Cell-like Maps Research Articles

An infinite-dimensional phenomenon in finite-dimensional metric topology

We show that there are homotopy equivalences $h:N\to M$ between closed manifolds which are induced by cell-like maps $p:N\to X$ and $q:M\to X$ but which are not homotopic to homeomorphisms. The phenomenon is based on construction of cell-like maps that kill certain $\mathbb L$-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension $>6$ we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.

A Vietoris–Begle theorem for connective Steenrod homology theories and cell-like maps between metric compacta

We prove that a cell-like map f:X→Y between metric compacta induces isomorphisms f⁎:h¯⁎st(X)→h¯⁎st(Y) for any connective homology theory h.

Approximating resolutions by cell-like maps with codimension-three point inverses

Let X be a resolvable generalized n-manifold and M be a (PL) n-manifold, where n≥6. We claim every resolution from M to X can be approximated, arbitrarily closely, by cell-like maps with (n−3)-dimensional point preimages. The key instrument of the proof is the disjoint Pontryagin disks property.

Simultaneous Z/p-acyclic resolutions of expanding sequences

We prove the following Theorem: Let X be a nonempty compact metrizable space, let $l_1 \leq l_2 \leq...$ be a sequence of natural numbers, and let $X_1 \subset X_2 \subset...$ be a sequence of nonempty closed subspaces of X such that for each k in N, $dim_{Z/p} X_k \leq l_k < \infty$. Then there exists a compact metrizable space Z, having closed subspaces $Z_1 \subset Z_2 \subset...$, and a surjective cell-like map $\pi: Z \to X$, such that for each k in N, (a) $dim Z_k \leq l_k$, (b) $\pi (Z_k) = X_k$, and (c) $\pi | {Z_k}: Z_k \to X_k$ is a Z/p-acyclic map. Moreover, there is a sequence $A_1 \subset A_2 \subset...$ of closed subspaces of Z, such that for each k, $dim A_k \leq l_k$, $\pi|{A_k}: A_k\to X$ is surjective, and for k in N, $Z_k\subset A_k$ and $\pi|{A_k}: A_k\to X$ is a UV^{l_k-1}-map. It is not required that X be the union of all X_k, nor that Z be the union of all Z_k. This result generalizes the Z/p-resolution theorem of A. Dranishnikov, and runs parallel to a similar theorem of S. Ageev, R. Jimenez, and L. Rubin, who studied the situation where the group was Z.

Correcting Taylor's cell-like map

Correcting Taylor's cell-like map

Some remarks on configuration spaces

This paper studies the homotopy type of the configuration spaces $F_n(X)$ by introducing the idea of configuration spaces of maps. For every map $f: X \to Y$, the configuration space $F_n(f)$ is the space of configurations in $X$ that have distinct images in $Y$. We show that the natural maps $F_n(X) \leftarrow F_n(f) \rightarrow F_n(Y)$ are homotopy equivalences when $f$ is a proper cell-like map between $d$-manifolds. We also show that the best approximation to $X \mapsto F_n(X)$ by a homotopy invariant functor is given by the $n$-fold product map.

Desingularizing homology manifolds

We prove that if X n , n ≥ 6, is a compact ANR homology n-manifold, we can blow up the singularities of X to obtain an ANR homology n-manifold with the disjoint disks property. More precisely, we show that there is an ANR homology n-manifold Y with the disjoint disks property and a cell-like map /: Y → X.

Cell-like resolutions in the strongly countable [formula omitted]-dimensional case

Suppose that X is a nonempty compact metrizable space and X 1⊂ X 2⊂⋯ is a sequence of nonempty closed subspaces such that for each k∈ N , dim Z X k⩽k<∞ . We show that there exists a compact metrizable space Z, having closed subspaces Z 1⊂Z 2⊂⋯ , and a surjective cell-like map π :Z→X , such that for each k∈ N , (a) dim Z k ⩽ k, (b) π( Z k )= X k , and (c) π|Z k :Z k→X k is a cell-like map. Moreover, there is a sequence A 0⊂ A 1⊂⋯ of closed subspaces of Z such that for each k, Z k ⊂ A k , dim A k ⩽ k, π|A k :A k→X is surjective, and for k∈ N , π|A k :A k→X is a UV k−1 -map.

Homogeneity, terminality and some mapping problems

The collection of terminal subcontinua in the theory of homogeneous continua and some problems concerning atomic mapping, cell-like mapping connected with homogeneity are studied.

On Dranishnikov's cell-like resolution

We prove the following theorem: Theorem 1. For a compactum X with c- dim Z/p X⩽n and c- dim Z (q) X⩽n for some distinct prime numbers p,q, and c- dim Z X⩽n+1 , where n>1, there exists an (n+1)-dimensional compactum Z with c- dim Z/p Z⩽n , c- dim Z (q) Z⩽n and a cell-like map f :Z→X . Moreover, giving the following theorem, we note that Theorem 1 cannot be true in the case of n=1. Theorem 2. For every pair p,q of distinct prime numbers there exists an infinite-dimensional compactum X such that c- dim Z/p X=1 , c- dim Z (q) X=1 and c- dim Z X=2 .

