Moore Space Research Articles

Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, vol. 78 (1980), pp. 429–435. - William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, vol. 273 (1982), pp. 365–373. - Alan Dow, Franklin D.

Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, vol. 78 (1980), pp. 429–435. - William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, vol. 273 (1982), pp. 365–373. - Alan Dow, Franklin D. Tall, and William A. R. Weiss. New proofs of the consistency of the normal Moore space conjecture I. Topology and its applications, vol. 37 (1990), pp. 33–51. - Zoltán Balogh. On collectionwise normality of locally compact, normal spaces. Transactions of the American Mathematical Society, vol. 323 (1991), pp. 389–411. - Volume 8 Issue 3

A Provisional Solution to the Normal Moore Space Problem

A Provisional Solution to the Normal Moore Space Problem

The Whitehead square of a lift of the Hopf map to a mod 2 Moore space

The Whitehead square of a lift of the Hopf map to a mod 2 Moore space

Compact‐calibres of regular and monotonically normal spaces

A topological space has calibre ω1 (resp., calibre (ω1, ω)) if every point‐countable (resp., point‐finite) collection of nonempty open sets is countable. It has compact‐calibre ω1 (resp., compact‐calibre (ω1, ω)) if, for every family of uncountably many nonempty open sets, there is some compact set which meets uncountably many (resp., infinitely many) of them. It has CCC (resp., DCCC) if every disjoint (resp., discrete) collection of nonempty open sets is countable. The relative strengths of these six conditions are determined for Moore spaces, regular first countable spaces, linearly‐ordered spaces, and arbitrary regular spaces. It is shown that the relative strengths for spaces with point‐countable bases are the same as those for Moore spaces, and the relative strengths for linearly‐ordered spaces are the same as those for arbitrary monotonically normal spaces.

Quasi-Metrizable Spaces Satisfying Certain Completeness Conditions

It is shown that a quasi-metrizable space possesses a $$\lambda $$ -base if and only if it admits a left $$K$$ -complete quasi-metric and that a quasi-metrizable Moore space possesses a $$\lambda $$ -base if and only if it admits a Smyth complete quasi-metric. Furthermore those quasi-metrizable Moore spaces are characterized that admit a right $$K$$ -complete (equivalently, bicomplete) quasi-metric. Finally, it is established that a topological space admits a Smyth complete quasi-uniformity if and only if it is quasi-sober.

On hereditary Baireness of the Vietoris topology

It is shown that a metrizable space X, with completely metrizable separable closed subspaces, has a hereditarily Baire hyperspace K(X) of nonempty compact subsets of X endowed with the Vietoris topology τ v . In particular, making use of a construction of Saint Raymond, we show in ZFC that there exists a non-completely metrizable, metrizable space X with hereditarily Baire hyperspace (K(X),τ v) ; thus settling a problem of Bouziad. Hereditary Baireness of (K(X),τ v) for a Moore space X is also characterized in terms of an auxiliary product space and the strong Choquet game. Finally, using a result of Kunen, a non-consonant metrizable space having completely metrizable separable closed subspaces is constructed under CH.

Homotopy classes that are trivial mod ℱ

If F is a collection of topological spaces, then a homotopy class \alpha in [X,Y] is called F-trivial if \alpha_* = 0: [A,X] --> [A,Y] for all A in F. In this paper we study the collection Z_F(X,Y) of all F-trivial homotopy classes in [X,Y] when F = S, the collection of spheres, F = M, the collection of Moore spaces, and F = \Sigma, the collection of suspensions. Clearly Z_\Sigma (X,Y) \subset Z_M(X,Y) \subset Z_S(X,Y), and we find examples of finite complexes X and Y for which these inclusions are strict. We are also interested in Z_F(X) = Z_F(X,X), which under composition has the structure of a semigroup with zero. We show that if X is a finite dimensional complex and F = S, M or \Sigma, then the semigroup Z_F(X) is nilpotent. More precisely, the nilpotency of Z_F(X) is bounded above by the F-killing length of X, a new numerical invariant which equals the number of steps it takes to make X contractible by successively attaching cones on wedges of spaces in F, and this in turn is bounded above by the F-cone length of X. We then calculate or estimate the nilpotency of Z_F(X) when F = S, M or \Sigma for the following classes of spaces: (1) projective spaces (2) certain Lie groups such as SU(n) and Sp(n). The paper concludes with several open problems.

