Lim colim versus colim lim. II: Derived limits over a pospace

\v{C}ech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_\beta$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets $K_\alpha\subset X$. We show that one can also go vice versa: in a sense, $H^n(X)$ can be reconstructed from $H^n(K_\alpha)$, and if $X$ is finite dimensional, $H_n(X)$ can be reconstructed from $H_n(P_\beta)$. The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral sequence, except that the derived limits have to be "corrected" so as to take into account a natural topology on the indexing set. The corrected derived limits coincide with the usual ones when the topology is discrete, and in general are applied not to an inverse system but to a "partially ordered sheaf". The "correction" of the derived limit functors in turn involves constructing a "correct" (metrizable) topology on the order complex $|P|$ of a partially ordered metrizable space $P$ (such as the hyperspace $K(X)$ of nonempty compact subsets of $X$ with the Hausdorff metric). It turns out that three natural approaches (by using the space of measurable functions, the space of probability measures, or the usual embedding $K(X)\to C(X;\mathbb R)$) all lead to the same topology on $|P|$.Comment: 30 page

On maps with unstable singularities

If a continuous map f: X->Q is approximable arbitrary closely by embeddings X->Q, can some embedding be taken onto f by a pseudo-isotopy? This question, called Isotopic Realization Problem, was raised by Shchepin and Akhmet'ev. We consider the case where X is a compact n-polyhedron, Q a PL m-manifold and show that the answer is 'generally no' for (n,m)=(3,6); (1,3), and 'yes' when: 1) m>2n, (n,m)\neq (1,3); 2) 2m>3(n+1) and the set {(x,y)|f(x)=f(y)} has an equivariant (with respect to the factor exchanging involution) mapping cylinder neighborhood in X\times X; 3) m>n+2 and f is the composition of a PL map and a TOP embedding. In doing this, we answer affirmatively (with a minor preservation) a question of Kirby: does small smooth isotopy imply small smooth ambient isotopy in the metastable range, verify a conjecture of Kearton-Lickorish: small PL concordance implies small PL ambient isotopy in codimension \ge 3, and a conjecture set of Repovs-Skopenkov.Comment: 46 pages, 5 figures, to appear in Topol Appl; some important footnotes added in version

Metrizable uniform spaces

Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spaces and A is a closed subset of X, we show that the adjunction space X\cup_f Y with the quotient uniformity (hence also with the topology thereof) is metrizable, by an explicit metric. This yields natural constructions of cone, join and mapping cylinder in the category of metrizable uniform spaces, which we show to coincide with those based on subspace (of a normed linear space); on product (with a cone); and on the isotropy of the l_2 metric. 2) We revisit Isbell's theory of uniform ANRs, as refined by Garg and Nhu in the metrizable case. The iterated loop spaces \Omega^n P of a pointed compact polyhedron P are shown to be uniform ANRs. Four characterizations of uniform ANRs among metrizable uniform spaces X are given: (i) the completion of X is a uniform ANR, and the remainder is uniformly a Z-set in the completion; (ii) X is uniformly locally contractible and satisfies the Hahn approximation property; (iii) X is uniformly \epsilon-homotopy dominated by a uniform ANR for each \epsilon>0; (iv) X is an inverse limit of uniform ANRs with "nearly splitting" bonding maps.Comment: 93 pages. v5: a little bit of new stuff added. Proposition 8.7, entire section 10, Lemma 14.11, Proposition 18.4. Possibly something els

Lifting generic maps to embeddings

Given a generic PL map or a generic smooth fold map $f:N^n\to M^m$, where $m\ge n$ and $2(m+k)\ge 3(n+1)$, we prove that $f$ lifts to a PL or smooth embedding $N\to M\times\mathbb R^k$ if and only if its double point locus $(f\times f)^{-1}(\Delta_M)\setminus\Delta_N$ admits an equivariant map to $S^{k-1}$. As a corollary we answer a 1990 question of P. Petersen on whether the universal coverings of the lens spaces $L(p,q)$, $p$ odd, lift to embeddings in $L(p,q)\times\mathbb R^3$. We also show that if a non-degenerate PL map $N\to M$ lifts to a topological embedding in $M\times\mathbb R^k$ then it lifts to a PL embedding in there. The Appendix extends the 2-multi-0-jet transversality over the usual compactification of $M\times M\setminus\Delta_M$ and Section 3 contains an elementary theory of stable PL maps.Comment: 37 pages. v4: Added a discussion of stable PL maps (in Section 3) and the general case of the extended 2-multi-0-jet transversality theorem (in the end of the Appendix