Contributions to the homotopy theory of mapping spaces

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Abstract

This thesis offers several contributions to the homotopy theory of mapping spaces. These contributions fall naturally into two groups. Accordingly the thesis is divided into two parts:

a) Part I Spaces of continuous maps;
b) Part II : Spaces of differentiable maps.

Part 1 is mainly devoted to a study of the (path-) components in a given space of continuous maps. We show that a rich variety of homotopy types is possible thereby extending earlier fragmentary results of G.W.Whitehead, Hu and Koh in the same direction. As a new type of result we produce lots of examples of components which have the same homotopy type for non-trivial reasons. In certain favourable cases, e.g. in a space of maps of a sphere into itself, we solve completely the homotopy problem for the set of components. The study of the components in a space of continuous maps is closely related to the study of certain evaluation fibrations. For these evaluation fibrations we obtain strong results on the fibre homotopy type. The methods to obtain the results mentioned above involve Whitehead products and in particular a fundamental theorem of G.W.Whitehead, which describes the boundary operator in the homotopy sequence of an evaluation fibration in terms of such products. In the final chapter of Part I, Chapter 4, we prove a variation of a theorem of Federer and, independently, Thom on the qualitative structure of the homotopy groups of a mapping space.

In Part II we study the homotopy properties of a space of differentiable embeddings or k-mersions from a compact smooth manifold into either an infinite dimensional smooth manifold or a closed expanding system of finite dimensional smooth manifolds of increasing dimension. Such a study is naturally motivated by the work of Eells, Elworthy and Mukherjea according to which many infinite dimensional smooth manifolds are homotopy direct limits of closed expanding systems of finite dimensional smooth submanifolds of increasing dimension. Briefly stated, we show that a space of embeddings or k-mersions with infinite dimensional target, or the appropriate induced limit space over an expanding system of finite dimensional manifolds in the target has the same homotopy type as the naturally associated space of continuous maps. Using these results we prove that the functors we get in the target, when we fix the domain in a space of embeddings or k-mersions, "commute" with the homotopy direct limits given by the results of Eells, Elworthy and Mukherjea. As an application of these limit theorems we classify the k-mersions from a compact smooth manifold into a smooth (infinite dimensional) pseudo Fredholm manifold in the spirit of the Smale-Hirsch theory for immersions in finite dimensions. The class of pseudo Fredholm manifolds contains e.g. the class of separable Hilbert manifolds. We obtain hereby a strict analogue to a theorem in finite dimensions due to S.D.Feit and independently Gromov.

As an application of our results for spaces of embeddings we construct models for the classifying space of a diffeomorphism group. Preliminary to all the limit arguments in Part II we study in Chapter 6 the limit space of a closed expanding sY6te~ of manifolds or ANR's. We show that such a limit space usually is not metrizable, in particular it is not an ANR. It has however the homotopy type of an ANR. This result is crucial, since we in many places want to apply a fundamental theorem of J.H.C. Whitehead, according to which a weak homotopy equivalence between ANR's is a homotopy equivalence.

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