The universal fibration with fibre \(X\) in rational homotopy theory. (English) Zbl 1445.55009
Given a simply connected \(CW\) complex \(X\) of finite type, let \(\mbox{aut}_1(X)\) denote the space of self-maps of \(X\) homotopic to the identity map. The group-like space \(\mbox{aut}_1(X)\) has a classifying space \(\mbox{Baut}_1(X)\). It appears as the base space of the
universal example \(p_\infty: UE \to \mbox{Baut}_1(X)\) of a fibration of simply connected \(CW\) complexes with fibre of the homotopy type of \(X\) in e.g., [J. D. Stasheff, Topology 2, 239–246 (1963; Zbl 0123.39705)] and offers a computational challenge in homotopy theory. When \(X\) is a finite complex, \(\mbox{Baut}_1(X)\) is of \(CW\) type and satisfies the localization identity \(\mbox{Baut}_1(X_P)\simeq \mbox{Baut}_1(X)_P\) for any collection of primes \(P\) by the work of J. P. May [Trans. Am. Math. Soc. 258, 127–146 (1980; Zbl 0429.55004)]. In rational homotopy theory, models for \(\mbox{Baut}_1(X_\mathbb{Q})\) are due to D. Sullivan [Publ. Math., Inst. Hautes Étud. Sci. 47, 269–331 (1977; Zbl 0374.57002)] and others.
Let \(X\) be a simply connected space with finite-dimensional rational homotopy groups and write \(p_\infty: UE \to \mbox{Baut}_1(X)\) for the universal fibration of simply connected spaces with fibre \(X\). First, the authors give a \(DG\) Lie algebra model for the evaluation map \(\omega: \mbox{aut}_1(\mbox{Baut}_1(X_\mathbb{Q}))\to \mbox{Baut}_1(X_\mathbb{Q})\) expressed in terms of derivations of the relative Sullivan model of \(p_\infty\). Then, formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space \(\mbox{Baut}_1(X_\mathbb{Q})\) are deduced. More generally, a description of the poset of evaluation subgroups \(G_\ast(\xi; X_\mathbb{Q})\subseteq \pi_\ast(\mbox{Baut}_1(X_\mathbb{Q}))\) parameterized by fibrations \(\xi\) with fibre \(X_\mathbb{Q}\) is obtained. Then, some examples and results on this poset complementing work of T. Yamaguchi [Topology Appl. 196, Part B, 1060–1076 (2015; Zbl 1334.55004)] are given. At the end, a non-realization result for the classifying space is proved: the rationalization \(\mathbb{C}P^n_\mathbb{Q}\) of the complex projective space \(\mathbb{C}P^n\) cannot be realized as \(\mbox{Baut}_1(X_\mathbb{Q})\) for \(n \le 4\) and \(X\) with finite-dimensional rational homotopy groups.
Let \(X\) be a simply connected space with finite-dimensional rational homotopy groups and write \(p_\infty: UE \to \mbox{Baut}_1(X)\) for the universal fibration of simply connected spaces with fibre \(X\). First, the authors give a \(DG\) Lie algebra model for the evaluation map \(\omega: \mbox{aut}_1(\mbox{Baut}_1(X_\mathbb{Q}))\to \mbox{Baut}_1(X_\mathbb{Q})\) expressed in terms of derivations of the relative Sullivan model of \(p_\infty\). Then, formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space \(\mbox{Baut}_1(X_\mathbb{Q})\) are deduced. More generally, a description of the poset of evaluation subgroups \(G_\ast(\xi; X_\mathbb{Q})\subseteq \pi_\ast(\mbox{Baut}_1(X_\mathbb{Q}))\) parameterized by fibrations \(\xi\) with fibre \(X_\mathbb{Q}\) is obtained. Then, some examples and results on this poset complementing work of T. Yamaguchi [Topology Appl. 196, Part B, 1060–1076 (2015; Zbl 1334.55004)] are given. At the end, a non-realization result for the classifying space is proved: the rationalization \(\mathbb{C}P^n_\mathbb{Q}\) of the complex projective space \(\mathbb{C}P^n\) cannot be realized as \(\mbox{Baut}_1(X_\mathbb{Q})\) for \(n \le 4\) and \(X\) with finite-dimensional rational homotopy groups.
