An induced map between rationalized classifying spaces for fibrations

2017/06/12 by Toshihiro Yamaguchi, Yamaguchi, Toshihiro
Mathematics · #55P62 #55R15 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55P62 #msc:55R15

paper · pdf · doi:10.48550/arxiv.1706.03450

21 pages

openalex publication_date 2017/06/12 · arxiv created 2018/08/01 · arxiv updated 2018/08/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let B aut₁X be the Dold-Lashof classifying space of orientable fibrations with fiber X. For a rationally weakly trivial map f:X→ Y, our strictly induced map a_f: (Baut₁X)₀→ (Baut₁Y)₀ induces a natural map from a X₀-fibration to a Y₀-fibration. It is given by a map between the differential graded Lie algebras of derivations of Sullivan models. We note some conditions that the map a_f admits a section and note some relations with the Halperin conjecture. Furthermore we give the obstruction class for a lifting of a classifying map h: B→ (Baut₁Y)₀ and apply it for liftings of G-actions on Y for a compact connected Lie group G as the case of B=BG and evaluating of rational toral ranks as r₀(Y)≤ r₀(X).

Related

• The universal fibration with fibre X in rational homotopy theory
• Sullivan minimal models of classifying spaces for non-formal spaces of small rank
• Lie models of homotopy automorphism monoids and classifying fibrations
• Algebraic models for classifying spaces of fibrations
• Rational Homotopy Type of the Classifying Space for Fibrewise Self-Equivalences