Measurements on domains and topology

The concept of a measurement on a continuous partial order has been recently introduced by Keye Martin. Measurement provides a uniform degree of approximation for elements of the kernal, i.e., those elements with measure zero. Measurement also induces a Scott topology for elements “near” the kernal. The results below are joint work with Keye Martin.IX is developable and T_1 iff it is the kernel of a measurement on a continuous poset, X is developable T1 and choquet complete iff it is the kernal of a measurement on a continuous cpo.IIFor each developable T1-space X, there exists a developable T1-space M(X) with a poset order < such that (1) X is the kernal of a measurement. (2) the topology on M(X) is exactly the topology induced by the measurement. (3) If X is T_2, then M(X) is a Moore space. (4) If X is a complete (or even semi-complete) Moore space then (M(X),<) is a cpo. (5) If X is the real line, then M(X) is a non-normal Moore space. (6) (MA + not CH) If X is a subspace of the real line and omega < card(X) < c, then M(X) is a normal nonmetrizable Moore space.IIIThe countable ordinals with the order topology is the top of a Scott domain, and is a G_delta set with respect to the Scott topology on the domain. [This answers several questions open questions in the area.]IVFinally, we give a new recursion induction theorem for cpo's using measurement theory.

The Postnikov Tower and the Steenrod problem

The Steenrod problem asks: given a G-module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a ZG-module is ZG-realizable if and only if it is ZHrealizable for all p-Sylow subgroups H, for all primes pl GI. Let G be a finite group. Let M be a finitely generated ZG-module. We say that M is a Steenrod representation if there exists a Moore space X with G-action such that the homology of X is isomorphic to M as a ZG-module. Recall that a Moore space is a topological space whose reduced homology vanishes in all dimensions except one. Without loss of generality, M will be assumed 2-free [1] from now on. In this paper, the following statement will be proved: M is a Steenrod representation as a ZG-module if and only if M is a Steenrod representation as a 7H-module for all p-Sylow subgroups H of G, for all primes pII GIC. This statement is inspired by papers by J. Arnold [1, 2] and P. Vogel [18]. Arnold [2] showed that if G is cyclic, then every finitely generated ZG-module is a Steenrod representation. And Vogel showed that if for any ZG-module M, we can always find a G-Moore space X realizes M, then G has only cyclic Sylow subgroups. Let us first consider a Moore space X whose homology is isomorphic to M as a 2-module. Let G(X) denote the space of self homotopy equivalences of X and let BG(X) [9] denote the classifying space of the H-space G(X) for a certain fibration defined by Dold and Lashof [10]; see also Stasheff [15]. Since a homotopy equivalence of X induces an automorphism on M, there is a map G(X) -* Aut(M). And the map G(X) -* Aut(M) induces another map a : BG(X) -' BAut(M). On the other hand, since M is a ZG-module, there is a map G -* Aut(M) (= GL(n, 2) if M = Zn), which induces .p : BG -) BAut(M). Following Cooke's theory [9], as pointed out in [12], M is a Steenrod representation as a ZG-module if and only if the lifting f exists in

Preservation of completeness under mappings in asymmetric topology

&lt;p&gt;The preservation of various completeness properties in the quasi-metric (and quasi-uniform) setting under open, closed and uniformly open mappings is investigated. In particular, it is noted that between quasi-uniform spaces the property that each costable filter has a cluster point is preserved under uniformly open continuous surjections. Furthermore in the realm of quasi-uniform spaces conditions under which almost uniformly open mappings are uniformly open are given which generalize corresponding classical results for uniform spaces. As a by-product it is shown that a quasi-metrizable Moore space admits a left K-complete quasi-metric if and only if it is a complete Aronszajn space.&lt;/p&gt;

Lifting to mod 2 Moore spaces

We try to investigate whether an element of order 2 in homotopy groups of spheres has a lift to an element of homotopy groups of mod 2 Moore spaces or not. A typical element of the affirmative property is Thomeier's element of π8n+7(S4n+3). And the Whitehead square [l2n+1,l2n+1] of the identity class l2n+1 of S2n+1 is a negative example except for n=0,1 and 3.

