Topology Proceedings 38 (2011) Pp. 29-82: on Homotopical and Homological $Z N$-Sets

Volume 38, 2011 Pages 29–82

http://topology.auburn.edu/tp/

On Homotopical and

Homological 푍푛-Sets

by Taras Banakh, Robert Cauty, and Alex Karassev

Electronically published on August 2, 2010

Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 38 (2011) Pages 29-82 E-Published on August 2, 2010

ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS

TARAS BANAKH, ROBERT CAUTY, AND ALEX KARASSEV

Abstract. A closed subset 퐴 ⊂ 푋 is called a homological (homotopical) 푍푛-set if for any 푘 < 푛 + 1 and any open set 푈 ⊂ 푋 the relative homology (homotopy) group 퐻푘(푈, 푈 ∖퐴) 푛 ( 휋푘(푈, 푈 ∖ 퐴) ) vanishes. A closed subset 퐴 of an LC -space 푋 is a homotopical 푍푛-set if and only if 퐴 is a 푍푛-set in 푋 in the sense that each map 푓 : [0, 1]푛 → 푋 can be uniformly approximated by maps into 푋∖퐴. Applying the Hurewicz isomorphism theorem, we prove that a homotopical 푍2-subset of 1 an LC -space is a homotopical 푍푛-set if and only if it is a homological 푍푛-set in 푋. From the K¨unnethformula we derive multiplication, division, and 푘-root formulas for homological ℤ 푍푛-sets. We prove that the set 풵푛(푋) of homological 푍푛- 푛 points of a metrizable separable 푙푐 -space 푋 is of type 퐺훿 in ℤ ℤ 푋. We introduce and study the classes 풵푛 (풵푛, respectively) ℤ ℤ of topological spaces 푋 with 풵푛(푋) = 푋 (풵푛(푋) = 푋, respectively) and prove multiplication, division, and 푘-root formulas for such classes. We also show that a (locally compact 푛 ℤ 푙푐 -)space 푋 ∈ 풵푛 has Steinke dimension t(푋) ≥ 푛 + 1 (has cohomological dimension dim퐺(푋) ≥ 푛 + 1 for any coeﬃcient ℤ group 퐺). A locally compact ANR-space 푋 ∈ 풵∞ is not a 퐶-space and has extension dimension e-dim푋 ∕≤ 퐿 for any non-contractible CW-complex 퐿.

In this paper we focus on applications of homological methods to studying 푍푛-sets in topological spaces. Being higher-dimensional

2010 Mathematics Subject Classiﬁcation. Primary 57N20; Secondary 54C50; 54C55; 54F35; 54F45; 55M10; 55M15; 55N10.

Key words and phrases. homological 푍푛-set, 푍푛-set. The substantial part of this paper was written during the stay of the first author at Nipissing University (North Bay, Ontario, Canada). ⃝c 2010 Topology Proceedings. 29 30 T. BANAKH, R. CAUTY, AND A. KARASSEV counterparts of closed nowhere dense subsets, 푍푛-sets are of crucial importance in infinite-dimensional and geometric topologies [6], [10], [16], [17], [31], [9] and play a role also in dimension theory [3] and theory of selections [39]. 푍푛-sets were introduced by H. Toruńczykin [35]. He defined a closed subset 퐴 of a topolog- 푛 ical space 푋 to be a 푍푛-set if any map 푓 : 퐼 → 푋 from the 푛-dimensional cube 퐼푛 = [0, 1]푛 can be uniformly approximated by 푛 maps into 푋 ∖ 퐴. Actually, 푍푛-sets work properly only in LC - spaces where they coincide with so-called homotopical 푍푛-sets. By definition, a closed subset 퐴 of a topological space 푋 is a homo- 푛 topical 푍푛-set if for any open cover 풰 of 푋, every map 푓 : 퐼 → 푋 can be approximated by a map 푓 ′ : 퐼푛 → 푋 ∖ 퐴, 풰-homotopic to 푓. In fact, homotopical 푍푛-sets are nothing else but closed locally 푛-negligible sets in the sense of Toruńczyk[35]. In section 3, we apply the Hurewicz isomorphism theorem to 1 characterize homotopical 푍푛-sets 퐴 in Tychonoff LC -spaces 푋 as homotopical 푍min{푛,2}-sets such that the relative homology groups 퐻푘(푈, 푈 ∖ 퐴) vanish for all 푘 < 푛 + 1 and all open sets 푈 ⊂ 푋. Having in mind this characterization of homotopical 푍푛-sets, we define a closed subset 퐴 of a topological space 푋 to be a homological 푍푛-set (more generally, a 퐺-homological 푍푛-set for a coefficient group 퐺) if 퐻푘(푈, 푈 ∖ 퐴) = 0 (퐻푘(푈, 푈 ∖ 퐴; 퐺) = 0, respectively) for all 푘 < 푛 + 1 and all open sets 푈 ⊂ 푋. Therefore, a homo- 1 topical 푍2-set in a Tychonoff LC -space 푋 is a homotopical 푍푛-set if and only if it is a homological 푍푛-set. It should be mentioned that under some restrictions on the space 푋 this characterization of 푍푛-sets has been exploited in mathematical literature [17, Propo- sition 4.2], [18], [28]. The homological characterization of 푍푛-sets makes it possible to apply powerful tools of algebraic topology for studying 푍푛-sets. Homological 푍푛-sets behave like usual 푍푛-sets: The union of two 퐺-homological 푍푛-sets is a 퐺-homological 푍푛-set and so is each closed subset of a 퐺-homological 푍푛-set. In section 4, applying the technique of irreducible homological barriers, we prove that a closed subset 퐴 ⊂ 푋 is a homological 푍푛-set in 푋 if each point 푎 ∈ 퐴 is a homological 푍푛-point in 푋 and each closed subset 퐵 ⊂ 퐴 with ∣퐵∣ > 1 can be separated by a homological 푍푛+1-set. This characterization makes it possible to apply Steinke’s separation dimension t(⋅) and its transfinite extension trt(⋅) to studying homological 푍푛-sets. In particular, we ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 31 prove that a closed subset 퐴 ⊂ 푋 with finite separation dimension 푑 = t(퐴) is a 퐺-homological 푍푛-set in 푋 if each point 푎 ∈ 퐴 is a 퐺-homological 푍푛+푑-point in 푋. An infinite version of this result asserts that a closed subset 퐴 ⊂ 푋 having transfinite separation dimension trt(퐴) is a 퐺-homological 푍∞-set in 푋 if and only if each point 푎 ∈ 퐴 is a 퐺-homological 푍∞-point in 푋. In section 5, we develop the Bockstein theory for 퐺-homological 푍푛-sets. The main result is Theorem 5.5 asserting that a subset 퐴 ⊂ 푋 is a 퐺-homological 푍푛-set in 푋 if and only if 퐴 is an 퐻-homological 푍푛-set for all groups 퐻 ∈ 휎(퐺), where 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝, 푅푝 : 푝 is prime} is the Bockstein family of 퐺. The main result of section 6 is Theorem 6.1: For a homological (homotopical) 푍푛-set 퐴 in a space 푋 and a homological (homotopical) 푍푚-set in 푌 the product 퐴 × 퐵 is a homological (homotopical) 푍푛+푚+1-set in 푋 × 푌 . For ANR’s the homotopical version of this result has been proved by T. O. Banakh and Kh. R. Trushchak in [8]. It is interesting that the multiplicative theorem for homological 푍푛-sets can be partly reversed, which leads to division and 푘-root theorems proved in section 8. In section 9, we apply the obtained results on 푍푛-sets to study 푍푛-points. We show that the set of 푍푛-points in a metrizable sepa- 푛 rable space 푋 always is a 퐺훿-set. Moreover, if 푋 is an LC -space, then the sets of homological and homotopical points also are 퐺훿 in 푋. In sections 10 and 11, we introduce and study the classes 풵푛, ℤ 풵푛 of spaces, all of whose points are homotopical (homological, ℤ respectively), 푍푛-points, and classes of spaces 풵푛 (풵푛, respectively) containing dense sets of homotopical (homological, respectively) 푍푛-points. Applying the results from the preceding sections, we prove multiplication, division, and 푘-root formulas for these classes. In sections 12 and 13, we study dimension properties of spaces ℤ from the class of 풵푛 spaces (i.e., all of whose points are homo- ℤ logical 푍푛-points). We show that for any space 푋 ∈ 풵푛 , the transfinite separation dimension trt(푋) ≥ 1 + 푛. If, in addition, 푋 is a locally compact LC푛-space, then 푋 has cohomological dimension dim퐺(푋) ≥ 푛 + 1 for any group 퐺. Also, the inequality e-dim(푋) ≤ 퐿 for a CW-complex 퐿 implies that 휋푘(퐿) = 0 for all ℤ 푘 ≤ 푛. Each locally compact locally contractible space 푋 ∈ 풵∞ is 32 T. BANAKH, R. CAUTY, AND A. KARASSEV infinite-dimensional in a rather strong sense: 푋 fails to be a 퐶-space and has extension dimension e-dim푋 ∕≤ 퐿 for any non-contractible CW-complex 퐿; see Theorem 13.1.

1. Preliminaries All topological spaces considered in this paper are Tychonoff; 퐼 stands for the closed interval [0, 1] and 푛 will denote a non-negative integer or infinity. In the paper we use singular homology 퐻∗(푋; 퐺) with coefficients in a non-trivial abelian group 퐺. If 퐺 = ℤ is the group of integers, we omit the symbol ℤ and write 퐻∗(푋) instead ˜ of 퐻∗(푋; ℤ). By 퐻∗(푋; 퐺), we denote the singular homology of 푋, reduced in dimension zero. Let 풰 be a cover of a space 푋. Two maps 푓, 푔 : 푍 → 푋 are called ∙ 풰-near (denoted by (푓, 푔) ≺ 풰) if for any 푧 ∈ 푍 there is 푈 ∈ 풰 with {푓(푧), 푔(푧)} ⊂ 푈; ∙ 풰-homotopic (denoted by 푓 ∼ 푔) if there is a homotopy 풰 ℎ : 푍 × [0, 1] → 푋 such that ℎ(푧, 0) = 푓(푧), ℎ(푧, 1) = 푔(푧), and ℎ({푧} × [0, 1]) ⊂ 푈 ∈ 풰 for all 푧 ∈ 푍. There is also a (pseudo)metric counterpart of these notions. Let 휌 be a continuous pseudometric on a space 푋 and 휀 > 0 be a real number. Two maps 푓, 푔 : 푍 → 푋 are called ∙ 휀-near if dist(푓, 푔) < 휀 where dist(푓, 푔) = sup 휌(푓(푧), 푔(푧)); 푧∈푍 ∙ 휀-homotopic if there is a homotopy ℎ : 푍 × [0, 1] → 푋 such that ℎ(푧, 0) = 푓(푧), ℎ(푧, 1) = 푔(푧), and diam휌ℎ({푧} × [0, 1]) < 휀 for all 푧 ∈ 푍. The following easy lemma helps to reduce the “cover” version of (homotopical) nearness to the “pseudometric” one. Lemma 1.1. For any open cover 풰 of a Tychonoff space 푋 and any compact set 퐾 ⊂ 푋 there is a continuous pseudometric 휌 on 푋 such that each 1-ball 퐵(푥, 1) = {푥′ ∈ 푋 : 휌(푥, 푥′) < 1} centered at a point 푥 ∈ 퐾 lies in some set 푈 ∈ 풰. Proof: Embed the Tychonoff space 푋 into a Tychonoff cube 퐼휅 for a suitable cardinal 휅. For each 푥 ∈ 퐾, find a finite index 퐹 (푥) set 퐹 (푥) ⊂ 휅 and an open subset 푊푥 ⊂ 퐼 whose preimage ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 33

−1 퐹 (푥) 푉푥 = pr퐹 (푥)(푊푥) under the projection pr퐹 (푥) : 푋 → 퐼 contains the point 푥 and lies in some 푈 ∈ 풰. By the compactness of 퐾, the open cover {푉푥 : 푥 ∈ 퐾} of 퐾 contains a∪ finite subcover {푉 , . . . , 푉 }. Now consider the finite set 퐹 = 푚 퐹 (푥 ) 푥1 푥푚 푖=1 푖 and note that each set 푉푥푖 is the preimage of the some open set 퐹 퐹 푊푖 ⊂ 퐼 under the projection pr퐹 : 푋 → 퐼 . Let 푑 be any metric on the finite-dimensional cube 퐼퐹 . By the compactness of ∪푚 퐶 = pr퐹 (퐾) ⊂ 푖=1 푊푖, there is 휀 > 0 such that each 휀-ball centered at a point 푧 ∈ 퐶 lies in some 푊푖. Finally, define the pseudo- ′ 1 ′ ′ metric 휌 on 푋 letting 휌(푥, 푥 ) = 휀 ⋅푑(pr퐹 (푥), pr퐹 (푥 )) for 푥, 푥 ∈ 푋. It is easy to see that each 1-ball centered at any point 푥 ∈ 퐾 lies in some 푈 ∈ 풰. □

Let us recall that a space 푋 is called an LC푛-space if for each point 푥 ∈ 푋, each neighborhood 푈 of 푥, and each 푘 < 푛 + 1 there is a neighborhood 푉 ⊂ 푈 of 푥 such that each map 푓 : ∂퐼푘 → 푉 from the boundary of the 푘-dimensional cube extends to a map 푓 : 퐼푘 → 푈. The following homotopy approximation theorem for LC푛-spaces is well known; see [26, p. 159] or [9].

Lemma 1.2. For any open cover 풰 of a paracompact LC푛-space 푋 and any 푘 < 푛 + 1, there is an open cover 풱 of 푋 such that any two 풱-near maps 푓, 푔 : 퐾 → 푋 from a simplicial complex 퐾 of dimension dim 퐾 ≤ 푘 are 풰-homotopic.

This lemma has a homological counterpart. A space 푋 is deﬁned to be an 푙푐푛-space if for each point 푥 ∈ 푋, each neighborhood 푈 of 푥, and each 푘 < 푛 + 1, there is a neighborhood 푉 ⊂ 푈 of ˜ ˜ 푥 such that the homomorphism 푖∗ : 퐻푘(푉 ) → 퐻푘(푈) of singular homologies induced by the inclusion map 푖 : 푉 → 푈 is trivial. It is known that each LC푛-space is an 푙푐푛-space; see [38]. The converse is true for LC1-spaces; see [38]. The proof of the following lemma can be found in [11, Lemma 5.4]

Lemma 1.3. For any paracompact 푙푐푛-space 푋 and any 푘 < 푛 + 1 there is an open cover 풰 of 푋 such that any two 풰-near maps 푓, 푔 : 퐾 → 푋 deﬁned on a simplicial complex 퐾 of dimension dim 퐾 ≤ 푘 induce the same homomorphisms 푓∗, 푔∗ : 퐻∗(퐾) → 퐻∗(푋) on homologies. 34 T. BANAKH, R. CAUTY, AND A. KARASSEV

In the sequel, we shall need another three homological properties of 푙푐푛-spaces. First, we recall one well-known fact from homological algebra; see [12, Lemma 16.3]. If the commutative diagram in the category of abelian groups 퐴 −−−−→ 퐴 ⏐2 ⏐3 ⏐ ⏐ 푖2y 푖3y 퐵 −−−−→ 퐵 −−−−→ 퐵 ⏐1 ⏐2 3 ⏐ ⏐ 푖1y 푖4y

퐶1 −−−−→ 퐶2 has exact middle row, then the finite generacy (or triviality) of the groups 푖1(퐵1) and 푖3(퐴3) implies the finite generacy (triviality) of the group 푖4 ∘ 푖2(퐴2). A subset 퐴 of a topological space 푋 is called precompact if 퐴 has compact closure in 푋. Lemma 1.4. Any precompact set 퐶 in a Tychonoff 푙푐푛-space 푋 with 푛 < ∞ has an open neighborhood 푈 such that the inclusion homomorphisms 퐻푘(푈) → 퐻푘(푋) have finitely-generated image for all 푘 ≤ 푛. Proof: We shall prove this lemma by induction on 푛. For 푛 = 0, the assertion of the lemma follows from the local connectedness of 푙푐0-spaces. Assume that for some 푛 the lemma has been proved for all 푘 < 푛. Consider the family 풦푛 of compact subsets 퐾 of 푋 having an open neighborhood 푈 such that the inclusion homomorphism 퐻푛(푈) → 퐻푛(푋) has finitely generated range. We claim that for any compact subsets 퐴, 퐵 ∈ 풦푛, the union 퐴 ∪ 퐵 ∈ 풦푛. To show this, fix open neighborhoods 푂퐴 and 푂퐵 of the compacta 퐴 and 퐵 such that the inclusion homomorphisms 퐻푛(푂퐴) → 퐻푛(푋) and 퐻푛(푂퐵) → 퐻푛(푋) have finitely generated ranges. By the inductive assumption, the compact set 퐴 ∩ 퐵 has a neighborhood 푂퐴∩퐵 ⊂ 푂퐴 ∩ 푂퐵 such that the inclusion homomorphism 퐻푛−1(푂퐴∩퐵) → 퐻푛−1(푂퐴 ∩ 푂퐵) has finitely generated range. Using the complete regularity of 푋, find two open neighborhoods 푈퐴 ⊂ 푂퐴 and 푈퐵 ⊂ 푂퐵 of the compacta 퐴 and 퐵 such that 푈퐴 ∩ 푈퐵 ⊂ 푂퐴∩퐵. The proof will be completed as soon as we check that the inclusion homomorphism 퐻푛(푈퐴∪푈퐵) → 퐻푛(푋) has ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 35

ﬁnitely-generated range. This follows from the diagram whose rows are Mayer-Vietoris sequences of the pairs {푈퐴, 푈퐵}, {푂퐴, 푂퐵}, and {푋, 푋}: / / 퐻푛(푈퐴) ⊕ 퐻푛(푈퐵) 퐻푛(푈퐴 ∪ 푈퐵) 퐻푛−1(푈퐴 ∩ 푈퐵)

푖2 푖3 / / 퐻푛(푂퐴) ⊕ 퐻푛(푂퐵) 퐻푛(푂퐴 ∪ 푂퐵) 퐻푛−1(푂퐴 ∩ 푂퐵)

푖1 푖4 / / 퐻푛(푋) ⊕ 퐻푛(푋) 퐻푛(푋 ∪ 푋) 퐻푛−1(푋 ∩ 푋)

Since the homomorphisms 푖1 and 푖3 have finitely generated ranges, so does the homomorphism 푖4 ∘ 푖2 : 퐻푛(푈퐴 ∪ 푈퐵) → 퐻푛(푋). This proves the additivity of the family 풦푛. To finish the proof of the lemma, it remains to check that each 푛 compact subset 퐶 of 푋 belongs to the family 풦푛. By the 푙푐 - property of 푋, each point 푥 ∈ 푋 has an open neighborhood 푂푥 ⊂ 푋 such that the inclusion homomorphism 퐻푛(푂푥) → 퐻푛(푋) has finitely generated range. Take any closed neighborhood 퐶푥 ⊂ 푂푥 of 푥 in the compact subset 퐶. It is clear that 퐶푥 ∈ 풦푛 for all 푥 ∈ 퐶. By the compactness of 퐶, the cover {퐶푥 : 푥 ∈ 퐶} has finite subcover 퐶 , . . . , 퐶 . Now the additivity of the family 풦 implies ∪푥1 푥푚 푛 that 퐶 = 퐶 ∈ 풦 . □ 푖≤푚 푥푖 푛 Lemma 1.5. Let 푋 be a locally compact 푙푐푛-space and 푉 ⊂ 푈 be open subsets of 푋 such that 푉 ⊂ 푈 and 푈 is compact. Then for any ¯ ¯ 푘 ≤ 푛 the inclusion homomorphism 퐻푘(푋, 푋 ∖ 푈) → 퐻푘(푋, 푋 ∖ 푉 ) has finitely generated range.

