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An infinite-dimensional phenomenon in finite-dimensional metric topology. (English) Zbl 1445.57018

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One clue of the paper under review is to understand precompact subsets of Riemannian manifolds in Gromov-Hausdorff space.
We say a metric space \(X\) is \(LCG(\rho)\) if there is a contractibility function \(\rho: [0,R)\to [0,\infty)\) with \(\rho(0)=0\), \(\rho(t)\geq t\), and continuous at 0, such that every ball of radius \(r< R\) in \(X\) is nullhomotopic in the concentric ball of radius \(\rho(r)\).
In [J. Differ. Geom. 31, No. 2, 387–395 (1990; Zbl 0696.55005)], P. Petersen showed that sufficiently Gromov-Hausdorff close \(n\)-dimensional \(LGC(\rho)\) spaces are homotopy equivalent. Results of T. A. Chapman and S. Ferry [Am. J. Math. 101, 583–607 (1979; Zbl 0426.57004)] imply that if \(M\) is a closed \(n\)-manifold with a fixed topological metric, \(n \geq 5\), with contractibility function \(\rho\), then there is an \(\epsilon > 0\) such that any \(LGC(\rho)\) \(n\)-manifold within \(\epsilon\) of \(M\) in Gromov-Hausdorff space is homeomorphic to it.
In the preprint of the current paper [“Cell-like maps and topological structure groups on manifolds”, Preprint, arXiv:math/0611004], the first and second author noticed that, for certain precompact collections of closed Riemannian manifolds with contractibility function \(\rho\), there are limit points \(X\) with the property that every \(\epsilon\) neighborhood of \(X\) contains manifolds of different homeomorphism types. Examples were constructed by considering cell-like maps \(c_1: N\to X\) and \(c_2: M\to X\) and deforming the metrics on \(N\) and \(M\) to yield two sequences of Riemannian manifolds converging to \(X\). The manifolds \(M\) and \(N\) are not necessarily homeomorphic (though they are homotopy equivalent). The authors then proceed to classify the homotopy equivalences \(N\to M\) obtainable by such deformations. These were identified with a subset of the structure set \(S(M)\) realized by cell-like maps. Here a homotopy equivalence \(f: N\to M\) between closed manifolds is realized by cell-like maps if there exist a space \(X\) and cell-like maps \(c_1: N\to X\), \(c_2: M\to X\) s.t. the diagram \[ \begin{tikzcd}[column sep=small] N\ar[rr, "{f}"]\ar[rd, "{c_1}"'] &&M\ar[dl, "{c_2}"]\\ &X \end{tikzcd} \] homotopy commutes. Denote this subset by \(S^{CE}(M)\). The authors give a thorough study of \(S^{CE}(M)\), which involves controlled topology, surgery theory and the second stage of the Postnikov tower of \(M\). Concrete examples of “malleable” and “immutable” manifolds are developed. Invariants which detect “malleability” are also discussed.

MSC:

57N65 Algebraic topology of manifolds
55P10 Homotopy equivalences in algebraic topology
53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57N60 Cellularity in topological manifolds
57R65 Surgery and handlebodies
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