Gottlieb elements and rational homotopy types realized as classifying spaces

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-07-15 DOI:10.1016/j.topol.2025.109525

Yang Bai , Xiugui Liu

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引用次数: 0

Abstract

In this paper, we are interested in the realizability problem as classifying spaces in rational homotopy theory. Namely, if a given space Y can appear as the classifying space

B {aut}_{1} (X)

up to rational homotopy for some space X. We prove the non-realization of

S^{2 n}

for

n \leq 5

under the hypothesis of X having non-vanishing Gottlieb elements above degree

2 n - 1

, which extends the results of Lupton and Smith. Moreover, we also show some realization and non-realization results for products of Eilenberg-Mac Lane spaces under the hypothesis of X having finitely many non-zero homotopy groups. Our proofs are based on a technique of constructing specific derivations of a Sullivan minimal algebra with a given Gottlieb element.

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Gottlieb元素和有理同伦类型实现为分类空间

本文研究了有理同伦理论中作为分类空间的可实现性问题。即，如果给定空间Y可以表现为对某空间X具有有理同伦的分类空间Baut1(X)，我们证明了在X具有2n−1次以上的非消失Gottlieb元的假设下，对于n≤5，S2n的不实现，推广了Lupton和Smith的结果。此外，我们还给出了在X有有限多个非零同伦群的假设下Eilenberg-Mac Lane空间积的一些可实现和不可实现的结果。我们的证明是基于构造具有给定戈特利布元素的沙利文最小代数的特定衍生的技术。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.