Proof: Take open sets 푊1 ⊂ 푊2 ⊂ 푊3 ⊂ 푋 such that 푊3 has ¯ compact closure in 푋 and 푈 ⊂ 푊1 ⊂ 푊 1 ⊂ 푊2 ⊂ 푊 2 ⊂ 푊3. The excision property for singular homology (see [25, Theorem ¯ 2.20]) implies that the inclusion homomorphisms 퐻푘(푊1, 푊1∖푈) → ¯ ¯ ¯ 퐻푘(푋, 푋 ∖ 푈) and 퐻푘(푊3, 푊3 ∖ 푉 ) → 퐻푘(푋, 푋 ∖ 푉 ) are isomor- phisms. Thus, it suﬃces to check that the inclusion homomorphism ¯ ¯ 퐻푘(푊1, 푊1 ∖ 푈) → 퐻푘(푊3, 푊3 ∖ 푉 ) has ﬁnitely generated range. This will be done with the help of the commutative diagram whose ¯ ¯ rows are the exact sequences of the pairs (푊1, 푊1∖푈), (푊2, 푊2∖푉 ), ¯ and (푊3, 푊3 ∖푉 ) and whose columns are inclusion homomorphisms in homologies: 36 T. BANAKH, R. CAUTY, AND A. KARASSEV

퐻 (푊 , 푊 ∖ 푈¯) −−−−→ 퐻 (푊 ∖ 푈¯) 푘 1 ⏐ 1 푘−1 ⏐ 1 ⏐ ⏐ 푖2y 푖3y 퐻 (푊 ) −−−−→ 퐻 (푊 , 푊 ∖ 푉¯ ) −−−−→ 퐻 (푊 ∖ 푉¯ ) 푘 ⏐ 2 푘 2 ⏐ 2 푘−1 2 ⏐ ⏐ 푖1y 푖4y ¯ 퐻푘(푊3) −−−−→ 퐻푘(푊3, 푊3 ∖ 푉 )

Lemma 1.4 implies that the homomorphisms 푖1 and 푖3 have finitely ¯ generated ranges (because 푊2 and 푊1 ∖ 푈 have compact closures ¯ in 푊3 and 푊2 ∖ 푉 , respectively). Consequently, the inclusion ho- ¯ ¯ momorphism 푖4 ∘ 푖2 : 퐻푘(푊1, 푊1 ∖ 푈) → 퐻푘(푊3, 푊3 ∖ 푉 ) also has finitely generated range. □ Lemma 1.6. Let 푋 be a locally compact 푙푐푛-space and 푥 be a point with 퐻푘(푋, 푋 ∖ {푥}) = 0 for some 푘 ≤ 푛. Then for any neighborhood 푈 ⊂ 푋 of 푥 there is a neighborhood 푉 ⊂ 푈 of 푥 such that the ¯ ¯ inclusion homomorphism 퐻푘(푋, 푋 ∖ 푈) → 퐻푘(푋, 푋 ∖ 푉 ) is trivial. Proof: Without loss of generality, the neighborhood 푈 has compact closure in 푋. By Lemma 1.5, there is a neighborhood 푊 ⊂ 푈 ¯ of 푥 such that the inclusion homomorphism 푖푈푊 : 퐻푘(푋, 푋∖푈) → ¯ 퐻푘(푋, 푋∖푊 ) has finitely generated range Im(푖푈푊 ). Pick up finitely many generators 푔1, . . . , 푔푚 of the group Im(푖푈푊 ). The triviality of the homology group 퐻푘(푋, 푋 ∖ {푥}) implies the triviality of the in- ¯ clusion homomorphism 푗 : 퐻푘(푋, 푋 ∖ 푊 ) → 퐻푘(푋, 푋 ∖ {푥}). Then for every 푖 ≤ 푚, the image 푗(푔푖) = 0 and we can find a neighborhood 푉푖 ⊂ 푊 of 푥 such that the image of 푔푖 under the inclusion homomor- ¯ phism 퐻푘(∩푋, 푋 ∖ 푊 ) → 퐻푘(푋, 푋 ∖ 푉푖) is trivial. For the neighborhood 푉 = 푖≤푚 푉푖 of 푥, the elements 푔1, . . . , 푔푚 have trivial images ¯ under the homomorphism 푖푊 푉 : 퐻푘(푋, 푋 ∖ 푊 ) → 퐻푘(푋, 푋 ∖ 푉 ). Since these elements generate the group Im(푖푈푊 ), the inclusion ho- ¯ ¯ momorphism 푖푈푉 = 푖푊 푉 ∘ 푖푈푊 : 퐻푘(푋, 푋 ∖ 푈) → 퐻푘(푋, 푋 ∖ 푉 ) is trivial. □

2. Characterizing locally 푛-negligible sets In this section we present a homological characterization of locally 푛-negligible sets. Following Toru´nczyk[35], we deﬁne a subset ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 37

퐴 of a space 푋 to be locally 푛-negligible if, given 푥 ∈ 푋, 푘 < 푛 + 1, and a neighborhood 푈 of 푥, there is a neighborhood 푉 ⊂ 푈 of 푥 such that for each 푓 :(퐼푘, ∂퐼푘) → (푉, 푉 ∖ 퐴), there is a homotopy 푘 푘 푘 (ℎ푡):(퐼 , ∂퐼 ) → (푈, 푈 ∖ 퐴) with ℎ0 = 푓 and ℎ1(퐼 ) ⊂ 푈 ∖ 퐴. Following [33], we shall say that a pair (푋, 퐴) is 푛-connected if 휋푖(푋, 퐴) = 0 for all 푖 ≤ 푛. Theorem 2.1. For a subset 퐴 of a Tychonoff space 푋, the following conditions are equivalent. (1) 퐴 is locally 푛-negligible; (2) given a simplicial pair (퐾, 퐿) with dim(퐾) ≤ 푛, a continuous pseudometric 휌 on 푋 and maps 휀 : ∣퐾∣ → (0, ∞) and 푓 : ∣퐾∣×{0}∪∣퐿∣×퐼 → 푋 with 휌(푓(푥, 0), 푓(푥, 푡)) < 휀(푥) and 푓(푥, 1) ∈/ 퐴 for all (푥, 푡) ∈ ∣퐿∣ × 퐼, there is 푓¯ : ∣퐾∣ × 퐼 → 푋 which extends 푓 and satisfies 휌(푓¯(푥, 푡), 푓(푥, 0)) < 휀(푥) and 푓¯(푥, 1) ∈/ 퐴 for all (푥, 푡) ∈ ∣퐾∣ × 퐼; (3) for each open set 푈 ⊂ 푋 and 푘 < 푛 + 1, the relative homotopy group 휋푘(푈, 푈 ∖ 퐴) vanishes; (4) each 푥 ∈ 푋 has a basis 픘푥 of open neighborhoods with 휋푘(푈, 푈 ∖ 퐴) = 0 for all 푈 ∈ 픘푥 and 푘 < 푛 + 1. If, in addition, 푋 is an LC1-space and 퐴 is locally 2-negligible in 푋, then conditions (1)–(4) are equivalent to (5) for each open 푈 ⊂ 푋 and 푘 < 푛 + 1, the relative homology group 퐻푘(푈, 푈 ∖ 퐴) vanishes; (6) each 푥 ∈ 푋 has a basis 픘푥 of open neighborhoods with 퐻푘(푈, 푈 ∖ 퐴) = 0 for all 푈 ∈ 픘푥 and 푘 < 푛 + 1. Proof: The equivalences (1) ⇔ (2) ⇔ (3) ⇔ (4) have been proved by Toruńczyk[35, Theorem 2.3] for normal spaces 푋. But because of Lemma 1.1, his proof also works for any Tychonoff 푋. Now assume that 퐴 is a locally 2-negligible set in an LC1-space 푋. The implication (5) ⇒ (6) is trivial while (3) ⇒ (5) easily follows from the Hurewicz isomorphism theorem [33, Ch. 7, Sec. 5, The- orem 4] (see also [25, Theorem 4.37]) because 푋 is an LC0-space and 퐴 is locally 1-negligible in 푋. It remains to prove the implication (6) ⇒ (1). Take any point 푥 ∈ 푋 and a neighborhood 푈 ⊂ 푋 of 푥. We lose no generality assuming that 푈 is connected. 38 T. BANAKH, R. CAUTY, AND A. KARASSEV

The LC1-property of 푋 yields a neighborhood 푉 ⊂ 푈 of 푥 such that any map 푓 : ∂퐼2 → 푉 extends to a map 푓¯ : 퐼2 → 푈. Moreover, by (6) we can choose the neighborhood 푉 so that 퐻푘(푉, 푉 ∖ 퐴) = 0 for all 푘 < 푛+1. Replacing 푉 by a connected component containing the point 푥, if necessary, we may assume that 푉 is connected. Then the LC0-property of 푋 implies that 푉 is path connected and the local 1-negligibility of 퐴 in 푋 implies that 푉 ∖ 퐴 is path-connected too. We claim that every map 푓 :(퐼푘, ∂퐼푘) → (푉, 푉 ∖퐴) with 푘 < 푛+1 is homotopic in 푈 to a map 푓˜ :(퐼푘, ∂퐼푘) → (푈 ∖ 퐴, 푈 ∖ 퐴) which will imply the local 푛-negligibility of 퐴 in 푋. This will be done by induction. For 푘 ≤ 2, the local 푘-negligibility of 퐴 in 푋 follows from the local 2-negligibility and there is nothing to prove. So we assume that the local (푘 − 1)-negligibility of 퐴 has been proved for some 푘 > 2. Then the implication (1) ⇒ (3) yields 휋푖(푉, 푉 ∖퐴) = 0 for all 푖 < 푘. This means that the pair (푉, 푉 ∖퐴) is (푘−1)-connected, which makes legal applying the relative Hurewicz isomorphism theorem [33, Ch. 7, Sec. 5, Theorem 4] to this pair. 푘 Fix any point ∗ ∈ ∂퐼 and let 푥0 = 푓(∗). Then the map 푓 can be considered as an element of the relative homotopy group 휋푘(푉, 푉 ∖ 퐴, ∗). The relative Hurewicz isomorphism theorem ap- ′ plied to the pair (푉, 푉 ∖ 퐴) implies that 휋푘(푉, 푉 ∖ 퐴, 푥0) = 0, ′ where 휋푘(푉, 푉 ∖ 퐴, 푥0) is the quotient group of the homotopy group 휋푘(푉, 푉 ∖ 퐴, 푥0) by the normal subgroup 퐺 generated by the elements [훾] ⋅ [훼] − [훼], [훼] ∈ 휋푘(푉, 푉 ∖ 퐴, 푥0), and [훾] ∈ 휋1(푉 ∖ 퐴, 푥0), where ⋅ : 휋1(푉 ∖ 퐴, 푥0) × 휋푘(푉, 푉 ∖ 퐴, 푥0) → 휋푘(푉, 푉 ∖ 퐴, 푥0) is the left action of the fundamental group on the relative 푘-th homotopy group, see [33, Ch. 7, Sec. 2] or [25]. It follows from ′ 0 = 휋푘(푉, 푉 ∖ 퐴, 푥0) = 휋푘(푉, 푉 ∖ 퐴, 푥0)/퐺 that the subgroup 퐺 coincides with 휋푘(푉, 푉 ∖ 퐴, 푥0). Now consider the homomorphism 푖∗ : 휋푘(푉, 푉 ∖ 퐴, 푥0) → 휋푘(푈, 푈 ∖ 퐴, 푥0) induced by the inclusion of pairs 푖 :(푉, 푉 ∖ 퐴) → (푈, 푈 ∖ 퐴). We claim that 푖∗ is a null-homomorphism. Since the group 휋푘(푉, 푉 ∖ 퐴, 푥0) is generated by the elements [훾] ⋅ [훼] − [훼], [훼] ∈ 휋푘(푉, 푉 ∖ 퐴, 푥0), and [훾] ∈ 휋1(푉 ∖ 퐴, 푥0), it suﬃces to check that 푖∗([훾] ⋅ [훼] − [훼]) = 0 for any such [훼] and [훾]. Let 푗∗ : 휋1(푉 ∖ 퐴, 푥0) → 휋1(푈 ∖ 퐴, 푥0) be the homomorphism induced by the inclusion 푗 :(푉, 푥0) → (푈, 푥0). The local 2-negligibility of 퐴 in 푈 and the choice of the set 푉 implies that ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 39

푗∗ = 0 and, thus, the action of the element 푗∗([훾]) ∈ 휋1(푈 ∖ 퐴, 푥0) on 휋푘(푈, 푈 ∖ 퐴, 푥0) is trivial. The naturality of the action of the fundamental group on the homotopy groups (see [33, Ch. 7, Sec. 3, Lemma 1]) implies that

푖∗([훾] ⋅ [훼] − [훼]) = 푗∗([훾]) ⋅ 푖∗([훼]) − 푖∗([훼]) = 푖∗([훼]) − 푖∗([훼]) = 0.

Thus, the homomorphism 푖∗ : 휋푘(푉, 푉 ∖퐴, 푥0) → 휋푘(푈, 푈 ∖퐴, 푥0) 푘 푘 is trivial, which means that the map 푓 :(퐼 , ∂퐼 , ∗) → (푉, 푉 ∖퐴, 푥0) is null homotopic in (푈, 푈 ∖ 퐴, 푥0), and this completes the proof of the local 푘-negligibility of 퐴 in 푋. □

3. Homotopical and homological 푍푛-sets

In this section, in order to study 푍푛-sets, we apply the characterization of locally 푛-negligible sets established in the preceding section. We recall the definition of a 푍푛-set and its homotopical and homological versions. Definition 3.1. A closed subset 퐴 of a topological space 푋 is defined to be 푛 ∙ a 푍푛-set if, for any open cover 풰 of 푋 and a map 푓 : 퐼 → 푋, there is a map 푓 ′ : 퐼푛 → 푋 ∖ 퐴 which is 풰-near to 푓; ∙ a homotopical 푍푛-set in 푋 if, for any open cover 풰 of 푋 and any map 푓 : 퐼푛 → 푋, there is a map 푓 ′ : 퐼푛 → 푋 ∖ 퐴 which is 풰-homotopic to 푓; ∙ a 퐺-homological 푍푛-set in 푋, where 퐺 is a coefficient group, if, for any open set 푈 ⊂ 푋 and any 푘 < 푛 + 1, the relative homology group 퐻푘(푈, 푈 ∖ 퐴; 퐺) = 0; ∙ a homological 푍푛-set in 푋 if it is a ℤ-homological 푍푛-set in 푋.

A point 푥 in a space 푋 is called a 푍푛-point if the singleton {푥} is a 푍푛-set in 푋. By analogy we deﬁne homotopical and homological 푍푛-points. The following theorem reveals interplay between various versions of 푍푛-sets. Theorem 3.2. Let 퐺 be a non-trivial Abelian group and 푋 be a Tychonoﬀ space.