A 4-dimensional 1-LCC Shrinking Theorem

This paper contains several shrinking theorems for decompositions of 4 -dimensional manifolds. Let f :M→X be a closed, cell-like mapping of a 4-manifold M onto a metric space X and let Y be a closed subset of X such that X−Y is a 4 -manifold and Y is locally simply co-connected in X . The main result states that f can be approximated by homeomorphisms if Y is a 1-dimensional ANR. The techniques of the proof also show that f can be approximated by homeomorphisms in case Y is an arbitrary 0-dimensional closed subset. Combining the two results gives the same conclusion in case Y contains a closed, 0-dimensional subset C such that Y−C is a 1 -dimensional ANR. The construction in the paper also gives a proof of a taming theorem for 1 -dimensional ANRs.

Countable dimensionality and dimension raising cell-like maps

We show that countable dimensionality is not preserved under hereditary shape equivalences between complete spaces if such maps can make “arbitrarily large dimension jumps” inside the class of countably dimensional compacta.

Cell-like maps and aspherical compacta

Cell-like maps and aspherical compacta

Universal finite-to-one map and universal countable dimensional spaces

There is a closed finite-to-one map t́f of a zero-dimensional, separable, metric absolute G δσ -set X onto a space Y such that for any closed, finite-to-one map f′: X′ → Y′ of separable, metric spaces, with dim X′ ⩽ 0, there exist embeddings i : X′ → X and j : Y′ → Y such that fi = jf. In particular, the space Y is universal for all separable metric spaces which are countable dimensional. We also show that finite-to-one maps produce naturally cell-like maps. Finally, using the method of absorbers we prove a topological characterization of the space σ × N>, where σ is Smirnov's universal strongly countable dimensional space and N is Nagata's universal countable dimensional space.

On cell-like mappings and the dimensional full-valuedness of compacta

On cell-like mappings and the dimensional full-valuedness of compacta

Universal Cell-Like Maps

The main results of the paper are the following: Theorem. Suppose $n \leq \infty$. There is a cell-like map $f:X \to Y$ of complete and separable metric spaces such that $\dim X \leq n$, and for any cell-like map $f’ :X’ \to Y’$ of (complete) separable metric spaces with $\dim X’ \leq n$ there exist (closed) embeddings $i:Y’ \to Y$ and $j:X’ \to {f^{ - 1}}(i(Y’ ))$ such that $fj = if’$. Corollary. Suppose $n < \infty$. There is a complete and separable metric space Y such that ${\dim _{\mathbf {Z}}}Y \leq n$, and any (complete) separable metric space $Y’$ with ${\dim _{\mathbf {Z}}}Y’ \leq n$ embeds as a (closed) subset of Y.

Real time video imaging of high-luminosity processes

We define shape bundles as a generalization of shape cell-bundles and prove that they are weak shape fibrations. We also show that shape cell-bundles coincide with cell-like maps and therefore with shape bundles in case the fibers are cells or the Hilbert cube.

𝐾-theory of Eilenberg-Mac Lane spaces and cell-like mapping problem

There exist cell-like dimension raising maps of 6 6 -dimensional manifolds. The existence of such maps is proved by using K K -theory of Eilenberg-Mac Lane complexes.

K-Theory of Eilenberg-Mac Lane Spaces and Cell-Like Mapping Problem

K-Theory of Eilenberg-Mac Lane Spaces and Cell-Like Mapping Problem

On 1-cycles and the finite dimensionality of homology 4-manifolds

IN HIS remarkable paper [ 121 A. N. DraniSnikov gave a method for constructing for every n 2 3 examples of infinite-dimensional compacta (i.e. compact metric spaces) X, with integral cohomological dimension c-dimzX, = n. It follows by a well-known result of R. D. Edwards [34] that there are therefore n-dimensional compacta Y, and cell-like surjections f.: Y. -+ X,, i.e. for every XE X,, the pre-image f;‘(x) has trivial shape. By the NiibelungPontrjagin embedding theorem [29] there exist embeddings 4.: Y, + IW2”+r which in turn yield upper semicontinuous decompositions G, of R2”+ ‘, with non-degeneracy set given by {4,/i *(x)IxeX,}, whose quotient spaces R 2n+‘/G contain X, and so are infinite dimenn sional. Thus cell-like maps can raise dimension on manifolds of dimensions 7 and above. Such phenomena arc impossible in Iw4 for q z$ 3. Cell-like images of topological qmanifolds are always Z-homology q-manifolds [36], and for these cohomoiogical and covering dimensions agree if q I; 3; for q 5 2 it is classical that homology q-manifolds are topological manifolds [37], for q = 3 see [353. Recently J. Dydak and J. J. Walsh [16] have shown that there exists an infinite-dimensional compactum X with c-dimzX = 2. The preceding argument shows that cell-like maps can also raise dimension on Iws and R6. For more on the cell-like mapping problem and its history, see the survey [25]. We are interested in dimension four, the remaining unsettled case of the cell-like mapping problem. A. N. DraniSnikov and E. V. Scepin conjectured Cl53 that cell-like maps cannot raise dimension on 4-manifolds. As evidence for this conjecture we prove: