Mild and tame homotopy theory
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 \(r\)- reduced CW complex \(X\): Take the Adams-Hilton model of \(X\) in \(\text{AHA}_ r(Z_ *)\) and find \({\mathcal M}(X)\) in \(\text{DGL}_ r(Z_ *)\) with \({\mathcal U}({\mathcal M}(X))\rightarrow{\mathcal A}(X)\) a weak equivalence. Thus is obtained a model \({\mathcal M}: \text{CW}_ r(Z_ *)\rightarrow\text{DGL}_ r(Z_ *)\) (which is unique up to homotopy). When the mild ring system is enlarged to a tame system \(T_ *\), the author gives a new proof of Dwyer's equivalence of homotopy categories by proving that her model \(\mathcal M\) is gentle and noting that \(\text{DGL}_ r(T_ *)\) has enough fibrant models. The main advantage of the author's approach seems to be a very real possibility of computability. While the Dwyer approach uses rather abstract simplicial methods alà Quillen, the author's use of the relatively small and `geometrically defined' Adams- Hilton model as a target for \({\mathcal M}(X)\) makes her model ripe for calculation. An instructive example of this is included before Theorem 6.7; namely, spaces are exhibited with the same tame homotopy type, but different mild homotopy types. (The reader should note that the reviewer thinks several misprints seem to occur in this example: (1) the degree of \(y\) in both models should be \(4N-1\) so that the degree \(-1\) differential behaves properly and (2) the restriction on primes \(p\) should be along the lines of \(2<p\leq N+1\), the first inequality following since \(Z_{4N-2}\) has 2 inverted and the second since \(T_{4N-2}\) has \(p\) inverted if \(p\leq{1\over 2}(4N-2-(2N-1)+3)=N+1)\).
scientific article; zbMATH DE number 166176
scientific article