(1) A subset 퐴 of 푋 is a homotopical 푍푛-set in 푋 if and only if 퐴 is closed and locally 푛-negligible set in 푋. 40 T. BANAKH, R. CAUTY, AND A. KARASSEV

(2) Each homotopical 푍푛-set in 푋 is a 푍푛-set in 푋. 푛 (3) If 푋 is an LC -space, then each 푍푛-set in 푋 is a homotopical 푍푛-set. (4) Each homotopical 푍푛-set in 푋 is a 퐺-homological 푍푛-set. (5) Each 퐺-homological 푍0-set in 푋 is a homotopical 푍0-set. (6) Each 퐺-homological 푍1-set in 푋 is a 푍1-set. Proof: The ﬁrst item follows immediately from Theorem 2.1 while the second is trivial. The third item was proved in [35, Corol- lary 3.3] and follows from Lemma 1.2. The fourth item can be easily derived from [13, Corollary 10.6] (of the relative Hurewicz theorem). The ﬁfth item follows from the fact that a relative homology group 퐻0(푈, 푈 ∖ 퐴; 퐺) vanishes if and only if each path-connected component of 푈 meets the set 푈 ∖ 퐴.

To prove the sixth item, assume that 퐴 is a 퐺-homological 푍1- set in 푋. Being a 퐺-homological 푍0-set, 퐴 is a homotopical 푍0-set in 푋. To show that 퐴 is a 푍1-set in 푋, ﬁx an open cover 풰 of 푋 and a map 푓 : 퐼 → 푋. Consider the open cover 푓 −1(풰) = {푓 −1(푈): 푈 ∈ 풰} of the interval 퐼 = [0, 1] and ﬁnd a sequence 0 = 푡0 < 푡1 < ⋅ ⋅ ⋅ < 푡푚 = 1 such that for each 푖 ≤ 푚, the interval −1 [푡푖−1, 푡푖] lies in 푓 (푈푖) for some set 푈푖 ∈ 풰. Let 푊푚 = 푈푚 and 푊푖 = 푈푖 ∩ 푈푖+1 for 푖 < 푚. Since 퐻0(푊푖, 푊푖 ∖ 퐴; 퐺) = 0, the path-connected component of 푊푖 containing the point 푓(푡푖) meets the set 푊푖 ∖ 퐴 at some point 푥푖. We claim that the points 푥푖−1 and 푥푖 lie in the same path- connected component of 푈푖 ∖ 퐴. Assuming the converse, we would get a nontrivial cycle 훼 = 푔 ⋅ 푥푖−1 − 푔 ⋅ 푥푖 in 퐻0(푈푖 ∖ 퐴; 퐺) with 푔 ∈ 퐺 being any non-zero element. On the other hand, this cycle is the boundary of an obvious 1-chain 훽 in 푈푖 and thus vanishes in the homology group 퐻0(푈푖; 퐺). But this contradicts the exact sequence 0 = 퐻1(푈푖, 푈푖 ∖ 퐴; 퐺) → 퐻0(푈푖 ∖ 퐴; 퐺) → 퐻0(푈푖; 퐺) for the pair (푈푖, 푈푖 ∖ 퐴). Therefore 푥푖−1 and 푥푖 lie in the same path-connected component of 푈푖 ∖퐴, ensuring the existence of a continuous map 푔푖 :[푡푖−1, 푡푖] → 푈푖 ∖ 퐴 with 푔푖(푡푖−1) = 푥푖−1 and 푔푖(푡푖) = 푥푖. The maps 푔푖 and 푖 ≤ 푚 compose a single continuous map 푔 : [0, 1] → 푋 ∖ 퐴 which is 풰-near to 푓, witnessing the 푍1-set property of 퐴. □ Combining Theorem 3.2(1) with Theorem 2.1, we get the following important characterization of homotopical 푍푛-sets. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 41

1 Theorem 3.3. A homotopical 푍2-set 퐴 in a Tychonoﬀ LC -space 푋 is a homotopical 푍푛-set in 푋 if and only if 퐴 is a homological 푍푛-set in 푋. Remark 3.4. Theorem 3.3 has been known as folklore, and its particular cases have appeared in literature; see e.g., [28], [18], [17]. It should also be mentioned that a substantial part of [17] is devoted to closed sets of inﬁnite codimension (coinciding with our homological 푍∞-sets). Next, we establish some elementary properties of 퐺-homological 푍푛-sets. From now on, 퐺 is a non-trivial abelian group.

Proposition 3.5. Let 퐴 and 퐵 be 퐺-homological 푍푛-sets in a topological space 푋.

(1) Any closed subset 퐹 ⊂ 퐴 is a 퐺-homological 푍푛-set in 푋. (2) The union 퐴 ∪ 퐵 is a 퐺-homological 푍푛-set in 푋.

Proof: (1) We should check that 퐻푘(푈, 푈 ∖퐹 ; 퐺) = 0 for all open sets 푈 ⊂ 푋 and all 푘 < 푛+1. The 퐺-homology 푍푛-set property of 퐴 yields 퐻푘(푈, 푈 ∖퐴; 퐺) = 0. Since (푈 ∖퐹, 푈 ∖퐴) = (푈 ∖퐹, (푈 ∖퐹 )∖퐴), we get also 퐻푘−1(푈 ∖ 퐹, 푈 ∖ 퐴; 퐺) = 0. Writing the exact sequence of the triple (푈, 푈 ∖ 퐹, 푈 ∖ 퐴) gives us 퐻푘(푈, 푈 ∖ 퐹 ; 퐺) = 0. (2) Given any 푘 < 푛 + 1 and any open set 푈 ⊂ 푋 consider the exact sequence of the triple (푈, 푈 ∖ 퐴, 푈 ∖ (퐴 ∪ 퐵)):

0 = 퐻푘(푈∖퐴, 푈∖(퐴∪퐵); 퐺) → 퐻푘(푈, 푈∖(퐴∪퐵); 퐺) → 퐻푘(푈, 푈∖퐴; 퐺) = 0 and conclude that 퐻푘(푈, 푈 ∖ (퐴 ∪ 퐵); 퐺) = 0. □

Next, we show that in the definition of a 퐺-homological 푍푛-set we can require that 푈 runs over some base for 푋. Proposition 3.6. Let ℬ be a base of the topology of a topological space 푋. A closed set 퐴 ⊂ 푋 is a 퐺-homological 푍푛-set in 푋 if and only if 퐻푘(푈, 푈 ∖ 퐴; 퐺) = 0 for all 푘 < 푛 + 1 and 푈 ∈ ℬ. Proof: The “only if” part of the theorem is trivial. To prove the “if” part, assume that 퐻푘(푈, 푈 ∖ 퐴; 퐺) = 0 for all 푘 < 푛 + 1 and all sets 푈 ∈ ℬ. Let 풰푘 be the family of all open subsets 푈 ⊂ 푋 such that 퐻푘(푈, 푈 ∖ 퐴; 퐺) = 0. By induction on 푘 < 푛 + 1, we shall show that 풰푘 consists of all open sets in 푋. First, we verify the case 푘 = 0. Take any non-empty open set 푈 ⊂ 푋. The equality 퐻0(푈, 푈 ∖ 퐴; 퐺) = 0 will follow as soon as we 42 T. BANAKH, R. CAUTY, AND A. KARASSEV show that each path-connected component 퐶 of 푈 intersects the set 푈 ∖퐴. Fix any point 푐 ∈ 퐶 and find a neighborhood 푉 ∈ ℬ of 푥 lying in 푈. The equality 퐻0(푉, 푉 ∖ 퐴; 퐺) = 0 implies that 푉 ∖ 퐴 meets each path-connected component of the set 푉 , in particular, the path-connected component 퐶′ of the point 푐 in 푉 . Since 퐶′ ⊂ 퐶, we conclude that 푈 ∖ 퐴 ⊃ 푉 ∖ 퐴 meets the component 퐶, which completes the proof of the case 푘 = 0. Assuming that for some positive 푘 < 푛 + 1 the family 풰푘−1 consists of all open subsets of 푋, we first show that the family 풰푘 is closed under finite unions. Indeed, given any two sets 푈, 푉 ∈ 풰푘 we can write down the piece of the relative Mayer-Vietoris exact sequence

퐻푘(푈, 푈 ∖ 퐴; 퐺) ⊕ 퐻푘(푉, 푉 ∖ 퐴; 퐺) →

퐻푘(푈 ∪ 푉, (푈 ∪ 푉 ) ∖ 퐴; 퐺) →

퐻푘−1(푈 ∩ 푉, 푈 ∩ 푉 ∖ 퐴; 퐺) and conclude that 푈 ∪ 푉 ∈ 풰푘 because 푈 ∩ 푉 ∈ 풰푘−1. Since 풰푘 contains the base ℬ, it contains all possible ﬁnite unions of sets of the base. Because the singular homology theory has compact support, 풰 contains all possible unions of sets from the base ℬ and consequently, 풰푘 consists of all open sets in 푋. □ Corollary 3.7. A subset 퐴 of a topological space 푋 is a 퐺-homological 푍푛-set in 푋 if and only if there is an open cover 풰 of 푋 such that for every 푈 ∈ 풰 the intersection 푈 ∩ 퐴 is a 퐺-homological 푍푛-set in 푈.

4. Detecting 퐺-homological 푍푛-sets with help of partitions In this section we apply the technique of irreducible homological barriers to detect 퐺-homological 푍푛-sets with help of their partitions. A closed subset 퐵 of a topological space 푋 is called an irreducible barrier for a non-zero homology element 훼 ∈ 퐻푛(푋, 푋 ∖ 퐵; 퐺) if for 퐵 every closed subset 퐴 ⊂ 퐵 with 퐴 ∕= 퐵 the image 푖퐴(훼) under the 퐵 inclusion homomorphism 푖퐴 : 퐻푛(푋, 푋 ∖ 퐵; 퐺) → 퐻푛(푋, 푋 ∖ 퐴; 퐺) is trivial. We shall say that a subset 퐴 of a topological space 푋 is separated by a subset 퐵 ⊂ 푋 if the complement 퐴 ∖ 퐵 is ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 43 disconnected. In dimension theory, closed separating sets are also referred to as partitions; see [23, 1.1.3]. We shall need two elementary properties of irreducible barriers, which were also exploited in [14] and [4]. Lemma 4.1. Let 푋 be a topological space and 퐺 be a non-trivial abelian group.

(1) Each closed subset 퐴 of 푋 with 퐻푛(푋, 푋 ∖ 퐴; 퐺) ∕= 0 for some 푛 ≥ 0 contains an irreducible barrier 퐵 ⊂ 퐴 for some element 훼 ∈ 퐻푛(푋, 푋 ∖ 퐵; 퐺). (2) If 퐴 is an irreducible barrier for some element 훼 ∈ 퐻푛(푋, 푋 ∖ 퐴; 퐺), then 퐻푛+1(푋, 푋 ∖ 퐵; 퐺) ∕= 0 for any closed subset 퐵 ⊂ 퐴 separating 퐴. Proof: (1) Given a closed subset 퐴 ⊂ 푋 and a non-zero element 훼 ∈ 퐻푛(푋, 푋 ∖ 퐴; 퐺) ∕= 0, consider the family ℬ of closed subsets 퐴 of 퐴 such that for every 퐵 ∈ ℬ the image 푖퐵(훼) under the inclu- 퐴 sion homomorphism 푖퐵 : 퐻푛(푋, 푋 ∖ 퐴; 퐺) → 퐻푛(푋, 푋 ∖ 퐵; 퐺) is not trivial. We claim that ℬ contains a minimal element. This will follow from Zorn’s Lemma as soon as we prove that the intersection ∩ 풞 of any linearly ordered subfamily 풞 ⊂ ℬ belongs to ℬ. Since singular homology has compact support, the homology group 퐻푛(푋, 푋 ∖∩ 풞; 퐺) is the direct limit of the groups 퐻푛(푋, 푋 ∖퐶; 퐺), 퐶 ∈ 풞; see [33, Ch. 4, Sec. 8, Theorem 13]. Since all the elements 퐴 퐴 푖퐶 (훼), 퐶 ∈ 풞, are not trivial, so neither is the element 푖∩퐶 (훼), which means that ∩ 풞 ∈ ℬ. Now, Zorn’s Lemma yields a minimal element 퐴 퐵 in ℬ. Let 훽 = 푖퐵(훼) ∕= 0. The minimality of 퐵 implies that for 퐴 퐵 any closed subset 퐶 ⊂ 퐵 with 퐶 ∕= 퐵, we get 푖퐶 (훼) = 푖퐶 (훽) = 0, which means that 퐵 is an irreducible barrier for 훽. (2) Assume that 퐴 is an irreducible barrier for some element 훼 ∈ 퐻푛(푋, 푋 ∖ 퐴; 퐺) and let 퐵 be a closed subset separating 퐴. Write 퐴 ∖ 퐵 = 푈 ∪ 푉 as the union of two disjoint non-empty open sets 푈, 푉 ⊂ 퐴. Let 퐶 = 푉 ∪ 퐵 = 퐴 ∖ 푈 and 퐷 = 푈 ∪ 퐵 = 퐴 ∖ 푉 . 퐴 퐴 The irreducibility of 퐴 for 훼 yields 푖퐶 (훼) = 푖퐷(훼) = 0. Now, writing a Mayer-Vietoris exact sequence for the pair (푋, 푋 ∖ 퐵) = (푋, 푋 ∖ 퐶 ∪ 푋 ∖ 퐷), we get ∂ 퐻푛+1(푋, 푋 ∖ 퐵; 퐺) −→ 퐻푛(푋, 푋 ∖ 퐴; 퐺) 푓 −→ 퐻푛(푋, 푋 ∖ 퐶; 퐺) ⊕ 퐻푛(푋, 푋 ∖ 퐷; 퐺). 44 T. BANAKH, R. CAUTY, AND A. KARASSEV

퐴 퐴 Since 푓(훼) = (푖퐶 (훼), −푖퐷(훼)) = (0, 0), 0 ∕= 훼 = ∂(훽) for some nontrivial element 훽 ∈ 퐻푛+1(푋, 푋 ∖ 퐵; 퐺). □ Irreducible barriers help us to prove the following theorem detecting 퐺-homological 푍푛-sets. Theorem 4.2. A closed subset 퐴 of a topological space 푋 is a 퐺-homological 푍푛-set in 푋 if each point of 퐴 is a 퐺-homological 푍푛-point in 푋 and each closed subset 퐵 of 퐴 with ∣퐵∣ > 1 can be separated by a 퐺-homological 푍푛+1-set. Proof: Assume that each point of a closed subset 퐴 ⊂ 푋 is a 퐺-homological 푍푛-point and each closed subset 퐵 ⊂ 퐴 with ∣퐵∣ > 1 can be separated by a 퐺-homological 푍푛+1-set 퐵 ⊂ 퐴 in 푋. Assuming that 퐴 fails to be a 퐺-homological 푍푛-set, find an open subset 푈 ⊂ 푋 and 푘 < 푛 + 1 with 퐻푘(푈, 푈 ∖ 퐴; 퐺) ∕= 0. By Lemma 4.1(1), the set 퐴 ∩ 푈 contains an irreducible barrier 퐵 for a non-zero element 훼 ∈ 퐻푘(푈, 푈 ∖ 퐵; 퐺). The set 퐵 must contain more than one point, since singletons are 퐺-homological 푍푛-sets in 푋. By our assumption the closure 퐵 in 푋 can be separated by a 퐺-homological 푍푛+1-set 퐶. Then 퐶 ∩ 푈 separates 퐵 ∩ 푈 = 퐵 and hence 퐻푘+1(푈, 푈 ∖ 퐶; 퐺) ∕= 0 by Lemma 4.1(2). But this is not possible because 퐶 is a 퐺-homological 푍푛+1-set in 푋. □ Theorem 4.2 will be applied to show that a subset 퐴 ⊂ 푋 is a 퐺-homological 푍푛-set in 푋 if each point 푎 ∈ 퐴 is a 푍푛+푑-point in 푋 where 푑 = trt(퐴) is the separation dimension of 퐴. The separation dimension t(⋅) was introduced by Günter Steinke [34] and later was extended to the transfinite separation dimension trt(⋅) by F. G. Arenas, V. A. Chatyrko, and M. L. Puertas [2] as follows: given a topological space 푋, we write ∙ trt(푋) = −1 if and only if 푋 = ∅; ∙ trt(푋) ≤ 훼 for an ordinal 훼 if any closed subset 퐵 ⊂ 푋 with ∣퐵∣ ≥ 2 can be separated by a closed subset 푃 ⊂ 퐵 with trt(푃 ) < 훼. A space 푋 is defined to be trt-dimensional if trt(푋) ≤ 훼 for some ordinal 훼. In this case, the ordinal trt(푋) = min{훼 : trt(푋) ≤ 훼} is called the (transfinite) separation dimension of 푋. If 푋 is not trt-dimensional, then we write trt(푋) = ∞ and assume that 훼 < ∞ for all ordinals 훼. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 45

By transfinite induction, one can show that trt(푋) ≤ trind(푋), where trind(푋) is the transfinite extension of the small inductive dimension ind(푋); see [2, Proposition 2.9]. This implies that each countable-dimensional completely-metrizable space is trt-dimensional (because trind(푋) < 휔1 [23, 7.1.9]). On the other hand, each trt-dimensional compact space is a 퐶-space; see [2, Proposition 4.7]. Observe that trt(푋) ≤ 0 if and only if 푋 is hereditarily disconnected space. The strongly infinite-dimensional totally disconnected Polish space 푋 constructed in [23, 6.2.4] has trt(푋) = 0 and trind (푋) = ∞ while a compactification 푐(푋) of 푋 with strongly countable-dimensional remainder (the famous example of Pol) has trt(푋) = 휔 and trind(푋) = ∞. Thus, (even on the compact level), the gap between trt(푋) and trind(푋) can be huge. However, for finite-dimensional metrizable compacta 푋, the separation dimension trt(푋) coincides with the usual dimension dim(푋) (and hence with trind(푋)); see [34].