Reviewer: Marek Golasiński (Olsztyn)
MSC:
| 55P62 | Rational homotopy theory |
| 55P10 | Homotopy equivalences in algebraic topology |
| 55R15 | Classification of fiber spaces or bundles in algebraic topology |
Keywords:
classifying space for fibrations; evaluation map; Gottlieb group; rationalization; derivations; minimal modelReferences:
| [1] | Booth, Peter; Heath, Philip; Morgan, Chris; Piccinini, Renzo, \(H\)-spaces of self-equivalences of fibrations and bundles, Proc. Lond. Math. Soc. (3), 49, 1, 111-127 (1984) · Zbl 0525.55005 · doi:10.1112/plms/s3-49.1.111 |
| [2] | Dold, Albrecht, Halbexakte Homotopiefunktoren. Lecture Notes in Mathematics (1966), Berlin: Springer-Verlag, Berlin · Zbl 0223.55010 |
| [3] | Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude, Rational Homotopy Theory, Graduate Texts in Mathematics (2001), New York: Springer-Verlag, New York · Zbl 0961.55002 |
| [4] | Félix, Yves; Lupton, Gregory; Smith, Samuel B., The rational homotopy type of the space of self-equivalences of a fibration, Homol. Homotopy Appl., 12, 2, 371-400 (2010) · Zbl 1214.55011 · doi:10.4310/HHA.2010.v12.n2.a13 |
| [5] | Gatsinzi, J-B, The homotopy Lie algebra of classifying spaces, J. Pure Appl. Algebra, 120, 3, 281-289 (1997) · Zbl 0884.55009 · doi:10.1016/S0022-4049(96)00037-0 |
| [6] | Gottlieb, DH, Evaluation subgroups of homotopy groups, Am. J. Math., 91, 729-756 (1969) · Zbl 0185.27102 · doi:10.2307/2373349 |
| [7] | Gregory Lupton and Samuel Bruce Smith, The evaluation subgroup of a fibre inclusion, Topol. Appl., 154, 6, 1107-1118 (2007) · Zbl 1121.55011 · doi:10.1016/j.topol.2006.11.001 |
| [8] | Gregory Lupton and Samuel Bruce Smith, Realizing spaces as classifying spaces, Proc. Am. Math. Soc., 144, 8, 3619-3633 (2016) · Zbl 1348.55010 · doi:10.1090/proc/13022 |
| [9] | May, J. Peter, Classifying spaces and fibrations, Mem. Am. Math. Soc., 1, 155, xiii+98 (1975) · Zbl 0321.55033 |
| [10] | May, J. Peter, Fibrewise localization and completion, Trans. Am. Math. Soc., 258, 1, 127-146 (1980) · Zbl 0429.55004 · doi:10.1090/S0002-9947-1980-0554323-8 |
| [11] | Meier, W., Rational universal fibrations and flag manifolds, Math. Ann., 258, 3, 329-340 (1981) · Zbl 0466.55012 · doi:10.1007/BF01450686 |
| [12] | Schlessinger, Mike, Stasheff, James: Deformation theory and rational homotopy type, preprint · Zbl 0576.17008 |
| [13] | Stasheff, James, A classification theorem for fibre spaces, Topology, 2, 239-246 (1963) · Zbl 0123.39705 · doi:10.1016/0040-9383(63)90006-5 |
| [14] | Sullivan, Dennis, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), 47, 269-331 (1978) · Zbl 0374.57002 · doi:10.1007/BF02684341 |
| [15] | Tanré, Daniel, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan (1983), Berlin: Springer-Verlag, Berlin · Zbl 0539.55001 |
| [16] | Xie, Sang, Liu, Jian, Liu, Xiugui: Spaces realized and non-realized as classifying spaces, preprint |
| [17] | Yamaguchi, Toshihiro, A fibre-restricted Gottlieb group and its rational realization problem, Topology and its Applications, 196, 1060-1076 (2015) · Zbl 1334.55004 · doi:10.1016/j.topol.2015.05.066 |
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.