James-Hopf Invariants, Anick’s Spaces, and the Double Loops on Odd Primary Moore Spaces

AbstractUsing spaces introduced by Anick, we construct a decomposition into indecomposable factors of the double loop spaces of odd primary Moore spaces when the powers of the primes are greater than the first power. If n is greater than 1, this implies that the odd primary part of all the homotopy groups of the 2n + 1 dimensional sphere lifts to a mod pr Moore space.

Monotone countable paracompactness

We study a monotone version of countable paracompactness, MCP, and of countable metacompactness, MCM. These properties are common generalizations of countable compactness and stratifiability and are shown to relate closely to the generalized metric g-functions of Hodel: MCM spaces coincide with β-spaces and, for q-spaces (hence first countable spaces) MCP spaces coincide with wN-spaces. A number of obvious questions are answered, for example: there are “monotone Dowker spaces” (monotonically normal spaces that are not MCP); MCP, Moore spaces are metrizable; first countable (or locally compact or separable) MCP spaces are collectionwise Hausdorff (in fact we show that wN-spaces are collectionwise Hausdorff). The extent of an MCP space is shown to be no larger than the density and the stability of MCP and MCM under various topological operations is studied.

Some results on property (a)

A space X has property (a) if for every open cover U of X and for each dense D⫅X there is a closed discrete F⫅D such that St(F, U)=X . In this paper, the relationship between property (a) and normality is investigated. A consistent example of a normal space without property (a) is constructed. A weakening of property (a) is defined and studied. Examples of nonmetrizable Moore spaces with property (a) are given. The question for which spaces X with property (a) the product X×(ω+1) also has property (a) is studied.

Sharp bases and weakly uniform bases versus point-countable bases

A base B for a topological space X is said to be sharp if for every x∈X and every sequence (U n) n∈ω of pairwise distinct elements of B with x∈U n for all n the set {⋂ i<nU i: n∈ω} forms a base at x. Sharp bases of T 0 -spaces are weakly uniform. We investigate which spaces with sharp bases or weakly uniform bases have point-countable bases or are metrizable. In particular, Davis, Reed, and Wage had constructed in a 1976 paper a consistent example of a Moore space with weakly uniform base, but without a point-countable base. They asked whether such an example can be constructed in ZFC. We partly answer this question by showing that under CH, every first-countable space with a weakly uniform base and at most ℵ ω isolated points has a point-countable base.

Mahowaldean families of elements in stable homotopy groups revisited

In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

A Moore space with a 𝜎-discrete 𝜋-base which cannot be densely embedded in any Moore space with the Baire property

The author answers a question raised in the literature about twenty five years ago and raised again more recently in Open Problems in Topology, by G. M. Reed, concerning the conjecture that every Moore space with a σ \sigma -discrete π \pi -base can be densely embedded in a Moore space having the Baire property. Even though closely related results have made this conjecture seem likely to be true, the author shows that, surprisingly, the conjecture is false.

A space with normal countable-to-one regular preimages

It is known that a space with every countable-to-one regular preimage normal must be at least ⩽ scc-cwN. We construct a model of ZFC + CH with a normal space X that has every countable-to-one regular preimage normal but is not ⩽ scc +-cwH, so that ZFC does not imply stronger separation properties for these spaces. X is also an example of a Moore space that is ⩽ ℵ 1- paracompact but not ⩽ ℵ 2- paracompact .

On the non-productivity of normality in Moore spaces

Under Martin’s Axiom and the denial of the Continuum Hypothesis, the authors give examples of normal Moore spaces whose squares are not normal.

A note on the homotopy type of ΩBSL 3(ℤ)and 2

It is known that for p-perfect groups G of finite virtual cohomological dimension and finite type mod-p cohomology, the p-completed classifying space BGandp has the property that ΩBGandp is a retract of the loop space on a simply-connected, [ ]p-finite, p-complete space. In this note we consider a particular example where this theorem applies, namely we study the homotopy type of BSL3(ℤ)and2. It particular we analyse ΩBSt3(ℤ)and2, a double cover of ΩBSL3(ℤ)and2, and obtain a splitting theorem for it in terms of 2-primary Moore spaces and fibres of degree 2r maps on spheres. We also give a formula for the Poincaré series of H∗(ΩBΓandp; [ ]p) for a general group Γ, as above, in terms of possibly simpler components. This formula is used to calculate the mod-2 homology of ΩBΓand2 for Γ=SL3(ℤ) or St3(ℤ) as modules over a certain tensor subalgebra.