Theorem 4.3. A closed subspace 퐴 of a space 푋 with 푚 = trt(퐴) < 휔 is a 퐺-homological 푍푛-set in 푋 provided each point 푎 ∈ 퐴 is a 퐺-homological 푍푛+푚-point in 푋. Proof: The proof is by induction of the number 푚 = trt(퐴). The assertion is trivial if 푚 = −1 (which means that 퐴 is empty). Assume that for some number 푚 the theorem has been proved for all sets 퐴 with trt(퐴) < 푚. Take a closed subset 퐴 ⊂ 푋 with trt(퐴) = 푚 and all points 푎 ∈ 퐴 being 퐺-homological 푍푛+푚-points in 푋. Assuming that 퐴 fails to be a 푍푛-set in 푋, ﬁnd an open set 푈 ⊂ 푋 and a number 푘 < 푛 + 1 with 퐻푘(푈, 푈 ∖ 퐴; 퐺) ∕= 0. By Lemma 4.1(1), the set 퐴 ∩ 푈 contains an irreducible barrier 퐵 ⊂ 퐴 ∩ 푈 for some non-zero element 훽 ∈ 퐻푘(푈, 푈 ∖ 퐵; 퐺). Since singletons are 퐺-homological 푍푛-sets in 푋, ∣퐵∣ > 1. Since trt(퐴) ≤ 푚, there is a partition 퐶 ⊂ 퐵 of 퐵 with 푑 = trt(퐶) < 푚. Then all points of the set 퐶 are 퐺-homological 푍푛+푑+1-points. Now the inductive assumption guarantees that 퐶 is a 퐺-homological 푍푛+1- set in 푈, which contradicts Lemma 4.1(2) because 퐶 separates the irreducible barrier 퐵. □

By the same method one can prove an inﬁnite version of this theorem. 46 T. BANAKH, R. CAUTY, AND A. KARASSEV

Theorem 4.4. A closed trt-dimensional subspace 퐴 of a space 푋 is a 퐺-homological 푍∞-set in 푋 if and only if each point 푎 ∈ 퐴 is a 퐺-homological 푍∞-point in 푋.

5. Bockstein Theory for 퐺-homological 푍푛-sets In this section, we study the interplay between 퐺-homological 푍푛-sets for various coefficient groups 퐺. Our principal instrument here is the Universal Coefficient Formula (UCF) expressing the homology with respect to an arbitrary coefficient group via homology with respect to the group ℤ of integers. The following its form is taken from [25, Ch. 3, Corollary 3A.4]. Lemma 5.1 (Universal Coefficient Formula). For each pair (푋, 퐴) and all 푛 ≥ 1, there is a natural exact sequence

0 → 퐻푛(푋, 퐴) ⊗ 퐺 → 퐻푛(푋, 퐴; 퐺) → 퐻푛−1(푋, 퐴) ∗ 퐺 → 0 and this sequence splits (though non-naturally). Here 퐺 ⊗ 퐻 and 퐺 ∗ 퐻 stand for the tensor and torsion products of the groups 퐺 and 퐻, respectively. We need some information about those products. First, some notation. By Π we denote the set of prime numbers. We recall that a group 퐺 is divisible if it is divisible by each number 푛 ∈ ℕ. The latter means that for any 푔 ∈ 퐺 there is 푥 ∈ 퐺 with 푛 ⋅ 푥 = 푔. By Tor(퐺) = {푥 ∈ 퐺 : ∃푛 ∈ ℕ with 푛푥 = 0}, we denote the torsion part of 퐺. It is well known that Tor(퐺) is the direct sum of 푝-torsion parts 푝-Tor(퐺) = {푥 ∈ 퐺 : ∃푘 ∈ ℕ 푝푘푥 = 0} where 푝 runs over prime numbers. The following useful result can be found in [22].

Lemma 5.2. Let 퐺0 and 퐺1 be non-trivial abelian groups and 푝 be a prime number.

(1) The tensor product 퐺0 ⊗ 퐺1 contains an element of inﬁnite order if and only if both groups 퐺0 and 퐺1 contain elements of inﬁnite order. (2) The torsion product 퐺0 ∗ 퐺1 contains an element of order 푝 if and only if 퐺0 and 퐺1 contain elements of order 푝. (3) The tensor product 퐺0⊗퐺1 contains an element of order 푝 if and only if for some 푖 ∈ {0, 1}, either 퐺푖 is not divisible by ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 47

푝 and 푝-Tor(퐺1−푖) is not divisible by 푝 or else 퐺푖/푝-Tor(퐺푖) is not divisible by 푝 and 푝-Tor(퐺1−푖) ∕= 0 is divisible by 푝. The following fact follows immediately from the UCF.

Proposition 5.3. Each homological 푍푛-set in a space 푋 is a 퐺- homological 푍푛-set in 푋 for any coeﬃcient group 퐺.

Next, we show that the study of 퐺-homological 푍푛-sets for an arbitrary coeﬃcient group 퐺 can be reduced to studying 퐻-homological 푍푛-sets for coeﬃcient groups 퐻 from the countable family of so-called Bockstein groups. There are many (more or less standard) notations for Bockstein groups. We follow those of [30] and [22]: ∙ ℚ is the group of rational numbers; ∙ ℤ푝 = ℤ/푝ℤ is the cyclic group of a prime order 푝; ∙ 푅푝 is the group of rational numbers whose denominator is not divisible by 푝. ∙ ℚ푝 = ℚ/푅푝 the quasicyclic 푝-group.

By 픅 = {ℚ, ℤ푝, ℚ푝, 푅푝 : 푝 ∈ Π}, we denote the family of all Bock- stein groups. To each abelian group 퐺, assign the Bockstein family 휎(퐺) ⊂ 픅 containing the group ∙ ℚ if and only if the group 퐺/Tor(퐺) ∕= 0 is divisible; ∙ 푅푝 if and only if 퐺/푝-Tor(퐺) is not divisible by 푝; ∙ ℤ푝 if and only if 푝-Tor(퐺) is not divisible by 푝; ∙ ℚ푝 if and only if 푝-Tor(퐺) ∕= 0 is divisible by 푝. In particular, 휎(퐺) = {퐺} for every Bockstein group 퐺 ∈ 픅. The Bockstein families are helpful because of the following fact which can be easily derived from Lemma 5.2. Lemma 5.4. For abelian groups 퐺 and 퐻, the tensor product 퐺⊗퐻 is trivial if and only if 퐺 ⊗ 퐵 is trivial for every group 퐵 ∈ 휎(퐻). Combining this lemma with the UCF, we will obtain the principal result of this section. Theorem 5.5. A closed subset 퐴 of a topological space 푋 is a 퐺-homological 푍푛-set in 푋 if and only if 퐴 is an 퐻-homological 푍푛-set in 푋 for every group 퐻 ∈ 휎(퐺). 48 T. BANAKH, R. CAUTY, AND A. KARASSEV

Proof: Assume that a closed subset 퐴 ⊂ 푋 fails to be a 퐺- homological 푍푛-set in 푋 and ﬁnd 푘 < 푛 + 1 and an open set 푈 ⊂ 푋 with 퐻푘(푈, 푈 ∖ 퐴; 퐺) ∕= 0. The UCF implies that either 퐻푘(푈, 푈 ∖ 퐴) ⊗ 퐺 ∕= 0 or 퐻푘−1(푈, 푈 ∖ 퐴) ∗ 퐺 ∕= 0. In the latter case, the group 퐻푘−1(푈, 푈∖퐴) ∗ 퐺 contains an element of a prime order 푝 and then both the groups 퐻푘−1(푈, 푈∖퐴) and 퐺 contain elements of order 푝 by Lemma 5.2(2). Consequently, 휎(퐺) contains a (quasi)cyclic 푝-group 퐻 ∈ {ℤ푝, ℚ푝} and then 퐻푘−1(푈, 푈 ∖ 퐴) ∗ 퐻 ∕= 0. Applying the UCF, we conclude that 퐻푘(푈, 푈 ∖ 퐴; 퐻) ∕= 0. Next, we assume that 퐻푘(푈, 푈 ∖ 퐴) ⊗ 퐺 ∕= 0. Then Lemma 5.2 implies that 퐻푘(푈, 푈 ∖ 퐴) ⊗ 퐻 ∕= 0 for some group 퐻 ∈ 휎(퐺). Applying the UCF, we conclude that 퐻푘(푈, 푈 ∖ 퐴; 퐻) ∕= 0, which means that 퐴 fails to be an 퐻-homological 푍푛-set in 푋. This proves the “if” part of the theorem. To prove the “only if” part, assume that 퐴 fails to be a 퐻- homological 푍푛-set in 푋 for some group 퐻 ∈ 휎(퐺). Then for some 푘 < 푛 + 1 and an open set 푈 ⊂ 푋, the group 퐻푘(푈, 푈 ∖ 퐴; 퐻) is not trivial. The UCF yields that either 퐻푘(푈, 푈 ∖ 퐴) ⊗ 퐻 ∕= 0 or 퐻푘−1(푈, 푈 ∖ 퐴) ∗ 퐻 ∕= 0. In the ﬁrst case, the tensor product 퐻푘(푈, 푈 ∖ 퐴) ⊗ 퐺 ∕= 0 by Lemma 5.2. In the second case, 퐻푘−1(푈, 푈 ∖ 퐴) ∗ 퐻 contains an element of a prime order 푝 and so do the groups 퐻푘−1(푈, 푈 ∖ 퐴) and 퐻. The inclusion 퐻 ∈ 휎(퐺) implies then that 퐺 contains an element of order 푝 and hence 퐻푘−1(푈, 푈 ∖ 퐴) ∗ 퐺 ∕= 0. In both cases, the UCF implies that 퐻푘(푈, 푈 ∖퐴; 퐺) ∕= 0, which means that 퐴 fails to be a 퐺-homological 푍푛-set in 푋. □

Next, we study the interplay between 퐺-homological 푍푛-sets for various Bockstein groups 퐺.

Theorem 5.6. Let 퐴 be a closed subset of a space 푋 and 푝 be a prime number.

(1) If 퐴 is an 푅푝-homological 푍푛-set in 푋, then 퐴 is a ℚ-homological and ℤ푝-homological 푍푛-set in 푋. (2) If 퐴 is a ℤ푝-homological 푍푛-set in 푋, then 퐴 is a ℚ푝-homological 푍푛-set in 푋. (3) If 퐴 is a ℚ푝-homological 푍푛+1-set in 푋, then 퐴 is a ℤ푝-homological 푍푛-set in 푋. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 49

(4) 퐴 is an 푅푝-homological 푍푛-set in 푋, provided that 퐴 is a ℚ-homological 푍푛-set in 푋 and a ℚ푝-homological 푍푛+1-set in 푋.

Proof: (1) Assuming that 퐴 is not a ℚ-homological 푍푛-set in 푋, find a number 푘<푛+1 and an open set 푈 ⊂ 푋 with 퐻푘(푈, 푈∖퐴; ℚ) ∕= 0. Since ℚ is torsion free, the UCF implies that 퐻푘(푈, 푈 ∖ 퐴) contains an element of infinite order and then 퐻푘(푈, 푈 ∖퐴)⊗푅푝 ∕= 0. Applying the UCF once more, we obtain that 퐻푘(푈, 푈 ∖퐴; 푅푝) ∕= 0, which means that 퐴 is not an 푅푝-homological 푍푛-set in 푋. Next, assume that 퐴 is not a ℤ푝-homological 푍푛-set in 푋 and find an integer number 푘 ≤ 푛 and an open set 푈 ⊂ 푋 with 퐻푘(푈, 푈 ∖ 퐴; ℤ푝) ∕= 0. The UCF implies that either 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0 or 퐻푘−1(푈, 푈 ∖ 퐴) ∗ ℤ푝 ∕= 0. In the latter case, the group 퐻푘−1(푈, 푈 ∖ 퐴) contains an element of order 푝 and by Lemma 5.2, 퐻푘−1(푈, 푈 ∖퐴)⊗푅푝 ∕= 0. The UCF implies that 퐻푘−1(푈, 푈 ∖퐴; 푅푝) ∕= 0, which means that 퐴 fails to be an 푅푝-homological 푍푘−1-set in 푋. So assume that 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0. If 퐻푘(푈, 푈 ∖ 퐴) contains an element of order 푝, then we can proceed as in the preceding case to prove that 퐴 fails to be an 푅푝-homological 푍푘-set in 푋. So we can assume that 푝-Tor(퐻푘(푈, 푈 ∖ 퐴)) = 0, which implies that the torsion part Tor(퐻푘(푈, 푈 ∖ 퐴)) is divisible by 푝. Taking into account that 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0, we can apply Lemma 5.2(3) to find an element 푎 ∈ 퐻푘(푈, 푈 ∖ 퐴) not divisible by 푝. This element cannot belong to the torsion part of 퐻푘(푈, 푈 ∖퐴). Hence, the group 퐻푘(푈, 푈 ∖ 퐴) contains an element of infinite order and so does the tensor product 퐻푘(푈, 푈 ∖ 퐴) ⊗ 푅푝 = 퐻푘(푈, 푈 ∖ 퐴; 푅푝). This means that 퐴 fails to be an 푅푝-homological 푍푘-set in 푋.

(2) Now assume that 퐴 fails to be a ℚ푝-homological 푍푛-set in 푋 and ﬁnd 푘 ≤ 푛 and an open set 푈 ⊂ 푋 with 퐻푘(푈, 푈∖퐴; ℚ푝) ∕= 0. Applying the UCF, we get 퐻푘(푈, 푈∖퐴)⊗ℚ푝 ∕= 0 or 퐻푘−1(푈, 푈∖퐴)∗ ℚ푝 ∕= 0. In the latter case, 퐻푘−1(푈, 푈 ∖ 퐴) ∗ ℤ푝 ∕= 0 and hence 퐻푘(푈, 푈 ∖퐴; ℤ푝) ∕= 0. If 퐻푘(푈, 푈 ∖퐴)⊗ℚ푝 ∕= 0, then, by Lemma 5.2, the group 퐻푘(푈, 푈∖퐴) contains an element not divisible by 푝 and then 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0 by Lemma 5.2. In both the cases, the UCF implies that 퐻푘(푈, 푈 ∖ 퐴; ℤ푝) ∕= 0, which means that 퐴 fails to be a ℤ푝-homological 푍푛-set in 푋. 50 T. BANAKH, R. CAUTY, AND A. KARASSEV

(3) Assume that 퐴 fails to be a ℤ푝-homological 푍푛-set in 푋 and find 푘 ≤ 푛 and an open set 푈 ⊂ 푋 with 퐻푘(푈, 푈 ∖ 퐴; ℤ푝) ∕= 0. The UCF yields 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0 or 퐻푘−1(푈, 푈 ∖ 퐴) ∗ ℤ푝 ∕= 0. In the latter case, 퐻푘−1(푈, 푈 ∖ 퐴) contains an element of order 푝 and then 퐻푘−1(푈, 푈∖퐴) ∗ ℚ푝 ∕= 0. Applying the UCF once more, we get 퐻푘(푈, 푈 ∖ 퐴; ℚ푝) ∕= 0, which means that 퐴 fails to be a ℚ푝-homological 푍푘-set. Now assume that 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℤ푝 ∕= 0. If 퐻푘(푈, 푈 ∖ 퐴) contains an element of order 푝, then 퐻푘(푈, 푈 ∖ 퐴) ∗ ℚ푝 ∕= 0 and 퐻푘+1(푈, 푈 ∖ 퐴; ℚ푝) ∕= 0 by the UCF. This means that 퐴 fails to be a ℚ푝-homological 푍푘+1-set in 푋. So it remains to consider the case when the group 퐻 = 퐻푘(푈, 푈 ∖ 퐴) has trivial 푝-torsion part 푝-Tor(퐻). Since 퐻 ⊗ ℤ푝 ∕= 0, the group 퐻 = 퐻/푝-Tor(퐻) contains an element 푎 not divisible by 푝; see Lemma 5.2. Then 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℚ푝 = 퐻 ⊗ ℚ푝 ∕= 0, according to Lemma 5.2. Now the UCF implies that 퐻푘(푈, 푈 ∖ 퐴; ℚ푝) ∕= 0, which means that 퐴 fails to be a ℚ푝-homological 푍푘-set in 푋. (4) Assume that 퐴 fails to be an 푅푝-homological 푍푛-set in 푋. Then, for some 푘 ≤ 푛 and an open set 푈 ⊂ 푋, the homology group 퐻푘(푈, 푈 ∖퐴; 푅푝) = 퐻푘(푈, 푈 ∖퐴)⊗푅푝 is not trivial. By Lemma 5.2, the group 퐻푘(푈, 푈 ∖ 퐴) contains an element of infinite order or an element of order 푝. In the first case, the group 퐻푘(푈, 푈 ∖ 퐴; ℚ) = 퐻푘(푈, 푈 ∖ 퐴) ⊗ ℚ is not trivial, which means that 퐴 fails to be a ℚ-homological 푍푘-set. In the second case, the subgroup 퐻푘(푈, 푈 ∖ 퐴) ∗ ℚ푝 ⊂ 퐻푘+1(푈, 푈 ∖ 퐴; ℚ푝) is not trivial, which means that 퐴 fails to be a ℚ푝-homological 푍푘+1-set in 푋. □ Combining Theorem 5.5 and Theorem 5.6, we get the following. Corollary 5.7. Let 퐴 be a closed subset of a topological space 푋.

