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Register a new userThe universal fibration with fibre $X$ in rational homotopy theory
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Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_infty colon UE to mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $ omega colon mathrm{aut}_1(mathrm{Baut}_1(X_{mathbb Q})) to mathrm{Baut}_1(X_{mathbb Q})$ expressed in terms of derivations of the relative Sullivan model of $p_infty$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $mathrm{Baut}_1(X_{mathbb Q})$ as a consequence. We also prove that ${mathbb C} P^n_{mathbb Q}$ cannot be realized as $mathrm{Baut}_1(X_{mathbb Q})$ for $n leq 4$ and $X$ with finite-dimensional rational homotopy groups.
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Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik-Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.
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Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model structure are not well understood calculationally. To illustrate, we exhibit some examples of cofibrant and fibrant posets and an example of a non-cofibrant finite poset.
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