Rational cohomologies of classifying spaces for homogeneous spaces of small rank. (English) Zbl 1361.55015
In rational homotopy theory, a simply connected CW-complex \(X\) is said to be (rationally) elliptic if \(\pi_*(X)\otimes\mathbb{Q}\) and \(H^*(X;\mathbb{Q})\) are both finite dimensional. Such a space is called an \(F_0\)-space if moreover \(\dim \pi_{\text{even}}(X)\otimes \mathbb{Q} = \dim \pi_{\text{odd}}(X)\otimes \mathbb{Q}\). The \( F_0 \)-spaces have pure minimal Sullivan models and are well known through the famous Halperin conjecture [S. Halperin, Trans. Am. Math. Soc. 230, 173–199 (1977; Zbl 0364.55014)] which states that the Serre spectral sequences of all fibrations \(X\rightarrow E\rightarrow B\) of simply-connected CW complexes degenerate at the \(E_2\) term for any \(F_0\)-space \(X\).
Recall that a fibration \(X\rightarrow E\rightarrow B\) degenerates at the \(E_2\)-term if and only if the induced map \(H^*(E;\mathbb{Q}) \rightarrow H^*(X; \mathbb{Q})\) is onto. Such a fibration is said to be totally non-cohomologous to zero (TNCZ for short).
Now, let aut\(_1X\) be the space of the identity component of self-homotopy equivalences of a space \(X\) and denote by \(\mathrm{Baut}_1(X)\) its Dold-Lashof classifying space. An equivalent version to the Halperin conjecture established in [W. Meier, Math. Ann. 258, 329–340 (1982; Zbl 0466.55012)] states that when \(X\) is an \(F_0\)-space, the rational cohomology algebra \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) is a finitely generated polynomial algebra.
Based on some examples of fibrations \(X\rightarrow E\rightarrow B\) exhibiting different behaviors of the TNCZ property even if \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) is a polynomial algebra, the authors propose the following:
Question If the Serre spectral sequence of any fibration with a rationally elliptic fibre \(X\) degenerates at the \(E_2\) term, then is \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) a polynomial algebra? In particular, does this hold when \(X\) is pure?.
The first main result (Theorem 1.3) gives an affirmative answer for fibrations on a sphere \(\mathbb{S}^n\) and fiber a pure space \(X\) (said of type \((2,3)\)) satisfying \(\dim \pi_{\text{even}}(X)\otimes \mathbb{Q} = 2\) and \(\dim \pi_{\text{odd}}(X)\otimes \mathbb{Q} = 3\).
Indeed, the authors make use of their result in [H. Nishinobu and T. Yamaguchi, Topol. Appl. 196, 290–307 (2015; Zbl 1329.55011)] where the only cases in which \(H^*(\mathrm{Baut}_1(X);\mathbb{Q})\) is not a polynomial algebra are exhibited. By assuming separately one of them, they construct non TNCZ fibrations \(X{\rightarrow} E\rightarrow \mathbb{S}^n\).
In [S. B. Smith, Contemp. Math. 274, 299–307 (2001; Zbl 0983.55009)], it is shown that for a pure formal space \(X\) of type \((2,3)\), \(H^*(\mathrm{Baut}_1(X); \mathbb{Q})\) is not necessarily a polynomial algebra. In their second main result (Theorem 1.6), the authors use minimal Sullivan models as a major tool to construct examples showing that this still holds for non-formal homogeneous spaces of type \((2,3)\).
Recall that a fibration \(X\rightarrow E\rightarrow B\) degenerates at the \(E_2\)-term if and only if the induced map \(H^*(E;\mathbb{Q}) \rightarrow H^*(X; \mathbb{Q})\) is onto. Such a fibration is said to be totally non-cohomologous to zero (TNCZ for short).
Now, let aut\(_1X\) be the space of the identity component of self-homotopy equivalences of a space \(X\) and denote by \(\mathrm{Baut}_1(X)\) its Dold-Lashof classifying space. An equivalent version to the Halperin conjecture established in [W. Meier, Math. Ann. 258, 329–340 (1982; Zbl 0466.55012)] states that when \(X\) is an \(F_0\)-space, the rational cohomology algebra \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) is a finitely generated polynomial algebra.
Based on some examples of fibrations \(X\rightarrow E\rightarrow B\) exhibiting different behaviors of the TNCZ property even if \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) is a polynomial algebra, the authors propose the following:
Question If the Serre spectral sequence of any fibration with a rationally elliptic fibre \(X\) degenerates at the \(E_2\) term, then is \(H^*(\mathrm{Baut}_1X; \mathbb{Q})\) a polynomial algebra? In particular, does this hold when \(X\) is pure?.
The first main result (Theorem 1.3) gives an affirmative answer for fibrations on a sphere \(\mathbb{S}^n\) and fiber a pure space \(X\) (said of type \((2,3)\)) satisfying \(\dim \pi_{\text{even}}(X)\otimes \mathbb{Q} = 2\) and \(\dim \pi_{\text{odd}}(X)\otimes \mathbb{Q} = 3\).
Indeed, the authors make use of their result in [H. Nishinobu and T. Yamaguchi, Topol. Appl. 196, 290–307 (2015; Zbl 1329.55011)] where the only cases in which \(H^*(\mathrm{Baut}_1(X);\mathbb{Q})\) is not a polynomial algebra are exhibited. By assuming separately one of them, they construct non TNCZ fibrations \(X{\rightarrow} E\rightarrow \mathbb{S}^n\).