(1) If 퐴 is a 퐺-homological 푍푛-set in 푋 for some coefficient group 퐺, then 퐴 is an 퐻-homological 푍푛 in 푋 for some divisible group 퐻 ∈ {ℚ, ℚ푝 : 푝 ∈ Π}. (2) 퐴 is an 퐹 -homological 푍푛-set in 푋 for a field 퐹 if and only if 퐴 is a 퐺-homological 푍푛-set in 푋 for the field 퐺 ∈ {ℚ, ℤ푝 : 푝 ∈ Π} with {퐺} = 휎(퐹 ). ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 51

6. Multiplication Theorem for homotopical and homological 푍푛-sets We discuss so-called multiplication theorems for homotopical and 푅-homological 푍푛 sets with 푅 being a principal ideal domain. The latter means that 푅 is a commutative ring with unit and without zero divisors in which each proper ideal is generated by a single element. A typical example of a principal ideal domain is the ring ℤ of integers. According to the K¨unnethformula [33, Ch. 5, Sec. 3, Theorem 10], for closed subsets 퐴 ⊂ 푋 and 퐵 ⊂ 푌 in topological spaces 푋 and 푌 and a principal ideal domain 푅, the relative homology group 퐻푛(푋 × 푌, 푋 × 푌 ∖ 퐴 × 퐵; 푅) is isomorphic to the direct sum of the 푅-modules

[퐻∗(푋, 푋 ∖ 퐴; 푅) ⊗푅 퐻∗(푌, 푌 ∖ 퐵; 푅)]푛 = ⊕ 퐻 (푋, 푋 ∖ 퐴; 푅) ⊗ 퐻 (푌, 푌 ∖ 퐵; 푅) and 푖+푗=푛 푖 푅 푗

[퐻∗(푋, 푋 ∖ 퐴; 푅) ∗푅 퐻∗(푌, 푌 ∖ 퐵; 푅)]푛−1 =

⊕푖+푗=푛−1 퐻푖(푋, 푋 ∖ 퐴; 푅) ∗푅 퐻푗(푌, 푌 ∖ 퐵; 푅).

Here 퐺⊗푅퐻 and 퐺∗푅퐻 stand for the tensor and torsion products of 푅-modules 퐺 and 퐻 over 푅. If 푅 = ℤ, then we omit the subscript and write 퐺 ⊗ 퐻 and 퐺 ∗ 퐻. It is known that the torsion product over a ﬁeld 퐹 always is trivial. In this case,

퐻푛(푋×푌, 푋×푌 ∖퐴×퐵; 퐹 ) = [퐻∗(푋, 푋∖퐴; 퐹 )⊗퐹 퐻∗(푌, 푌 ∖퐵; 퐹 )]푛. With help of the Künnethformula and Theorem 5.5, we shall prove multiplication formulas for homological and homotopical 푍푛- sets. Theorem 6.1. Let 퐴 ⊂ 푋 and 퐵 ⊂ 푌 be closed subsets in Ty- chonoff spaces 푋 and 푌 , and let 퐺 be a coefficient group.

(1) If 퐴 is a 퐺-homological 푍푛-set in 푋 and 퐵 is a 퐺-homological 푍푚-set in 푌 , then 퐴 × 퐵 is a 퐺-homological 푍푛+푚-set in 푋 × 푌 . Moreover, if 휎(퐺) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}, then 퐴 × 퐵 is a 퐺-homological 푍푛+푚+1-set in 푋 × 푌 . (2) If 퐴 is a homotopical 푍푛-set in 푋 and 퐵 is a homotopical 푍푚-set in 푌 , then 퐴 × 퐵 is a homotopical 푍푛+푚+1-set in 푋 × 푌 . 52 T. BANAKH, R. CAUTY, AND A. KARASSEV

Proof: (1) In light of Theorem 5.5, it suffices to prove the first item only for a Bockstein group 퐺 ∈ {ℚ, ℤ푝, 푅푝, ℚ푝 : 푝 ∈ Π}. Assume that 퐴 is a 퐺-homological 푍푛-set in 푋 and 퐵 is a 퐺- homological 푍푚-set in 푌 . First, we prove the second part of the first item, assuming that 퐺 ∈ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π} is a principal ideal domain. We have to check that 퐴 × 퐵 is a 퐺-homological 푍푛+푚+1-set in 푋 × 푌 , which means that 퐻푘(푊, 푊 ∖ 퐴 × 퐵; 퐺) = 0 for every 푘 ≤ 푛 + 푚 + 1 and every open set 푊 ⊂ 푋 × 푌 . By Lemma 3.6, it suffices to consider the case 푊 = 푈 × 푉 for some open sets 푈 ⊂ 푋 and 푉 ⊂ 푌 . By the Künneth formula, the homology group 퐻푘(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵; 퐺) is isomorphic to the direct sum of the groups

⊕푖+푗=푘퐻푖(푈, 푈 ∖ 퐴; 퐺) ⊗퐺 퐻푗(푉, 푉 ∖ 퐵; 퐺) and

⊕푖+푗=푘−1퐻푖(푈, 푈 ∖ 퐴; 퐺) ∗퐺 퐻푗(푉, 푉 ∖ 퐵; 퐺). Observe that for every 푖 and 푗 with 푖+푗 ≤ 푛+푚+1, either 푖 ≤ 푛 or 푗 ≤ 푚. In the first case, the group 퐻푖(푈, 푈 ∖퐴; 퐺) is trivial (because 퐴 is a 퐺-homological 푍푛-set in 푋); in the second case, the group 퐻푗(푉, 푉 ∖ 퐵; 퐺) = 0. Hence, the above tensor and torsion products are trivial and so is the group 퐻푘(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵; 퐺). Next, assume that 퐺 = ℚ푝 for a prime 푝. We need to prove that 퐴 × 퐵 is a 퐺-homological 푍푛+푚-set in 푋 × 푌 . As in the preceding case, this reduces to showing that for every 푘 ≤ 푛+푚 and open sets 푈 ⊂ 푋 and 푉 ⊂ 푌 ; the homology group 퐻푘(푈×푉, 푈×푉 ∖퐴×퐵; ℚ푝) is trivial. By the Universal Coefficient Formula (UCF), this group is isomorphic to the direct sum of the groups 퐻푘(푈 ×푉, 푈 ×푉 ∖퐴×퐵) ⊗ ℚ푝 and 퐻푘−1(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵) ∗ ℚ푝. So it suffices to prove the triviality of these two groups. The triviality of the group 퐻푘−1(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵) ∗ ℚ푝 will follow as soon as we prove that the group 퐻푘−1(푈 ×푉, 푈 ×푉 ∖퐴×퐵) has no 푝-torsion. Assuming the converse and using the Künneth formula, we conclude that either the group ⊕푖+푗=푘−1퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푗(푉, 푉 ∖ 퐵) or the group ⊕푖+푗=푘−2퐻푖(푈, 푈 ∖ 퐴) ∗ 퐻푗(푉, 푉 ∖ 퐵) contains an element of order 푝. In the latter case, there are 푖 and 푗 with 푖 + 푗 = 푘 − 2 such that 퐻푖(푈, 푈 ∖ 퐴) ∗ 퐻푗(푉, 푉 ∖ 퐵) contains an element of order 푝. By ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 53

Lemma 5.2, the groups 퐻푖(푈, 푈 ∖ 퐴) and 퐻푗(푉, 푉 ∖ 퐵) contain elements of order 푝 and then 퐻푖(푈, 푈 ∖퐴)∗ℚ푝 and 퐻푗(푈, 푈 ∖퐵)∗ℚ푝 are not trivial, so neither are the homology groups 퐻푖+1(푈, 푈 ∖ 퐴; ℚ푝) and 퐻푗+1(푉, 푉 ∖퐵; ℚ푝) by the UCF. Since 퐴 is a ℚ푝-homological 푍푛- set in 푋 and 퐵 is a ℚ푝-homological 푍푚-set in 푌 , the non-triviality of the two latter homology groups implies that 푖 ≥ 푛 and 푗 ≥ 푚 and hence, 푘 − 2 = 푖 + 푗 ≥ 푛 + 푚 ≥ 푘, which is a contradiction. Next, we consider the case when, for some 푖 and 푗 with 푖 + 푗 = 푘 − 1, the group 퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푗(푉, 푉 ∖ 퐵) contains an element of order 푝. To simplify notation, let 퐻푖 = 퐻푖(푈, 푈 ∖ 퐴) and 퐻푗 = 퐻푗(푉, 푉 ∖ 퐵). Since 퐻푖 ⊗ 퐻푗 contains an element of order 푝, we can apply Lemma 5.2(3) to conclude that either 퐻푖 or 퐻푗 contains an element of order 푝. Without loss of generality, 퐻푖 contains an element of order 푝. It follows from the UCF and the fact that 퐴 is a ℚ푝-homological 푍푛-set in 푋 that 푖 ≥ 푛. Then 푗 = 푘 − 1 − 푖 ≤ 푛 + 푚 − 1 − 푖 < 푚. Since 퐵 is a ℚ푝-homological 푍푚-set in 푌 , the UCF implies that 퐻푗 = 퐻푗(푉, 푉 ∖ 퐵) has no 푝- torsion. Taking into account that 퐻푖 has 푝-torsion and 퐻푖 ⊗ 퐻푗 contains an element of order 푝, we can apply Lemma 5.2(3) to conclude that 퐻푗 = 퐻푗/푝-Tor(퐻푗) is not divisible by 푝 and hence 퐻푗(푉, 푉 ∖ 퐵) ⊗ ℚ푝 = 퐻푗 ⊗ ℚ푝 ∕= 0 by Lemma 5.2. The UCF implies now that 퐻푗(푉, 푉 ∖ 퐵; ℚ푝) ∕= 0, which contradicts the fact that 퐵 is a ℚ푝-homological 푍푚-set (because 푗 ≤ 푚). This completes the proof of the triviality of the group 퐻푘−1(푈 ×푉, 푈 ×푉 ∖퐴×퐵)∗ℚ푝. Next, we check the triviality of the group

퐻푘(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵) ⊗ ℚ푝.

By the K¨unnethformula, the group 퐻푘(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵) is isomorphic to the direct sum of the groups ⊕푖+푗=푘퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푗(푉, 푉 ∖ 퐵) and ⊕푖+푗=푘−1퐻푖(푈, 푈 ∖ 퐴) ∗ 퐻푗(푉, 푉 ∖ 퐵). The second sum is a torsion group and hence, its tensor product with ℚ푝 is trivial. So, it suﬃces to prove that (퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푗(푉, 푉 ∖ 퐵)) ⊗ ℚ푝 = 0 for any 푖 and 푗 with 푖 + 푗 = 푘. Since 푘 ≤ 푛 + 푚, either 푖 ≤ 푛 or 푗 ≤ 푚. If 푖 ≤ 푛, we can use the fact that 퐴 is a ℚ푝-homological 푍푛-set in 푋 to conclude that the homology group 퐻푖(푈, 푈 ∖ 퐴; ℚ푝) is trivial and so is the tensor product 퐻푖(푈, 푈 ∖ 퐴) ⊗ ℚ푝 according 54 T. BANAKH, R. CAUTY, AND A. KARASSEV

(to the UCF. The associativity) of the tensor product implies that 퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푗(푉, 푉 ∖ 퐵) ⊗ ℚ푝 is trivial as well. If 푗 ≤ 푚, then the triviality of the above tensor product follows from the ℚ푝-homological 푍푚-set property of 퐵.

(2) Let 퐴 be a homotopical 푍푛-set in 푋 and let 퐵 be a homotopical 푍푚-set in 푌 . It will be convenient to uniformize the notations and put 푋0 = 푋, 푋1 = 푌 , 퐴0 = 퐴, 퐴1 = 퐵, 푘0 = 푛, 푘1 = 푚, and 푘 = 푘0 + 푘1 + 1. So we need to prove that the product 퐴0 × 퐴1 is a homotopical 푍푘-set in 푋0 × 푋1 provided 퐴푖 is a homotopical 푍 -set in 푋 for 푖 ∈ {0, 1}. 푘푖 푖 푘 Let 풰 be an open cover of 푋0 × 푋1 and 푓 = (푓0, 푓1): 퐼 → 푋0 × 푋1 be a map of the 푘-dimensional cube. Then the sets 푘 푓푖(퐼 ) ⊂ 푋푖, 푖 ∈ {0, 1}, are compact and so is their product. A 푘 standard compactness argument yields ﬁnite covers 풰푖 of 푓푖(퐼 ) by open subsets of 푋푖 for 푖 ∈ {0, 1} such that for any sets 푈0 ∈ 풰0 and 푈1 ∈ 풰1, the product 푈0 × 푈1 lies in some set 푈 ∈ 풰. By Lemma 1.1, each space 푋푖 has a continuous pseudometric 휌푖 ′ ′ such that any 1-ball 퐵(푥, 1) = {푥 ∈ 푋푖 : 휌푖(푥, 푥 ) < 1} centered at 푘 a point 푥 ∈ 푓푖(퐼 ) lies in some set 푈 ∈ 풰푖. Then the pseudometric 휌 = max{휌0, 휌1}, i.e., ′ ′ ′ ′ 휌((푥0, 푥1), (푥0, 푦1)) = max{휌0(푥0, 푥0), 휌1(푥1, 푥1)}

on 푋0 × 푋1 has a similar feature: any 1-ball 퐵(푎, 1) centered at a 푘 푘 point 푎 ∈ 푓0(퐼 ) × 푓1(퐼 ) lies in some set 푈 ∈ 풰. Let 푇 be a triangulation of 퐼푘 so ﬁne that the image 푓(휎) of any simplex 휎 ∈ 푇 has 휌-diameter < 1/3. Let 퐾0 be the 푘0- dimensional skeleton of the triangulation 푇 and 퐾1 be the dual skeleton consisting of all simplexes of the barycentric subdivision of 푇 that do not meet the skeleton 퐾0. It is well known (and easy 푘 to see) that 퐾1 has dimension 푘1 = 푘 −푘0 −1 and each point 푧 ∈ 퐼 lying in a simplex 휎 ∈ 푇 can be written as 푧 = (1−휆(푧))푧0 +휆(푧)푧1 for some points 푧푖 ∈ 퐾푖, 푖 ∈ {0, 1}, and some real number 휆(푧) ∈ [0, 1]. This number is uniquely determined by the point 푧 and is equal to zero if and only if 푧 ∈ 퐾0 and equal to 1 if and only if 푧 ∈ 퐾1. Moreover, the point 푧푖 is uniquely determined by 푧 if and 푘 only if 푧∈ / 퐾1−푖 for 푖 ∈ {0, 1}. This means that the cube 퐼 has the structure of a subset of the join 퐾0 ∗ 퐾1. This structure allows 푘 푘 us to write the cube 퐼 as 퐼 = 퐾≤1/2 ∪ 퐾≥1/2, where 퐾≤1/2 = 푘 푘 {푧 ∈ 퐼 : 휆(푧) ≤ 1/2} and 퐾≥1/2 = {푧 ∈ 퐼 : 휆(푧) ≥ 1/2}. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 55

