## Rational homotopy type of the classifying space for fibrewise self-equivalences

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- by Urtzi Buijs and Samuel B. Smith PDF
- Proc. Amer. Math. Soc.
**141**(2013), 2153-2167 Request permission

## Abstract:

Let $p \colon E \to B$ be a fibration of simply connected CW complexes with finite base $B$ and fibre $F$. Let ${\mathrm {aut}}_1(p)$ denote the identity component of the space of all fibre-homotopy self-equivalences of $p$. Let ${\mathrm {Baut}}_1(p)$ denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for ${\mathrm {Baut}}_1(p)$, connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by ${\mathrm {Baut}}_1(p)$ and also to obtain conditions under which the monoid ${\mathrm {aut}}_1(p)$ is a double loop-space after rationalization.## References

- M. F. Atiyah and R. Bott,
*The Yang-Mills equations over Riemann surfaces*, Philos. Trans. Roy. Soc. London Ser. A**308**(1983), no. 1505, 523–615. MR**702806**, DOI 10.1098/rsta.1983.0017 - Urtzi Buijs, Yves Félix, and Aniceto Murillo,
*Lie models for the components of sections of a nilpotent fibration*, Trans. Amer. Math. Soc.**361**(2009), no. 10, 5601–5614. MR**2515825**, DOI 10.1090/S0002-9947-09-04870-3 - Urtzi Buijs, Yves Félix, and Aniceto Murillo,
*$L_\infty$ models of based mapping spaces*, J. Math. Soc. Japan**63**(2011), no. 2, 503–524. MR**2793109** - Peter Booth, Philip Heath, Chris Morgan, and Renzo Piccinini,
*$H$-spaces of self-equivalences of fibrations and bundles*, Proc. London Math. Soc. (3)**49**(1984), no. 1, 111–127. MR**743373**, DOI 10.1112/plms/s3-49.1.111 - M. C. Crabb and W. A. Sutherland,
*Counting homotopy types of gauge groups*, Proc. London Math. Soc. (3)**81**(2000), no. 3, 747–768. MR**1781154**, DOI 10.1112/S0024611500012545 - Pierre Deligne, Phillip Griffiths, John Morgan, and Dennis Sullivan,
*Real homotopy theory of Kähler manifolds*, Invent. Math.**29**(1975), no. 3, 245–274. MR**382702**, DOI 10.1007/BF01389853 - Albrecht Dold and Richard Lashof,
*Principal quasi-fibrations and fibre homotopy equivalence of bundles*, Illinois J. Math.**3**(1959), 285–305. MR**101521** - Yves Félix, Stephen Halperin, and Jean-Claude Thomas,
*Rational homotopy theory*, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR**1802847**, DOI 10.1007/978-1-4613-0105-9 - Yves Félix, Gregory Lupton, and Samuel B. Smith,
*The rational homotopy type of the space of self-equivalences of a fibration*, Homology Homotopy Appl.**12**(2010), no. 2, 371–400. MR**2771595**, DOI 10.4310/HHA.2010.v12.n2.a13 - Yves Félix and John Oprea,
*Rational homotopy of gauge groups*, Proc. Amer. Math. Soc.**137**(2009), no. 4, 1519–1527. MR**2465678**, DOI 10.1090/S0002-9939-08-09721-9 - J.-B. Gatsinzi,
*The homotopy Lie algebra of classifying spaces*, J. Pure Appl. Algebra**120**(1997), no. 3, 281–289. MR**1468920**, DOI 10.1016/S0022-4049(96)00037-0 - Daniel H. Gottlieb,
*On fibre spaces and the evaluation map*, Ann. of Math. (2)**87**(1968), 42–55. MR**221508**, DOI 10.2307/1970593 - Stephen Halperin,
*Rational fibrations, minimal models, and fibrings of homogeneous spaces*, Trans. Amer. Math. Soc.**244**(1978), 199–224. MR**515558**, DOI 10.1090/S0002-9947-1978-0515558-4 - Peter Hilton, Guido Mislin, and Joe Roitberg,
*Localization of nilpotent groups and spaces*, North-Holland Mathematics Studies, No. 15, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR**0478146** - John R. Klein, Claude L. Schochet, and Samuel B. Smith,
*Continuous trace $C^*$-algebras, gauge groups and rationalization*, J. Topol. Anal.**1**(2009), no. 3, 261–288. MR**2574026**, DOI 10.1142/S179352530900014X - Akira Kono,
*A note on the homotopy type of certain gauge groups*, Proc. Roy. Soc. Edinburgh Sect. A**117**(1991), no. 3-4, 295–297. MR**1103296**, DOI 10.1017/S0308210500024732 - Akira Kono and Shuichi Tsukuda,
*4-manifolds $X$ over $B\textrm {SU}(2)$ and the corresponding homotopy types $\textrm {Map}(X,B\textrm {SU}(2))$*, J. Pure Appl. Algebra**151**(2000), no. 3, 227–237. MR**1776430**, DOI 10.1016/S0022-4049(99)00069-9 - J. Peter May,
*Classifying spaces and fibrations*, Mem. Amer. Math. Soc.**1**(1975), no. 1, 155, xiii+98. MR**370579**, DOI 10.1090/memo/0155 - H. Scheerer,
*On rationalized $H$- and co-$H$-spaces. With an appendix on decomposable $H$- and co-$H$-spaces*, Manuscripta Math.**51**(1985), no. 1-3, 63–87. MR**788673**, DOI 10.1007/BF01168347 - Claude L. Schochet and Samuel B. Smith,
*Localization of grouplike function and section spaces with compact domain*, Homotopy theory of function spaces and related topics, Contemp. Math., vol. 519, Amer. Math. Soc., Providence, RI, 2010, pp. 189–202. MR**2648714**, DOI 10.1090/conm/519/10242 - H. Shiga and M. Tezuka,
*Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians*, Ann. Inst. Fourier (Grenoble)**37**(1987), no. 1, 81–106 (English, with French summary). MR**894562**, DOI 10.5802/aif.1078 - Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 269–331 (1978). MR**646078**, DOI 10.1007/BF02684341 - Daniel Tanré,
*Homotopie rationnelle: modèles de Chen, Quillen, Sullivan*, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). MR**764769**, DOI 10.1007/BFb0071482 - Shuichi Tsukuda,
*A remark on the homotopy type of the classifying space of certain gauge groups*, J. Math. Kyoto Univ.**36**(1996), no. 1, 123–128. MR**1381543**, DOI 10.1215/kjm/1250518608

## Additional Information

**Urtzi Buijs**- Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain
- Email: ubuijs@ub.edu
**Samuel B. Smith**- Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
- MR Author ID: 333158
- Email: smith@sju.edu
- Received by editor(s): July 26, 2011
- Received by editor(s) in revised form: September 18, 2011
- Published electronically: December 13, 2012
- Additional Notes: The first author was partially supported by the Ministerio de Ciencia e Innovación grant MTM2010-15831 and by the Junta de Andalucía grant FQM-213.
- Communicated by: Brooke Shipley
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**141**(2013), 2153-2167 - MSC (2010): Primary 55P62, 55Q15
- DOI: https://doi.org/10.1090/S0002-9939-2012-11560-6
- MathSciNet review: 3034442