In [S. B. Smith, Contemp. Math. 274, 299–307 (2001; Zbl 0983.55009)], it is shown that for a pure formal space \(X\) of type \((2,3)\), \(H^*(\mathrm{Baut}_1(X); \mathbb{Q})\) is not necessarily a polynomial algebra. In their second main result (Theorem 1.6), the authors use minimal Sullivan models as a major tool to construct examples showing that this still holds for non-formal homogeneous spaces of type \((2,3)\).
Reviewer: Youssef Rami (Meknès)
MSC:
| 55P62 | Rational homotopy theory |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
References:
| [1] | Amann M.: Non-formal homogeneous spaces. Math. Z. 274, 1299-1325 (2013) · Zbl 1273.57014 · doi:10.1007/s00209-012-1117-6 |
| [2] | Deligne P., Griffith P., Morgan J., Sullivan D.: Real homotopy theory of Kähler manifolds. Inventiones Math. 29, 245-274 (1975) · Zbl 0312.55011 · doi:10.1007/BF01389853 |
| [3] | Dold A., Lashof R.: Principal quasi-fibrations and fibre homotopy equivalence of bundles. Ill. J. Math. 3, 285-305 (1959) · Zbl 0088.15301 |
| [4] | Félix, Y.; Halperin, S.; Thomas, J.C.: Rational Homotopy Theory, Graduate Texts in Mathematics, vol. 205. Springer, Berlin (2001) · Zbl 0961.55002 |
| [5] | Félix Y., Oprea J., Tanré D.: Algebraic Models in Geometry, vol. 17. Oxford G.T.M., Oxford (2008) · Zbl 1149.53002 |
| [6] | Greub W., Halperin S., Vanstone R.: Connection, Curvature and Cohomology III. Academic Press, Cambridge (1976) · Zbl 0372.57001 |
| [7] | 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 |
| [8] | Kuribayashi, K.: On the rational cohomology of the total space of the universal fibration with an elliptic fibre. Contemp. Math. 519, 165-179 (2010) (Y. Félix, G. Lupton and S.B. Smith Ed.) · Zbl 1227.55010 |
| [9] | Kuribayashi K.: Rational visibility of Lie group in the monoid of self-homotopy equivalences of a homogeneous space. HHA 13, 349-379 (2011) · Zbl 1247.55006 · doi:10.4310/HHA.2011.v13.n1.a14 |
| [10] | Lupton G.: Note on a conjecture of Stephen Halperin’s. Springer LNM 1440, 148-163 (1990) · Zbl 0708.55008 |
| [11] | Lupton G., Smith B.S.: Realizing spaces as classifying spaces. Proc. Amer. Math. Soc. 144, 3619-3633 (2016) · Zbl 1348.55010 · doi:10.1090/proc/13022 |
| [12] | Meier W.: Rational universal fibrations and flag manifolds. Math. Ann. 258, 329-340 (1982) · Zbl 0466.55012 · doi:10.1007/BF01450686 |
| [13] | Mimura,M.; Toda, H.: Topology of Lie groups, I and II. AMS Translations of Mathematical Monographs, vol. 91. American Mathematical Society, Providence, RI (1991) · Zbl 0757.57001 |
| [14] | Nishinobu H., Yamaguchi T.: Sullivan minimal models of classifying spaces for non-formal spaces of small rank. Topol. Appl. 196, 290-307 (2015) · Zbl 1329.55011 · doi:10.1016/j.topol.2015.10.003 |
| [15] | Parent, P.E.: Formal and non-formal homogeneous spaces of small rank. University of Ottawa, Thesis (1996) · Zbl 1273.57014 |
| [16] | Shiga H., Tezuka M.: Rational fibrations, homogeneous spaces with positive Euler characteristics and Jacobians. Ann. Inst. Fourier (Grenoble) 37, 81-106 (1987) · Zbl 0608.55006 · doi:10.5802/aif.1078 |
| [17] | Smith S.B.: Rational type of classifying spaces for fibrations. Contemp. Math. 274, 299-307 (2001) · Zbl 0983.55009 · doi:10.1090/conm/274/04472 |
| [18] | Smith S.B.: The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces. J. Pure Appl. Algebra 160, 333-343 (2001) · Zbl 0980.55008 · doi:10.1016/S0022-4049(00)00082-7 |
| [19] | Stasheff J.: A classification theorem for fibre spaces. Topology 2, 239-246 (1963) · Zbl 0123.39705 · doi:10.1016/0040-9383(63)90006-5 |
| [20] | Sullivan D.: Infinitesimal computations in topology. IHES 47, 269-331 (1978) · Zbl 0374.57002 · doi:10.1007/BF02684341 |
| [21] | Tanré, D.: Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Note in Math, vol. 1025. Springer, Berlin (1983) · Zbl 0539.55001 |
| [22] | Yamaguchi T.: When is the classifying space for elliptic fibrations rank one?. Bull. Korean Math. Soc. 42, 521-525 (2005) · Zbl 1083.55009 · doi:10.4134/BKMS.2005.42.3.521 |
| [23] | Yamaguchi T.: An example of a fiber in fibrations whose Serre spectral sequences collapse. Czechoslov. Math. J. 55, 997-1001 (2005) · Zbl 1081.55010 · doi:10.1007/s10587-005-0083-0 |
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.