Let ℓ : [0, 1] → [0, 1] be the piecewise linear map determined by the conditions ℓ(0) = 0 = ℓ(1/2) and ℓ(1) = 1. Combined with the joint structure 퐾0∗퐾1, the map ℓ induces two piecewise-linear maps 푘 푘 ℎ푖 : 퐼 → 퐼 , 푖 ∈ {0, 1}, assigning to each point 푧 = 휆0푧0 + 휆1푧1 ′ ′ with 푧푖 ∈ 퐾푖, 휆0 + 휆1 = 1 the point ℎ푖(푧) = 휆0푧0 + 휆1푧1 where ′ ′ ′ 휆푖 = ℓ(휆푖) and 휆1−푖 = 1 − 휆푖. The crucial property of the maps ℎ푖 is that ℎ0(퐾≤1/2) ⊂ 퐾0, ℎ1(퐾≥1/2) ⊂ 퐾1 and both ℎ0 and ℎ1 are 풮-homotopic to the identity map of 퐼푘 with respect to the cover 풮 of 퐼푘 by maximal simplexes of the triangulation 푇 . Applying Theorem 2.1 to the homotopical 푍 -set 퐴 ⊂ 푋 , ﬁnd 푘푖 푖 푖 a map 푔푖 : 퐾푖 → 푋푖 ∖퐴푖, 1/6-homotopic to the map 푓푖∣퐾푖 : 퐾푖 → 푋푖 with respect to the pseudometric 휌푖. Since 퐾푖 is a subcomplex of 퐼푘, we may apply Borsuk’s homotopy extension theorem (see [33, Ch. 1, Exercise D.2]) and extend the map 푔푖 : 퐾푖 → 푋푖 to a 푘 map푔 ¯푖 : 퐼 → 푋푖, 1/6-homotopic to 푓푖. Finally, consider the map ˜ ˜ ˜ 푘 ˜ ˜ 푓 = (푓0, 푓1): 퐼 → 푋0 × 푋1, where 푓푖 =푔 ¯푖 ∘ ℎ푖. We claim that 푓 is ˜ 푘 풰-homotopic to 푓 and 푓(퐼 ) ∩ (퐴0 × 퐴1) = ∅. The 풮-homotopy of the maps ℎ푖 to the identity implies the푔 ¯푖(풮)- homotopy of푔 ¯푖 ∘ ℎ푖 to푔 ¯푖. Now observe that for each simplex 휎 of the triangulation 푇 , we get 1 1 2 diam(¯푔푖(휎)) ≤ diam(푓푖(휎)) + 2dist(푓푖, 푔¯푖) < 3 + 2 6 = 3 . ˜ Consequently, 푓푖 is 2/3-homotopic to푔 ¯푖. Since푔 ¯푖 is 1/6-homotopic ˜ ˜ to 푓푖, we get that 푓푖 is 1-homotopic to 푓푖 and consequently, 푓 is 1-homotopic to 푓. Now the choice of the pseudometric 휌 implies that 푓˜ is 풰-homotopic to 푓. ˜ 푘 So it remains to prove that 푓(푧) ∈/ 퐴1 ×퐴2 for every point 푧 ∈ 퐼 . ˜ Indeed, if 푧 ∈ 퐾≤1/2, then ℎ0(푧) ∈ 퐾0 and 푓0(푧) =푔 ¯0 ∘ ℎ0(푧) ∈ ˜ 푔¯0(퐾0) ⊂ 푋0 ∖ 퐴0. A similar argument yields 푓1(푧) ∈/ 퐴1, provided 푧 ∈ 퐾≥1/2. □

7. Functions 푧(퐴, 푋) and 푧퐺(퐴, 푋) It will be convenient to write Theorem 6.1 in terms of the functions 푧(퐴, 푋) = sup{푛 ∈ 휔 : 퐴 is a homotopical 푍푛-set in 푋} and 퐺 푧 (퐴, 푋) = sup{푛 ∈ 휔 : 퐴 is a 퐺-homological 푍푛-set in 푋} defined for a closed subset 퐴 of a space 푋 and a coefficient group 퐺. In this definition, we put sup ∅ = −1. So 푧퐺(퐴, 푋) = −1 if and 56 T. BANAKH, R. CAUTY, AND A. KARASSEV only if 퐴 is not a 퐺-homotopical 푍0-set in 푋. For a point 푥 ∈ 푋 of a topological space 푋, we write 푧퐺(푥, 푋) instead of 푧퐺({푥}, 푋). Rewriting theorems 3.2, 3.3, 4.3, 5.5, and 5.6 in the terms of the functions 푧(퐴, 푋) and 푧퐺(퐴, 푋), we obtain the following. Theorem 7.1. Let 퐴 be a closed subset of a topological space 푋, and 퐺 be a coefficient group. Then ℤ 퐺 퐻 (1) 푧(퐴, 푋) ≤ 푧 (퐴, 푋) ≤ 푧 (퐴, 푋) = min퐻∈휎(퐺) 푧 (퐴, 푋); (2) 푧(퐴, 푋) = 푧ℤ(퐴, 푋) if 푋 is an LC1-space and 푧(퐴, 푋) ≥ 2; 퐺 퐺 (3) 푧 (퐴, 푋) + trt(퐴) ≥ min푎∈퐴 푧 (푎, 푋); (4) min{푧ℚ(퐴, 푋), 푧ℚ푝 (퐴, 푋) − 1} ≤ 푧푅푝 (퐴, 푋) and 푧푅푝 (퐴, 푋) ≤ min{푧ℚ(퐴, 푋), 푧ℤ푝 (퐴, 푋)}; (5) 푧ℤ푝 (퐴, 푋) ≤ 푧ℚ푝 (퐴, 푋) ≤ 푧ℤ푝 (퐴, 푋) + 1. Also, Theorem 6.1 can be rewritten in the form of multiplication formulas. Theorem 7.2 (Multiplication Formulas). Let 퐴 and 퐵 be closed subsets in Tychonoff spaces 푋 and 푌 , and let 퐺 be a coefficient group. Then (1) 푧퐺(퐴 × 퐵, 푋 × 푌 ) ≥ 푧퐺(퐴, 푋) + 푧퐺(퐵, 푌 ); (2) 푧퐺(퐴 × 퐵, 푋 × 푌 ) ≥ 푧퐺(퐴, 푋) + 푧퐺(퐵, 푌 ) + 1 if 휎(퐺) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}; (3) 푧(퐴 × 퐵, 푋 × 푌 ) ≥ 푧(퐴, 푋) + 푧(퐵, 푌 ) + 1.

8. Division and 푘-root formulas for homological 푍푛-sets It turns out that inequalities in Theorem 7.2 can be partly reversed, which leads to so-called division and 푘-root formulas. We start with division formulas for Bockstein coeﬃcient groups. Lemma 8.1. Let 퐴 ⊂ 푋, 퐵 ⊂ 푌 , and 퐶 ⊂ 푍 be closed subsets in topological spaces 푋, 푌 , and 푍, and let 푝 be a prime number. Then (1) 푧퐹 (퐴 × 퐵, 푋 × 푌 ) = 푧퐹 (퐴, 푋) + 푧퐹 (퐵, 푋) + 1 for a ﬁeld 퐹 ; (2) 푧ℚ푝 (퐴 × 퐵, 푋 × 푌 ) ≤ 푧ℚ푝 (퐴, 푋) + 푧ℤ푝 (퐵, 푌 ) + 2; (3) 푧푅푝 (퐴, 푋)+1 ≥ min{푧푅푝 (퐴×퐵, 푋×푌 )−푧ℚ(퐵, 푌 ), 푧푅푝 (퐴×퐶, 푋×푍)−푧ℚ푝 (퐶, 푍)} if max{푧ℚ(퐵, 푌 ), 푧ℚ푝 (퐶, 푍)} < ∞. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 57

Proof: (1) Let 퐹 be a field. By Theorem 6.1(1), 푧퐹 (퐴 × 퐵, 푋 × 푌 ) ≥ 푧퐹 (퐴, 푋) + 푧퐹 (퐵, 푌 ) + 1. So it remains to prove the reverse inequality, which is trivial if one of the numbers 푛 = 푧퐹 (퐴, 푋) or 푚 = 푧퐹 (퐵, 푌 ) is infinite. So assume that 푛, 푚 < ∞ and find open sets 푈 ⊂ 푋 and 푉 ⊂ 푌 such that the homology groups 퐻푛+1(푈, 푈 ∖ 퐴; 퐹 ) and 퐻푚+1(푉, 푉 ∖ 퐵; 퐹 ) are not trivial. Their tensor product 퐻푛+1(푈, 푈 ∖ 퐴; 퐹 ) ⊗퐹 퐻푚+1(푉, 푉 ∖ 퐵; 퐹 ) over the field 퐹 is not trivial as well. Now the Künnethformula implies that the group 퐻푛+푚+2(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵; 퐹 ) is not trivial, which means that 퐴×퐵 is not an 퐹 -homological 푍푛+푚+2-set in 푋 ×푌 and hence, 푧퐹 (퐴 × 퐵, 푋 × 푌 ) ≤ 푛 + 푚 + 1 = 푧퐹 (퐴, 푋) + 푧퐹 (퐵, 푌 ) + 1. (2) follows from (1) and Theorem 7.1(5):

푧ℚ푝 (퐴 × 퐵, 푋 × 푌 ) ≤ 푧ℤ푝 (퐴 × 퐵, 푋 × 푌 ) + 1 = 푧ℤ푝 (퐴, 푋) + 푧ℤ푝 (퐵, 푌 ) + 2 ≤ 푧ℚ푝 (퐴, 푋) + 푧ℤ푝 (퐵, 푌 ) + 2.

(3) Let 푛 = 푧푅푝 (퐴, 푋) and find an open set 푈 ⊂ 푋 and 푖 ≤ 푛+1 with 퐻푖(푈, 푈 ∖퐴; 푅푝) ∕= 0. The Universal Coefficient Formula (UCF) yields 퐻푖(푈, 푈 ∖ 퐴) ⊗ 푅푝 ∕= 0. By Lemma 5.2, the group 퐻푖(푈, 푈 ∖ 퐴) contains an element of infinite order or of order 푝. In the first case, use the fact that 퐵 is not a ℚ-homological 푍푚+1- set in 푌 for 푚 = 푧ℚ(퐵, 푌 ) < ∞ to find an open set 푉 ⊂ 푌 with 퐻푚+1(푉, 푉 ∖ 퐵; ℚ) ∕= 0. By the UCF, 퐻푚+1(푉, 푉 ∖ 퐵) ⊗ ℚ ∕= 0 and hence, 퐻푚+1(푉, 푉 ∖ 퐵) contains an element of infinite order and so does the tensor product 퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푚+1(푉, 푉 ∖ 퐵), which lies in the homology group 퐻푖+푚+1(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐵) according to the Künnethformula. Then the tensor product

퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐵)⊗푅푝 = 퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐵; 푅푝) is not trivial. This means that 퐴 × 퐵 fails to be an 푅푝-homological 푅 푍푖+푚+1-set in 푋×푌 and thus 푧 푝 (퐴×퐵, 푋×푌 ) ≤ 푖+푚 ≤ 1+푛+푚. Consequently, 푧푅푝 (퐴, 푋) + 1 = 푛 + 1 ≥ 푧푅푝 (퐴 × 퐵, 푋 × 푌 ) − 푧ℚ(퐵, 푌 ). Next, assume that 퐻푖(푈, 푈∖퐴) contains an element of order 푝 and use the fact that the set 퐶 fails to be a ℚ푝-homological 푍푚+1-set in 푍 for the number 푚 = 푧ℚ푝 (퐶, 푍) < ∞ to ﬁnd an open set 푉 ⊂ 푍 with 퐻푚+1(푉, 푉 ∖ 퐶; ℚ푝) ∕= 0. Then either 퐻푚+1(푉, 푉 ∖ 퐶) ⊗ ℚ푝 ∕= 0 or 퐻푚(푉, 푉 ∖ 퐶) ∗ ℚ푝 ∕= 0. In the latter case, 퐻푚(푉, 푉 ∖ 퐶) 58 T. BANAKH, R. CAUTY, AND A. KARASSEV contains an element of order 푝 and so does the torsion product 퐻푖(푈, 푈∖퐴)∗퐻푚(푉, 푉 ∖퐶) which lies in 퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐶) by the K¨unnethformula. Applying Lemma 5.2, we see that

퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐶)⊗푅푝 = 퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐶; 푅푝) is not trivial. This means that 퐴 × 퐶 is not an 푅푝-homological 푍푖+푚+1-set in 푋 × 푍. Finally, assume that 퐻푚+1(푉, 푉 ∖ 퐶) ⊗ ℚ푝 ∕= 0. By Lemma 5.2, the group 퐻푚+1(푉, 푉 ∖ 퐶)/푝-Tor(퐻푚+1(푉, 푉 ∖ 퐶)) is not divisible by 푝 and hence, the tensor product 퐻푖(푈, 푈 ∖ 퐴) ⊗ 퐻푚+1(푉, 푉 ∖ 퐶) contains an element of order 푝. By the K¨unnethformula, the latter tensor product lies in 퐻푖+푚+1(푈 × 푉, 푈 × 푉 ∖ 퐴 × 퐶). Applying Lemma 5.2 again, we obtain that the tensor product

퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴×퐶)⊗푅푝 = 퐻푖+푚+1(푈×푉, 푈×푉 ∖퐴⊗퐶; 푅푝) is not trivial. In both cases, 퐴×퐶 is not an 푅푝-homological 푍푖+푚+1- set in 푋 × 푍 and hence, 푧푅푝 (퐴 × 퐶, 푋 × 푍) ≤ 푖 + 푚 ≤ 푛 + 1 + 푧ℚ푝 (퐶, 푍). Then 푧푅푝 (퐴, 푋) + 1 = 푛 + 1 ≥ 푧푅푝 (퐴 × 퐶, 푋 × 푍) − 푧ℚ푝 (퐶, 푍). □ Now we can establish division formulas in the general case. First, to each coeﬃcient group 퐺, assign two families 푑(퐺), 휑(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π} as follows. Put ∙ 푑(ℚ) = 휑(ℚ) = {ℚ}, ∙ 푑(ℤ푝) = 휑(ℤ푝) = {ℤ푝}, ∙ 푑(ℚ푝) = 휑(ℚ푝) = {ℤ푝}, ∙ 푑(푅푝) = {ℚ, ℚ푝}, 휑(푅푝) = {ℚ, ℤ푝} and let ∪ ∪ 푑(퐺) = 퐻∈휎(퐺) 푑(퐻) and 휑(퐺) = 퐻∈휎(퐺) 휑(퐻).

In particular, 푑(ℤ) = {ℚ, ℚ푝 : 푝 ∈ Π} is the family of divisible Bockstein groups and 휑(ℤ) = {ℚ, ℤ푝 : 푝 ∈ Π} is the family of Bockstein ﬁelds. Theorem 8.2 (Division Formulas). Let 퐴 ⊂ 푋 and 퐵 ⊂ 푌 be closed subsets in topological spaces and let 퐺 be a coeﬃcient group. Then 퐺 퐺 퐹 (1) 푧 (퐴 × 퐵, 푋 × 푌 ) ≤ 2 + 푧 (퐴, 푋) + sup퐹 ∈휑(퐺) 푧 (퐵, 푌 ); 퐺 퐺 퐻 (2) 푧 (퐴 × 퐵, 푋 × 푌 ) ≤ 1 + 푧 (퐴, 푋) + sup퐻∈푑(퐺) 푧 (퐵, 푌 ) provided 휎(퐺) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 59

Proof: (1) The inequality 푧퐺(퐴 × 퐵, 푋 × 푌 ) ≤ 2 + 푧퐺(퐴, 푋) + 퐹 퐺 sup퐹 ∈휑(퐺) 푧 (퐵, 푌 ) is trivial if 푛 = 푧 (퐴, 푋) is infinite. So assume that 푛 is finite and using Theorem 7.1(1), find a Bockstein group 퐷 ∈ 휎(퐺) with 푧퐷(퐴, 푋) = 푛. Theorem 5.5 implies that 푧퐺(퐴×퐵, 푋 × 푌 ) ≤ 푧퐷(퐴 × 퐵, 푋 × 푌 ). If 퐷 is a field, then {퐷} = 휑(퐷) ⊂ 휑(퐺) and, by Lemma 8.1, 푧퐺(퐴 × 퐵, 푋 × 푌 ) ≤ 푧퐷(퐴 × 퐵, 푋 × 푌 ) = 1+푧퐷(퐴, 푋)+푧퐷(퐵, 푌 ) ≤ 1+푧퐺(퐴, 푋)+ sup 푧퐻 (퐵, 푌 ). 퐻∈휑(퐺)

If 퐷 = 푅푝 for some 푝, then 휑(푅푝) = {ℚ, ℤ푝} ⊂ 휑(퐺) and by Lemma 8.1, we get

푧퐺(퐴 × 퐵, 푋 × 푌 ) ≤ 푧푅푝 (퐴 × 퐵, 푋 × 푌 ) ≤ 1 + 푧푅푝 (퐴, 푋) + max{푧ℚ(퐵, 푌 ), 푧ℚ푝 (퐵, 푌 )} ≤ 1 + 푧푅푝 (퐴, 푋) + max{푧ℚ(퐵, 푌 ), 푧ℤ푝 (퐵, 푌 ) + 1} ≤ 2 + 푧퐷(퐴, 푋) + max{푧ℚ(퐵, 푌 ), 푧ℤ푝 (퐵, 푌 )} ≤ 2 + 푧퐺(퐴, 푋) + sup 푧퐹 (퐵, 푌 ). 퐹 ∈휑(퐺)

If 퐷 = ℚ푝 for some 푝, then, according to Lemma 8.1(2),

푧퐺(퐴 × 퐵, 푋 × 푌 ) ≤ 푧ℚ푝 (퐴 × 퐵, 푋 × 푌 ) ≤ 2+푧ℚ푝 (퐴, 푋)+푧ℤ푝 (퐵, 푌 ) ≤ 2+푧퐺(퐴, 푋)+ sup 푧퐹 (퐵, 푌 ). 퐹 ∈휑(퐺)

(2) If 휎(퐺) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}, then the last case in the preceding item is excluded and we can repeat the preceding argument to prove the inequality 푧퐺(퐴 × 퐵, 푋 × 푌 ) ≤ 1 + 푧퐺(퐴, 푋) + sup 푧퐻 (퐵, 푌 ). □ 퐻∈푑(퐺) Theorem 8.3. Let 퐴 ⊂ 푋 and 퐵 ⊂ 푌 be closed nowhere dense subsets in Tychonoﬀ spaces 푋 and 푌 . Then (1) 1+푧(퐴, 푋)+푧(퐵, 푌 ) ≤ 푧(퐴×퐵, 푋×푌 ) ≤ 푧ℤ(퐴×퐵, 푋×푌 ); (2) 푧(퐴 × 퐵, 푋 × 푌 ) = 푧ℤ(퐴 × 퐵, 푋 × 푌 ) provided 푋 and 푌 are LC1-spaces. Proof: (1) follows from Theorem 6.1(2) and Theorem 7.1(1). 60 T. BANAKH, R. CAUTY, AND A. KARASSEV

(2) Assume that 푋 and 푌 are LC1-spaces and consider three cases. Case (i): 푧ℤ(퐴, 푋) = 푧ℤ(퐵, 푋) = 0. In this case, 푧(퐴, 푋) = 푧ℤ(퐴, 푋) = 푧ℚ(퐴, 푋) and 푧(퐵, 푌 ) = 푧ℤ(퐵, 푌 ) = 푧ℚ(퐵, 푌 ). So by Lemma 8.1(1) and Theorem 6.1(2),

푧(퐴, 푋) + 푧(퐵, 푋) + 1 = 푧ℚ(퐴, 푋) + 푧ℚ(퐵, 푌 ) + 1 = 푧ℚ(퐴×퐵, 푋 ×푌 ) ≥ 푧(퐴×퐵, 푋 ×푌 ) ≥ 푧(퐴, 푋)+푧(퐵, 푌 )+1.

