### Classes d'applications d'un espace dans un groupe topologique, d'après Shih Weishu

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Our point of departure is J. Neisendorfer's localization theorem which reveals a subtle connection between some simply connected finite complexes and their connected covers. We show that even though the connected covers do not forget that they came from a finite complex their homotopy-theoretic properties are drastically different from those of finite complexes. For instance, connected covers of finite complexes may have uncountable genus or nontrivial SNT sets, their Lusternik-Schnirelmann category...

Using the theory of resolving classes, we show that if X is a CW complex of finite type such that $map\u204e(X,{S}^{2n+1})\phantom{\rule{3.33333pt}{0ex}}\ast $ for all sufficiently large n, then map⁎(X,K) ∼ ∗ for every simply-connected finite-dimensional CW complex K; and under mild hypotheses on π₁(X), the same conclusion holds for all finite-dimensional complexes K. Since it is comparatively easy to prove the former condition for X = Bℤ/p (we give a proof in an appendix), this result can be applied to give a new, more elementary proof of the Sullivan conjecture....

Are there any kinds of self maps on the loop structure whose induced homomorphic images are the Lie brackets in tensor algebra? We will give an answer to this question by defining a self map of $\Omega \Sigma K(\mathbb{Z},2d)$, and then by computing efficiently some self maps. We also study the topological rationalization properties of the suspension of the Eilenberg-MacLane spaces. These results will be playing a powerful role in the computation of the same $n$-type problems and giving us an information about the rational homotopy...

We describe, for any compact connected Lie group G and any prime p, the monoid of self maps $B{G}_{{}^{p}}$ → $B{G}_{{}^{p}}$ which are rational equivalences. Here, $B{G}_{{}^{p}}$ denotes the p-adic completion of the classifying space of G. Among other things, we show that two such maps are homotopic if and only if they induce the same homomorphism in rational cohomology, if and only if their restrictions to the classifying space of the maximal torus of G are homotopic.

This paper is a study of the Gray index of phantom maps. We give a new, tower theoretic, definition of the Gray index, which allows us to study the naturality properties of the Gray index in some detail. McGibbon and Roitberg have shown that if f* is surjective on rational cohomology, then the induced map on phantom sets is also surjective. We show that if f* is surjective just in dimension k, then f induces a surjection on a certain subquotient of the phantom set. If the condition...

We investigate the mapping class groups of diffeomorphisms fixing a frame at a point for general classes of 3-manifolds. These groups form the equivalent to the groups of large gauge transformations in Yang-Mills theories. They are also isomorphic to the fundamental groups of the spaces of 3-metrics modulo diffeomorphisms, which are the analogues in General Relativity to gauge-orbit spaces in gauge theories.

A phantom mapping h from a space Z to a space Y is a mapping whose restrictions to compact subsets are homotopic to constant mappings. If the mapping h is not homotopic to a constant mapping, one speaks of an essential phantom mapping. The definition of (essential) phantom pairs of mappings is analogous. In the study of phantom mappings (phantom pairs of mappings), of primary interest is the case when Z and Y are CW-complexes. In a previous paper it was shown that there are no essential phantom...

Two mappings from a CW-complex to a 1-dimensional CW-complex are homotopic if and only if their restrictions to finite subcomplexes are homotopic.