Mild and tame homotopy theory
type: Item
instance of: scholarly article
title: Mild and tame homotopy theory
zbMATH Open document ID: 0777.55008
DOI: 10.1016/0022-4049(93)90003-C
author: Kathryn Hess
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: In 1969 Quillen showed that rational homotopy theory was (at least in principle) completely expressible in algebraic terms. Quillen's approach was shown by \textit{W. G. Dwyer} [Topology 18, 321-338 (1979; Zbl 0436.55007)] to extend to the class of spaces having enough divisibility in homotopy to prevent stable \(k\)-invariants from appearing; the so- called tame spaces. More precisely, Dwyer showed that the homotopy category of \(r\)-connected tame spaces was equivalent to the homotopy category of \(r\)-reduced differential graded Lie algebras. Since Dwyer's paper, the `Quillen-Dwyer' approach to tame homotopy theory has been explored in depth by \textit{H. Scheerer} and \textit{D. Tanré} [C. R. Acad. Sci., Paris, Sér. I 307, No. 14, 783-785 and No. 15, 799-801 (1988; Zbl 0658.55005 and Zbl 0658.55006); ibid. 308, No. 6, 181-183 and No. 15, 475-478 (1989; Zbl 0666.55009 and Zbl 0676.55013)]. The present paper approaches tame homotopy theory (and, in fact, a generalization called mild homotopy theory requiring fewer inverted primes) from a viewpoint heavily influenced by the ideas of \textit{D. Anick} [J. Am. Math. Soc. 2, No. 3, 417-453 (1989; Zbl 0681.55006) and Adv. Math. 77, No. 1, 116-136 (1989; Zbl 0684.55010)]. The author introduces the notion of a gentle functor between categories which, in the presence of enough fibrant models, induces an equivalence of homotopy categories. She then proceeds to show that the universal enveloping algebra functor \(\mathcal U\) from \(r\)- reduced differential graded Lie algebras \((\text{DGL}_ r(Z_ *))\) to \(r\)-reduced adjusted Hopf algebras up to homotopy \((\text{AHA}_ r(Z_ *))\) is gentle. Here, algebras are over \(\mathbb{Z}\) with weak equivalences \(Z_ *\)-isomorphisms, where \(Z_ *\) is the mild ring system mentioned above. Although a lack of enough fibrant models precludes the direct application of a gentle functor's salient property, the following prescription may be carried through to obtain a mild model of an \(…
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 166176
zbMATH Keywords: rational homotopy theory
zbMATH Keywords: homotopy category
zbMATH Keywords: differential graded Lie algebras
zbMATH Keywords: mild homotopy theory
zbMATH Keywords: Adams-Hilton model
reviewed by: John Oprea
MaRDI profile type: Publication
cites work: Hopf Algebras up to Homotopy
cites work: An R-local Milnor-Moore theorem
cites work: Q3472864
cites work: On the chain algebra of a loop space
cites work: Minimal models in homotopy theory
cites work: Tame homotopy theory
cites work: Rational homotopy theory
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description: scientific article; zbMATH DE number 166176