ℤ Case (ii): 푧 (퐴, 푋) > 0. In this case, 퐴 is a homological 푍1-set in 1 푋 and a homotopical 푍1-set in the LC -space 푋, by Theorem 3.2(6) and (3). Set 퐵, being nowhere dense in the LC1-space 푌 , is a homotopical 푍0-set in 푌 . Then 퐴 × 퐵, being the product of a homotopical 푍1-set and a homotopical 푍0-set, is a homotopical 푍2- set in 푋 × 푌 by Theorem 6.1(2). By Theorem 3.3, 퐴 × 퐵 is a homotopical 푍푛-set in 푋 × 푌 if and only if it is a homological 푍푛- set in 푋 ×푌 , which implies the desired equality 푧(퐴 ×퐵, 푋 ×푌 ) = 푧ℤ(퐴 × 퐵, 푋 × 푌 ). Case (iii): 푧ℤ(퐵, 푌 ) > 0 can be considered by analogy. □ Theorem 8.3 implies the following. Corollary 8.4. A subset 퐴 of a Tychonoﬀ LC1-space 푋 is a homological 푍푛-set in 푋 if and only if 퐴×{0} is a homotopical 푍푛+1-set in 푋 × [−1, 1]. Next we turn to the 푘-root theorem. Theorem 8.5 (푘-Root Theorem). Let 퐴 be a closed subset in a topological space 푋, 푘 ∈ ℕ, and 퐺 be a coeﬃcient group. Then (1) 푘 ⋅ 푧퐺(퐴, 푋) ≤ 푧퐺(퐴푘, 푋푘) ≤ 푘 ⋅ 푧퐺(퐴, 푋) + 2푘 − 2; (2) 푘 ⋅ 푧퐺(퐴, 푋) + 푘 − 1 ≤ 푧퐺(퐴푘, 푋푘) provided 휎(퐺) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}; (3) 푧퐺(퐴푘, 푋푘) ≤ 푘 ⋅ 푧퐺(퐴, 푋) + 푘, provided 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π}; (4) 푧퐺(퐴푘, 푋푘) = 푘 ⋅ 푧퐺(퐴, 푋) + 푘 − 1, provided 휎(퐺) ⊂ {ℚ, ℤ푝 : 푝 ∈ Π}. Proof: The items of this theorem will be proved in the following order: (2), (4), (3), (1). There is nothing to prove if 푘 = 1. So we assume that 푘 ≥ 2. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 61

(2) follows by induction from Theorem 7.2(2).

(4) Assume that 휎(퐺) ⊂ {ℚ, ℤ푝 : 푝 ∈ Π}. By Theorem 7.1(1), there is a Bockstein group 퐹 ∈ 휎(퐺) with 푧퐹 (퐴, 푋) = 푧퐺(퐴, 푋). Since 퐹 is a ﬁeld, we may apply Lemma 8.1(1) 푘 − 1 times and get the equality 푧퐹 (퐴푘, 푋푘) = 푘 ⋅ 푧퐹 (퐴, 푋) + 푘 − 1. Then

푘 ⋅ 푧퐺(퐴, 푋) + 푘 − 1 = 푘 ⋅ 푧퐹 (퐴, 푋) + 푘 − 1 = 푧퐹 (퐴푘, 푋푘) ≥ 푧퐺(퐴푘, 푋푘) ≥ 푘 ⋅ 푧퐺(퐴, 푋) + 푘 − 1, (the last inequality follows from the preceding item) which yields the equality from the fourth item of the theorem.

(3) Assume that 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π} and, using The- orem 7.1(1), ﬁnd a Bockstein group 퐻 ∈ 휎(퐺) with 푧퐺(퐴, 푋) = 푧퐻 (퐴, 푋). For this group, we also get 푧퐺(퐴푘, 푋푘) ≤ 푧퐻 (퐴푘, 푋푘) according to Theorem 7.1(1). If 퐻 ∈ {ℚ, ℤ푝 : 푝 ∈ Π}, then 푧퐺(퐴푘, 푋푘) ≤ 푧퐻 (퐴푘, 푋푘) = 푘⋅푧퐻 (퐴, 푋)+푘−1 = 푘⋅푧퐺(퐴, 푋)+푘−1 by the preceding case. So it remains to consider the case 퐻 = ℚ푝 for a prime 푝. Ap- plying Theorem 7.1(5) and the second item of this theorem, we get

푧퐺(퐴푘, 푋푘) ≤ 푧ℚ푝 (퐴푘, 푋푘) ≤ 푧ℤ푝 (퐴푘, 푋푘) + 1 = 푘 ⋅ 푧ℤ푝 (퐴, 푋) + 푘 ≤ 푘 ⋅ 푧ℚ푝 (퐴, 푋) + 푘 = 푘 ⋅ 푧퐺(퐴, 푋) + 푘.

(1) The inequality 푘 ⋅ 푧퐺(퐴, 푋) ≤ 푧퐺(퐴푘, 푋푘) follows by induction from Theorem 7.2(1). To prove the inequality 푧퐺(퐴푘, 푋푘) ≤ 푘⋅푧퐺(퐴, 푋)+2푘−2, apply Theorem 7.1(1) to ﬁnd a group 퐻 ∈ 휎(퐺) 퐻 퐺 such that 푧 (퐴, 푋) = 푧 (퐴, 푋). If 퐻 ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π}, then 푧퐺(퐴푘, 푋푘) ≤ 푧퐻 (퐴푘, 푋푘) ≤ 푘 ⋅ 푧퐻 (퐴, 푋) + 푘 = 푘 ⋅ 푧퐺(퐴, 푋) + 푘 ≤ 푧퐺(퐴, 푋) + 2푘 − 2.

So it remains to consider the case of 퐻 = 푅푝 for a prime 푝 and prove that 푧퐺(퐴푘, 푋푘) ≤ 푘 ⋅ 푧퐺(퐴, 푋) + 2푘 − 2. Assuming the 푘 푘 converse, we conclude that 퐴 is a 퐺-homological 푍푘푛+푘−1-set in 푋 for 푛 = 푧퐺(퐴, 푋)+1 = 푧푅푝 (퐴, 푋)+1. It follows from the deﬁnition 푅 of 푧 푝 (퐴, 푋) = 푛 − 1 that the homology group 퐻푛(푈, 푈 ∖ 퐴; 푅푝) = 퐻푛(푈, 푈 ∖퐴)⊗푅푝 is not trivial for some open set 푈 ⊂ 푋. Applying 62 T. BANAKH, R. CAUTY, AND A. KARASSEV

Lemma 5.2, we get that 퐻푛(푈, 푈 ∖ 퐴) contains an element 푎 that has either infinite order or order 푝. If 푎 has infinite order, then the tensor product 퐻푛(푈, 푈 ∖ 퐴) ⊗ 퐻푛(푈, 푈 ∖ 퐴) contains an element of infinite order and so does the 2 2 2 group 퐻2푛(푈 , 푈 ∖ 퐴 ) according to the Künnethformula. Now, 푘 푘 푘 we can show by induction that the group 퐻푘푛(푈 , 푈 ∖퐴 ) contains 푘 푘 푘 an element of infinite order, which implies that 퐻푘푛(푈 , 푈 ∖ 퐴 ) ⊗ 푘 푘 푘 푘 푅푝 = 퐻푘푛(푈 , 푈 ∖ 퐴 ; 푅푝) ∕= 0 and hence, 퐴 fails to be an 푅푝- 푘 homological 푍푘푛-set in 푋 . Then we get a contradiction: 푧퐺(퐴푘, 푋푘) ≤ 푧푅푝 (퐴푘, 푋푘) < 푘푛 ≤ 푘푛 + 푘 − 1 ≤ 푧퐺(퐴푘, 푋푘) . If 푎 is of order 푝, then the torsion product 퐻푛(푈, 푈∖퐴) ∗ 퐻푛(푈, 푈∖퐴) contains an element of order 푝 and, according to the Künnethfor- 2 2 2 mula, so does the group 퐻2푛+1(푈 , 푈 ∖ 퐴 ). By induction, we can 푖 푖 푖 show that the homology group 퐻푖(푛+1)−1(푈 , 푈 ∖ 퐴 ) contains an 푘 푘 푘 element of order 푝. For 푖 = 푘, the group 퐻푘푛+푘−1(푈 , 푈 ∖퐴 ) con- 푘 푘 푘 tains an element of order 푝 and hence, 퐻푘푛+푘−1(푈 , 푈 ∖퐴 )⊗푅푝 = 푘 푘 푘 퐻푘푛+푘−1(푈 , 푈 ∖ 퐴 ; 푅푝) is not trivial by Lemma 5.2. This means 푘 that 퐴 is not an 푅푝-homological 푍푘푛+푘−1-set in 푋 and thus, 푧퐺(퐴푘, 푋푘) ≤ 푧푅푝 (퐴푘, 푋푘) ≤ 푘푛 + 푘 − 2 < 푧퐺(퐴푘, 푋푘), which is a contradiction. □

9. 푍푛-points

A point 푥 of a space 푋 is defined to be a 푍푛-point if its singleton {푥} is a 푍푛-set in 푋. By analogy we define homotopical and 퐺-homological 푍푛-points. It is clear that all results proved in the preceding sections for 푍푛-sets concern also 푍푛-points. For 푍푛- points there are, however, some simplifications. In particular, the excision property for singular homology theory (see in [33, Ch. 4, Sec. 6, Theorem 4]) allows us to characterize homological 푍푛-points as follows. Proposition 9.1. For a point 푥 of a regular topological space 푋 and a coefficient group 퐺, the following conditions are equivalent.

(1) 푥 is a 퐺-homological 푍푛-point in 푋; (2) 퐻푘(푋, 푋 ∖ {푥}; 퐺) = 0 for all 푘 < 푛 + 1; (3) there is an open neighborhood 푈 ⊂ 푋 of 푥 such that 퐻푘(푈, 푈 ∖ {푥}; 퐺) = 0 for all 푘 < 푛 + 1. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 63

In this section, given a topological space 푋 and a coeﬃcient group 퐺 we shall study the Borel complexity of the sets

∙ 풵푛(푋) of all homotopical 푍푛-points in 푋, and 퐺 ∙ 풵푛 (푋) of all 퐺-homological 푍푛-points in 푋. Theorem 9.2. Let 푋 be a metrizable separable space and 퐺 be a coeﬃcient group. Then

(1) the set of 푍푛-points in 푋 is a 퐺훿-set in 푋; (2) the set 풵푛(푋) of homotopical 푍푛-points is a 퐺훿-set in 푋 provided 푋 is an LC푛-space; (3) the set of 퐺-homological 푍푛-points is a 퐺훿-set in 푋 if ∣퐻푘(푈; 퐺)∣ ≤ ℵ0 for all open sets 푈 ⊂ 푋 and all 푘 < 푛 + 1; 퐺 (4) the set 풵푛 (푋) of 퐺-homological 푍푛-points is a 퐺훿-set in 푋 provided 푋 is an 푙푐푛-space; (5) the set of homotopical 푍푛-points is a 퐺훿-set in 푋 if 푋 is 2 an LC -space and ∣퐻푘(푈)∣ ≤ ℵ0 for all open subsets 푈 ⊂ 푋 and all 푘 < 푛 + 1. Proof: (1) Let 휌 be a metric on 푋 generating the topology of 푋. Since the space 푋 is metrizable and separable, so is the function space 퐶(퐼푛, 푋) endowed with the sup-metric 휌˜(푓, 푔) = sup 휌(푓(푧), 푔(푧)). 푧∈퐼푛 ∞ 푛 So, we may ﬁx a countable dense set {푓푖}푖=1 in 퐶(퐼 , 푋). Observe that a closed subset 퐴 ⊂ 푋 fails to be a 푍푛-set if and only if there are a function 푓 ∈ 퐶(퐼푘, 푋) and 휀 > 0 such that for each function 푔 ∈ 퐶(퐼푛, 푋) with휌 ˜(푓, 푔) < 휀 the image 푔(퐼푛) meets the set 퐴. The density of {푓푖} implies that휌 ˜(푓, 푓푖) < 휀/2 for some 푖 ∈ 휔. Consequently, 푔(퐼푘) ∩ 퐴 ∕= ∅ for all 푔 ∈ 퐶(퐼푛, 푋) with 휌˜(푔, 푓푖) < 휀/2. Now for each 푖, 푘 ∈ ℕ, consider the set 푛 푛 퐹푖,푘 = {푥 ∈ 푋 : ∀푔 ∈ 퐶(퐼 , 푋)휌 ˜(푔, 푓푖) < 1/푘 ⇒ 푥 ∈ 푔(퐼 )}. It is easy to see that the set 퐹푖,푘 is closed in 푋. Moreover, the ∪∞ preceding discussion implies that 퐹 = 푖,푘=1 퐹푖,푘 is the set of all points of 푋 that fail to be 푍푛-points in 푋. Then its complement is a 퐺훿-set coinciding with the set of all 푍푛-points in 푋. 푛 (2) If 푋 is an LC -space, then each 푍푛-point is a homotopical 푍푛-point in 푋 and consequently, the set 풵푛(푋) of homotopical 푍푛-points coincides with the 퐺훿-set 푋 ∖ 퐹 of all 푍푛-points. 64 T. BANAKH, R. CAUTY, AND A. KARASSEV

(3) Assume that the homology groups 퐻푘(푈; 퐺) are countable for all 푘 < 푛 + 1 and all open sets 푈 ⊂ 푋. Let ℬ = {푈푖 : 푖 ∈ 휔} be a countable base of the topology for 푋 with 푈0 = ∅. For every 푖 ∈ 휔 and 푘 < 푛 + 1, use the countability of the groups 퐻푘(푋 ∖ 푈 푖; 퐺) to ﬁnd a countable sequence (훼푖,푗)푗∈휔 of cycles in 푋 ∖ 푈 푖 whose representatives [훼푖,푗] exhaust all non-zero elements of the groups ¯ 퐻푘(푋 ∖ 푈푖; 퐺), 푘 < 푛 + 1. For every 푗 ∈ 휔, consider the open set 푊0,푗 = {푥 ∈ 푋 : 훼0,푗 is homologous to some cycle in 푋 ∖ {푥}}. Also for every 푖 ∈ ℕ and 푗 ∈ 휔, consider the open set 푊푖,푗 = {푥 ∈ 푈푖 : if 훼푖,푗 is null- homological in 푋, then it is null-homological∩ in 푋∖{푥}}. It remains to prove that the 퐺훿-set 푍 = 푖,푗∈휔 푊푖,푗 coincides with the set 퐺 풵푛 (푋) of all 퐺-homological 푍푛-points in 푋. It is clear that 푍 contains all 퐺-homological 푍푛-points of 푋. Now take any point 푥 ∈ 푍. Assuming that 푥 is not a 퐺-homological 푍푛-point, ﬁnd 푘 < 푛 + 1 such that 퐻푘(푋, 푋 ∖ {푥}; 퐺) ∕= 0. The exact sequence

퐻푘(푋 ∖ {푥}; 퐺) → 퐻푘(푋; 퐺) → 퐻푘(푋, 푋 ∖ {푥}; 퐺) →

퐻푘−1(푋 ∖ {푥}; 퐺) → 퐻푘−1(푋; 퐺) of the pair (푋, 푋 ∖ {푥}) now implies that for 푚 = 푘 or 푚 = 푘 − 1 the inclusion homomorphism 푖 : 퐻푚(푋 ∖ {푥}; 퐺) → 퐻푚(푋; 퐺) fails to be an isomorphism. If 푖 is not onto, then, for some 푗 ∈ 휔, the element 훼0,푗 is homologous to no cycle in 푋 ∖ {푥}. This means that 푥∈ / 푊0,푗 and thus, 푥∈ / 푍, which is a contradiction. If 푖 is not injective, then there is a 푘-cycle 훼 in 푋 ∖ {푥} which is homologous to zero in 푋 but not in 푋 ∖ {푥}. Since 훼 has compact support, there is a basic neighborhood 푈푖 of 푥 such that 훼 is supported by the set 푋 ∖ 푈 푖. Find 푗 ∈ 휔 such that the cycle 훼푖,푗 is homologous to 훼 in 푋 ∖ 푈 푖. It follows that 훼푖,푗 is homologous to zero in 푋 but not in 푋 ∖ {푥}. Then 푥∈ / 푊푖,푗 and hence 푥∈ / 푍. This contradiction completes the proof of the third item. 푛 (4) Assuming that 푋 is an 푙푐 -space, we shall∩ show that the 퐺 퐺 퐻 set 풵푛 (푋) is of type 퐺훿 in 푋. Since 풵푛 (푋) = 퐻∈휎(퐺) 풵푛 (푋), 퐻 it suffices to check that 풵푛 (푋) is a 퐺훿-set in 푋 for every countable group 퐻. This will follow from the preceding item as soon as we show that for every open set 푈 ⊂ 푋, the homology groups 퐻푖(푈; 퐻), 푖 ≤ 푛, are at most countable. In its turn, this will follow ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 65 from the Universal Coefficient Formula as soon as we check that the homology groups 퐻푖(푈), 푖 ≤ 푛, are at most countable. Fix a countable family 풦 of compact polyhedra containing a topological copy of each compact polyhedron. For each polyhedron 퐾 ∈ 풦, fix a countable dense set ℱ퐾 in the function space 퐶(퐾, 푈). Note that for each polyhedron 퐾 ∈ 풦, the homology group 퐻∗(퐾) is finitely generated and hence at most countable. For every homology element 훼 ∈ 퐻푖(푈) with 푖 ≤ 푛, there is a continuous map 푓 : 퐾 → 푈 of a compact polyhedron 퐾 ∈ 풦 such that 훼 ∈ 푓∗(퐻푖(퐾)). Moreover, according to Lemma 1.3, we can assume that 푓 ∈ ℱ퐾 . Then the homology group ∪ ∪ 퐻푖(푈) = 푓∗(퐻푖(퐾))

