×

Every finite group is the group of self-homotopy equivalences of an elliptic space. (English) Zbl 1308.55005

In this paper Cristina Costoya and Antonio Viruel give a positive answer to one of the main questions concerning the group of homotopy self-equivalences of a given space. They prove that every finite group \(G\) can be realized as the group of self-homotopy equivalences of infinitely many (non homotopy equivalent) rational elliptic spaces. The basic tool is the theorem of Frucht: For every finite group \(G\) there exist infinitely many non-isomorphic connected finite graphs whose automorphim group is isomorphic to \(G\). Thus to each graph \({\mathcal G}\) they associate a Sullivan minimal model \((\land V,d)\) whose group of self-homotopy equivalences is isomorphic to the group of automorphisms of \({\mathcal G}\). More precisely \((\land V,d) = (\land (x_1, x_2, y_1, y_2, y_3,z) \otimes \land (x_v, z_v), d)\) where the \(x_v\) and the \(z_v\) are indexed by the vertices of the graph and the differential of \(z_v\) takes into account the names of the edges having \(v\) as a summit. As a corollary, there are infinitely many rational spaces that are homotopically rigid. In a second step the authors specify that the elliptic rational spaces can be chosen to be rationalization of inflexible manifolds.

MSC:

55P10 Homotopy equivalences in algebraic topology
55P62 Rational homotopy theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amann, M., Mapping degrees of self-maps of simply-connected manifolds. Preprint, 2011. arXiv:1109.0960 [math.AT].
[2] Arkowitz, M., The group of self-homotopy equivalences—a survey, in Groups of Self- Equivalences and Related Topics (Montreal, PQ, 1988), Lecture Notes in Math., 1425, pp. 170-203. Springer, Berlin-Heidelberg, 1990.
[3] Arkowitz, M., Problems on self-homotopy equivalences, in Groups of Homotopy Self-Equivalences and Related Topics (Gargnano, 1999), Contemp. Math., 274, pp. 309-315. Amer. Math. Soc., Providence, RI, 2001. · Zbl 0973.55005
[4] Arkowitz M., Lupton G.: Rational obstruction theory and rational homotopy sets. Math. Z. 235, 525-539 (2000) · Zbl 0968.55005 · doi:10.1007/s002090000144
[5] Barge J.: Structures diffërentiables sur les types d’homotopie rationnelle simplement connexes. Ann. Sci. École Norm. Sup. 9, 469-501 (1976) · Zbl 0348.57016
[6] Benkhalifa M.: Rational self-homotopy equivalences and Whitehead exact sequence. J. Homotopy Relat. Struct. 4, 111-121 (2009) · Zbl 1190.55006
[7] Benkhalifa M.: Realizability of the group of rational self-homotopy equivalences. J. Homotopy Relat. Struct. 5, 361-372 (2010) · Zbl 1278.55021
[8] Bollobás, B., Modern Graph Theory. Graduate Texts in Mathematics, 184. Springer, New York, 1998. · Zbl 0364.55014
[9] Copeland A. H. Jr., Shar A. O.: Images and pre-images of localization maps. Pacific J. Math. 57, 349-358 (1975) · Zbl 0294.55005 · doi:10.2140/pjm.1975.57.349
[10] Costoya, C. & Viruel, A., Faithful actions on commutative differential graded algebras and the group isomorphism problem. To appear in Q. J. Math. · Zbl 1356.20005
[11] Crowley, D. & Lӧh, C., Functorial semi-norms on singular homology and (in)flexible manifolds. Preprint, 2011. arXiv:1103.4139 [math.GT]. · Zbl 0431.55005
[12] Federinov J., Felix Y.: Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences. Topology Appl. 154, 2425-2433 (2007) · Zbl 1138.55005 · doi:10.1016/j.topol.2007.04.005
[13] Félix, Y., La dichotomie elliptique-hyperbolique en homotopie rationnelle. Astérisque, 176 (1989). · Zbl 0691.55001
[14] Félix, Y., Problems on mapping spaces and related subjects, in Homotopy Theory of Function Spaces and Related Topics, Contemp. Math., 519, pp. 217-230. Amer. Math. Soc., Providence, RI, 2010. · Zbl 1208.55007
[15] Félix, Y., Halperin, S. & Thomas, J. C., Rational Homotopy Theory. Graduate Texts in Mathematics, 205. Springer, New York, 2001. · Zbl 0294.55005
[16] Frucht R.: Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math. 6, 239-250 (1939) · JFM 64.0596.02
[17] Frucht R.: Graphs of degree three with a given abstract group. Canadian J. Math., 1, 365-378 (1949) · Zbl 0034.25802 · doi:10.4153/CJM-1949-033-6
[18] Gromov, M., Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, 152. Birkhäuser, Boston, MA, 1999. · Zbl 0953.53002
[19] Halperin S.: Finiteness in the minimal models of Sullivan. Trans. Amer. Math. Soc. 230, 173-199 (1977) · Zbl 0364.55014 · doi:10.1090/S0002-9947-1977-0461508-8
[20] Kahn D. W.: Realization problems for the group of homotopy classes of self-equivalences. Math. Ann. 220, 37-46 (1976) · Zbl 0305.55016 · doi:10.1007/BF01354527
[21] Kahn, D. W., Some research problems on homotopy-self-equivalences, in Groups of Self-Equivalences and Related Topics (Montreal, PQ, 1988), Lecture Notes in Math., 1425, pp. 204-207. Springer, Berlin-Heidelberg, 1990.
[22] Maruyama K.: Finite complexes whose self-homotopy equivalence groups realize the infinite cyclic group. Canad. Math. Bull. 37, 534-536 (1994) · Zbl 0841.55005 · doi:10.4153/CMB-1994-077-8
[23] Nešetřil, J., Homomorphisms of derivative graphs. Discrete Math., 1 (1971/72), 257-268. · Zbl 0227.05109
[24] Oka S.: Finite complexes whose self-homotopy equivalences form cyclic groups. Mem. Fac. Sci. Kyushu Univ. Ser. A 34, 171-181 (1980) · Zbl 0431.55005
[25] Puppe, V., Simply connected 6-dimensional manifolds with little symmetry and algebras with small tangent space, in Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud., 138, pp. 283-302. Princeton Univ. Press, Princeton, NJ, 1995. · Zbl 0929.57026
[26] Rutter, J.W., Spaces of Homotopy Self-Equivalences. Lecture Notes in Mathematics, 1662. Springer, Berlin-Heidelberg, 1997.
[27] Sullivan D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269-331 (1977) · Zbl 0374.57002 · doi:10.1007/BF02684341
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.