRNUMBER = {0864505},
      MRREVIEWER = {Hung-Hsi Wu},
      ZBLNUMBER = {0651.53001},
      }
  • [math.AT/0212321] Go to document I. Madsen and M. Weiss, "The stable moduli space of Riemann surfaces: Mumford’s conjecture," Ann. of Math., vol. 165, iss. 3, pp. 843-941, 2007.
    @article {math.AT/0212321,
      author = {Madsen, Ib and Weiss, Michael},
      TITLE = {The stable moduli space of {R}iemann surfaces: {M}umford's conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {165},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {843--941},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {14H10 (14F43 19D06 55P47)},
      MRNUMBER = {2335797},
      MRREVIEWER = {Ulrike Tillmann},
      DOI = {10.4007/annals.2007.165.843},
      ZBLNUMBER = {1156.14021},
      }
  • [math.GT/0601310] Go to document N. Wahl, "Homological stability for the mapping class groups of non-orientable surfaces," Invent. Math., vol. 171, iss. 2, pp. 389-424, 2008.
    @article {math.GT/0601310,
      author = {Wahl, Nathalie},
      TITLE = {Homological stability for the mapping class groups of non-orientable surfaces},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {171},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {389--424},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57M99 (55P47 55R35 55T99 57R50)},
      MRNUMBER = {2367024},
      MRREVIEWER = {J{é}r{ô}me Scherer},
      DOI = {10.1007/s00222-007-0085-7},
      ZBLNUMBER = {1140.55007},
      }
  • [math.AT/0605249] Go to document S. Galatius, I. Madsen, U. Tillmann, and M. Weiss, "The homotopy type of the cobordism category," Acta. Math., vol. 202, iss. 2, pp. 195-239, 2009.
    @article {math.AT/0605249,
      author = {Galatius, Søren and Madsen, Ib and Tillmann, Ulrike and Weiss, Michael},
      TITLE = {The homotopy type of the cobordism category},
      JOURNAL = {Acta. Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {202},
      YEAR = {2009},
      NUMBER = {2},
      PAGES = {195--239},
      ISSN = {0001-5962},
      MRNUMBER = {2506750},
      ZBLNUMBER = {pre05836542},
      DOI = {10.1007/s11511-009-0036-9},
      }
  • [MR1856399] Go to document I. Madsen and U. Tillmann, "The stable mapping class group and $Q(\mathbbm{C} P^\infty_+)$," Invent. Math., vol. 145, iss. 3, pp. 509-544, 2001.
    @article {MR1856399,
      author = {Madsen, Ib and Tillmann, Ulrike},
      TITLE = {The stable mapping class group and {$Q(\mathbbm{C} P^\infty_+)$}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {145},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {509--544},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {55P47 (55R12 57M99 57N05)},
      MRNUMBER = {1856399},
      MRREVIEWER = {Michael S. Weiss},
      DOI = {10.1007/PL00005807},
      ZBLNUMBER = {1050.55007},
      }
  • [MR1823966] D. Handel, "Some homotopy properties of spaces of finite subsets of topological spaces," Houston J. Math., vol. 26, iss. 4, pp. 747-764, 2000.
    @article {MR1823966,
      author = {Handel, David},
      TITLE = {Some homotopy properties of spaces of finite subsets of topological spaces},
      JOURNAL = {Houston J. Math.},
      FJOURNAL = {Houston Journal of Mathematics},
      VOLUME = {26},
      YEAR = {2000},
      NUMBER = {4},
      PAGES = {747--764},
      ISSN = {0362-1588},
      CODEN = {HJMADZ},
      MRCLASS = {55Q52 (54B15 55P65 55R80)},
      MRNUMBER = {1823966},
      MRREVIEWER = {Leonard R. Rubin},
      ZBLNUMBER = {0988.54015},
      }
  • [MR1474157] Go to document U. Tillmann, "On the homotopy of the stable mapping class group," Invent. Math., vol. 130, iss. 2, pp. 257-275, 1997.
    @article {MR1474157,
      author = {Tillmann, Ulrike},
      TITLE = {On the homotopy of the stable mapping class group},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {130},
      YEAR = {1997},
      NUMBER = {2},
      PAGES = {257--275},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {57M99 (19L20 55P47 55Q50 55S12 55U99)},
      MRNUMBER = {1474157},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.1007/s002220050184},
      ZBLNUMBER = {0891.55019},
      }
  • [MR0721773] Go to document S. M. Gersten, "On fixed points of certain automorphisms of free groups," Proc. London Math. Soc., vol. 48, iss. 1, pp. 72-90, 1984.
    @article {MR0721773,
      author = {Gersten, S. M.},
      TITLE = {On fixed points of certain automorphisms of free groups},
      JOURNAL = {Proc. London Math. Soc.},
      FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
      VOLUME = {48},
      YEAR = {1984},
      NUMBER = {1},
      PAGES = {72--90},
      ISSN = {0024-6115},
      CODEN = {PLMTAL},
      MRCLASS = {20E05 (05C25 57M05)},
      MRNUMBER = {0721773},
      MRREVIEWER = {G. Peter Scott},
      DOI = {10.1112/plms/s3-48.1.72},
      ZBLNUMBER = {0502.20020},
      }
  • [MR0338129] Go to document D. Quillen, "Higher algebraic $K$-theory. I," in Algebraic $K$-Theory, I: Higher $K$-Theories, New York: Springer-Verlag, 1973, vol. 341, pp. 85-147.
    @incollection {MR0338129,
      author = {Quillen, Daniel},
      TITLE = {Higher algebraic {$K$}-theory. {I}},
      BOOKTITLE = {Algebraic $K$-Theory, {\rm I}: Higher $K$-Theories},
      VENUE={Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972},
      PAGES = {85--147},
      SERIES={Lecture Notes in Math.},
      VOLUME={341},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1973},
      MRCLASS = {18F25},
      MRNUMBER = {0338129},
      MRREVIEWER = {Stephen M. Gersten},
      ZBLNUMBER = {0292.18004},
      DOI = {10.1007/BFb0067053},
      }
  • [MR1503449] Go to document A. P. Morse, "The behavior of a function on its critical set," Ann. of Math., vol. 40, iss. 1, pp. 62-70, 1939.
    @article {MR1503449,
      author = {Morse, Anthony P.},
      TITLE = {The behavior of a function on its critical set},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {40},
      YEAR = {1939},
      NUMBER = {1},
      PAGES = {62--70},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {Contributed Item},
      MRNUMBER = {1503449},
      DOI = {10.2307/1968544},
      ZBLNUMBER = {0020.01205},
      }

Authors

Søren Galatius

Stanford University
Stanford, CA