퐾∈풦 푓∈ℱ퐾 is countable, being the countable union of ﬁnitely generated groups. (5) Follows from items (2) and (4) and the characterization of homotopical 푍푛-sets given by Theorem 3.3. □

10. On spaces all of whose points are 푍푛-points In this section, we introduce three classes of Tychonoff spaces related to 푍푛-points: ∙ 풵푛 the class of spaces 푋 with 푋 = 풵푛(푋), 퐺 퐺 ∙ 풵푛 the class of spaces 푋 with 푋 = 풵푛 (푋), 퐺 퐺 ∙ ∪퐺풵푛 the union of classes 풵푛 over all coefficient groups. For example, ℝ푛+1 belongs to all of these classes, while ℝ푛 belongs to none of them. The classes 풵푛 play an important role in studying the general position properties from [9]. By LC푛 (푙푐푛, respectively), we shall denote the class of metrizable LC푛-spaces (푙푐푛-spaces, respectively). The following corollary describes the relation between the introduced classes and can be easily derived from theorems 3.2, 3.3, 5.5, and 5.6. Corollary 10.1. Let 푛 ∈ 휔 ∪ {∞} and 퐺 be a coefficient group. Then ∩ ℤ 퐺 퐻 (1) 풵푛 ⊂ 풵푛 ⊂ 풵푛 = 퐻∈휎(퐺) 풵푛 ; ℤ 퐺 (2) 풵0 = 풵0 = 풵0 ; 1 퐺 (3) LC ∩ 풵1 ⊂ 풵1; 1 ℤ (4) LC ∩ 풵2 ∩ 풵푛 ⊂ 풵푛; 66 T. BANAKH, R. CAUTY, AND A. KARASSEV ∪ 퐺 ℚ ℚ푝 (5) ∪퐺풵 = 풵 ∪ 풵푛 ; 푛 ∩ 푛 푝∈Π ℤ 푅푝 (6) 풵푛 = 푝∈Π 풵푛 ; 푅푝 ℚ ℤ푝 ℤ푝 ℚ푝 ℤ푝 ℚ ℚ푝 (7) 풵푛 ⊂ 풵푛 ∩ 풵푛 , 풵푛 ⊂ 풵푛 ⊂ ℤ푛−1, and 풵푛 ∩ 풵푛+1 ⊂ 푅푝 풵푛 for every prime number 푝. For a better visual presentation of our subsequent results, let us introduce the following operations on subclasses 풜, ℬ ⊂ Top of the class Top of topological spaces: 풜 × ℬ = {퐴 × 퐵 : 퐴 ∈ 풜, 퐵 ∈ ℬ}, 풜 = {푋 ∈ Top : ∃퐵 ∈ ℬ with 푋 × 퐵 ∈ 풜}, ℬ √ 풜푘 = {퐴푘 : 퐴 ∈ 풜} and 푘 풜 = {퐴 ∈ Top : 퐴푘 ∈ 풜}. Theorem 7.2 implies the following three multiplication formulas 퐺 for the classes 풵푛 and 풵푛 . Theorem 10.2 (Multiplication Formulas). Let 푛, 푚 ∈ 휔 ∪ ∞ and 푋 and 푌 be Tychonoff spaces.

(1) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚, then 푋 × 푌 ∈ 풵푛+푚+1:

풵푛 × 풵푚 ⊂ 풵푛+푚+1.

퐺 퐺 (2) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚 for a coeﬃcient group 퐺, then 퐺 푋 × 푌 ∈ 풵푛+푚: 퐺 퐺 퐺 풵푛 × 풵푚 ⊂ 풵푛+푚. 푅 푅 (3) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚 for a coeﬃcient group 푅 with 푅 휎(푅) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}, then 푋 × 푌 ∈ 풵푛+푚+1: 푅 푅 푅 풵푛 × 풵푚 ⊂ 풵푛+푚+1.

퐺 The multiplication formulas for the classes 풵푛 can be reversed. Theorem 10.3 (Division Formulas). Let 푛, 푚 ∈ 휔 ∪ ∞ and 푋 and 푌 be Tychonoﬀ spaces. (1) If 푋 × 푌 ∈ 풵퐺 for a coeﬃcient group 퐺, then either 푛+∪푚+1 퐺 퐹 푋 ∈ 풵푛 or 푌 ∈ 퐹 ∈휑(퐺) 풵푚. This can be written as 풵퐺 ∪ 푛+푚+1 ⊂ 풵퐹 . 퐺 퐹 ∈휑(퐺) 푚 Top ∖ 풵푛 ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 67

(2) If 푋 × 푌 ∈ 풵푅 for a coeﬃcient group 푅 with 휎(푅) ⊂ 푛+푚 ∪ 푅 퐻 {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}, then either 푋 ∈ 풵푛 or 푌 ∈ 퐻∈푑(푅) 풵푚 . This can be written as

풵푅 ∪ 푛+푚 ⊂ 풵퐻 . 푅 퐻∈푑(푅) 푚 Top ∖ 풵푛

∪ 퐺 퐹 Proof: (1) Assume that 푋∈ / 풵푛 and 푌∈ / 퐹 ∈휑(퐺) 풵푚. Then there is a point 푥 ∈ 푋 with 푧퐺(푥, 푋) < 푛, and for every ﬁeld 퐹 퐹 ∈ 휑(퐺), there is a point 푦퐹 ∈ 푌 with 푧 (푦퐹 , 푌 ) < 푚. By Theorem 5.5, 푧퐺(푥, 푋) = 푧퐻 (푥, 푋) for some group 퐻 ∈ 휎(퐺). If 퐻 is a ﬁeld, then 퐻 ∈ 휑(퐻) ⊂ 휑(퐺), and by Lemma 8.1, 퐻 퐻 퐻 푧 ((푥, 푦퐻 ), 푋 × 푌 ) = 푧 (푥, 푋) + 푧 (푦퐻 , 푌 ) + 1 ≤ (푛 − 1) + (푚 − 1) + 1 = 푛 + 푚 − 1, which means that (푥, 푦퐻 ) is not an 퐻-homological 푍푛+푚-set in 퐻 퐺 푋 × 푌 and thus, 푋 × 푌∈ / 풵푛+푚 ⊃ 풵푛+푚. If 퐻 = 푅푝 for some 푝, then {ℚ, ℚ푝} = 푑(푅푝) and ℤ푝 ∈ 휑(퐺). By Lemma 8.1(3), 푧퐻 (푥, 푋) + 1 ≥ 퐻 ℚ min{푧 ((푥, 푦ℚ), 푋 × 푌 ) − 푧 (푦ℚ, 푌 ), 퐻 ℚ푝 푧 ((푥, 푦ℤ푝 ), 푋 × 푌 ) − 푧 (푦ℤ푝 , 푌 )}. Therefore, either 퐻 퐻 ℚ 푧 ((푥, 푦ℚ), 푋 × 푌 ) ≤ 1 + 푧 (푥, 푋) + 푧 (푦ℚ, 푌 ) ≤ (푛 − 1) + (푚 − 1) + 1 or 퐻 퐻 ℚ푝 푧 ((푥, 푦ℤ푝 ), 푋 × 푌 ) ≤ 1 + 푧 (푥, 푋) + 푧 (푦ℤ푝 , 푌 ) ℤ푝 ≤ 1 + (푛 − 1) + 푧 (푦ℤ푝 , 푌 ) + 1 = 푛 + 푚. 퐻 In both cases, 푋 × 푌∈ / 풵푛+푚+1. If 퐻 = ℚ푝, then ℤ푝 ∈ 휑(퐺). By Lemma 8.1, 퐻 퐻 ℤ푝 푧 ((푥, 푦ℤ푝 ), 푋 × 푌 ) ≤ 푧 (푥, 푋) + 푧 (푦ℤ푝 , 푌 ) + 2 ≤ (푛 − 1) + (푚 − 1) + 2 = 푛 + 푚, 퐻 which implies that 푋 × 푌∈ / 풵푛+푚+1. ∪ 푅 퐹 (2) Assume that 푋∈ / 풵푛 and 푌∈ / 퐹 ∈푑(푅) 풵푚 for a group 푅 with 휎(푅) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}. Then there is a point 푥 ∈ 푋 with 푅 푧 (푥, 푋) < 푛 and for every group 퐻 ∈ 푑(푅) there is a point 푦퐻 ∈ 푌 퐻 푅 퐻 with 푧 (푦퐻 , 푌 ) < 푚. By Theorem 5.5, 푧 (푥, 푋) = 푧 (푥, 푋) for some group 퐻 ∈ 휎(푅) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}. Theorem 5.5 implies 푅 퐻 that 풵푛+푚 ⊂ 풵푛+푚. 68 T. BANAKH, R. CAUTY, AND A. KARASSEV

If 퐻 is a ﬁeld, then repeating the reasoning from the preceding item, we can prove that 퐻 퐻 퐻 푧 ((푥, 푦퐻 ), 푋 × 푌 ) = 푧 (푥, 푋) + 푧 (푦퐻 , 푌 ) + 1 ≤ (푛 − 1) + (푚 − 1) + 1 = 푛 + 푚 − 1, 퐻 which means that 푋 × 푌∈ / 풵푛+푚. If 퐻 = 푅푝 for some 푝, then {ℚ, ℚ푝} = 푑(푅푝) ⊂ 푑(퐺). By Lemma 8.1(3), 푧퐻 (푥, 푋) + 1 ≥ 퐻 ℚ min{푧 ((푥, 푦ℚ), 푋×푌 ) − 푧 (푦ℚ, 푌 ), 퐻 ℚ푝 푧 ((푥, 푦ℤ푝 ), 푋×푌 ) − 푧 (푦ℚ푝 , 푌 )}. Therefore, either 퐻 퐻 ℚ 푧 ((푥, 푦ℚ), 푋 × 푌 ) ≤ 1 + 푧 (푥, 푋) + 푧 (푦ℚ, 푌 ) ≤ (푛 − 1) + (푚 − 1) + 1 = 푛 + 푚 − 1 or 퐻 퐻 ℚ푝 푧 ((푥, 푦ℤ푝 ), 푋 × 푌 ) ≤ 1 + 푧 (푥, 푋) + 푧 (푦ℚ푝 , 푌 ) ≤ 1 + (푛 − 1) + (푚 − 1) = 푛 + 푚 − 1. 퐻 푅 In both cases, 푋 × 푌∈ / 풵푛+푚 ⊃ 풵푛+푚. □ 퐺 Finally, we prove 푘-root formulas for the classes 풵푛 . Theorem 10.4 (푘-Root Formulas). Let 푛 ∈ 휔 ∪ ∞, 푘 ∈ ℕ, 푋 be a topological space, and 퐺 be a coeﬃcient group. 푘 퐺 퐺 (1) If 푋 ∈ 풵푘푛+푘−1, then 푋 ∈ 풵푛 : √ 푘 퐺 퐺 풵푛푘+푘−1 ⊂ 풵푛 .

푘 퐺 (2) If 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π} and 푋 ∈ 풵푘푛+1, then 퐺 푋 ∈ 풵푛 : √ 푘 퐺 퐺 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π} ⇒ 풵푛푘+1 ⊂ 풵푛 .

푘 퐺 퐺 (3) If 휎(퐺) ⊂ {ℚ, ℤ푝 : 푝 ∈ Π} and 푋 ∈ 풵푘푛, then 푋 ∈ 풵푛 : √ 푘 퐺 퐺 휎(퐺) ⊂ {ℚ, ℤ푝 : 푝 ∈ Π} ⇒ 풵푛푘 = 풵푛 .

퐺 Proof: Assume that 푋∈ / 풵푛 and ﬁnd a point 푥 ∈ 푋 with 푧퐺({푥}, 푋) < 푛. (1) Applying Theorem 8.5(1), we get 푧퐺({푥}푘, 푋푘) < 푘⋅푧퐺(푥, 푋)+ 푘 퐺 2푘 − 1 ≤ 푘(푛 − 1) + 2푘 − 1 = 푘푛 + 푘 − 1 and hence 푋 ∈/ 풵푘푛+푘−1. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 69

(2) If 휎(퐺) ⊂ {ℚ, ℤ푝, ℚ푝 : 푝 ∈ Π}, then we can apply Theo- rem 8.5(3) to conclude that 푧퐺({푥}푘, 푋푘) ≤ 푘 ⋅ 푧퐺(푥, 푋) + 푘 ≤ 푘 퐺 푘(푛 − 1) + 푘 = 푘푛 and 푋 ∈/ 풵푘푛+1. (3) If 휎(퐺) ⊂ {ℚ, ℤ푝 : 푝 ∈ Π}, then we can apply Theorem 8.5(4) to conclude that 푧퐺({푥}푘, 푋푘) = 푘 ⋅ 푧퐺(푥, 푋) + 푘 − 1 ≤ 푘(푛 − 1) + 푘 퐺 푘 − 1 = 푘푛 − 1 and hence 푋 ∈/ 풵푘푛. □ 11. On spaces containing a dense set of (homological) 푍푛-points In this section we consider classes of Tychonoff spaces containing dense sets of (homological) 푍푛-points. More precisely, let ∙ 풵푛 be the class of spaces 푋 with dense set 풵푛(푋) of homotopical 푍푛-points in 푋; 퐺 퐺 ∙ 풵푛 be the class of spaces 푋 with dense set 풵푛 (푋) of 퐺-homological 푍푛-points; 퐺 퐺 ∙ ∪퐺풵푛 be the union of classes 풵푛 over all coefficient groups. The classes 풵푛 play an important role in [9], a paper devoted to general position properties. It is clear that 풵푛 ⊂ 풵푛. On the other hand, a dendrite 퐷 with dense set of end-points belongs to 풵∞ but not to 풵1. By Br we shall denote the class of metrizable separable Baire spaces. The following proposition describes the relation between the introduced classes. Proposition 11.1. Let 푛 ∈ 휔 ∪ {∞} and 퐺 be a coefficient group. Then ℤ 퐺 ∩ 퐻 (1) 풵푛 ⊂ 풵푛 ⊂ 풵푛 ⊂ 퐻∈휎(퐺) 풵푛 ; 푛 ∩ 퐻 퐺 (2) Br ∩ 푙푐 ∩ 퐻∈휎(퐺) 풵푛 ⊂ 풵푛 ; 퐺 (3) 풵0 = 풵0 ; 1 퐺 (4) LC ∩ 풵1 ⊂ 풵1; 1 ℤ (5) LC ∩ 풵2 ∩ 풵푛 ⊂ 풵푛; 푛 ℤ (6) LC ∩ Br ∩ 풵2 ∩ 풵푛 ⊂ 풵푛. Proof: (1) Follows from Theorem 3.2(4), Proposition 5.3, and Theorem 5.5. ∩ 퐻 푛 (2) If 푋 ∈ 퐻∈휎(퐺) 풵푛 is a metrizable separable Baire 푙푐 -space, 퐻 then for every group 퐻 ∈ 휎(퐺), the set 풵푛 (푋) of 퐻-homological 70 T. BANAKH, R. CAUTY, AND A. KARASSEV

푍푛-points is a dense 퐺훿-set in X∩ according to Theorem 9.2(4). 퐻 By Theorem 5.5, the intersection 퐻∈휎(퐺) 풵푛 (푋) coincides with 퐺 풵푛 (푋) and is dense in 푋, being the countable intersection of dense 퐺 퐺훿-sets in the Baire space 푋. Hence, 푋 ∈ 풵푛 . (3)–(5) Follow immediately from Theorem 3.2 and Theorem 3.3. ℤ (6) Assume that 푋 ∈ 풵2 ∩ 풵 is a metrizable separable Baire 푛 ℤ LC -space. By Theorem 9.2, the sets 풵2(푋) and 풵푛 (푋) are dense ℤ 퐺훿 in 푋. Since 푋 is Baire, the intersection 풵2(푋) ∩ 풵푛 (푋) is dense 퐺훿 in 푋. By Theorem 3.3, the latter intersection consists of homotopical 푍푛-points. So 푋 ∈ 풵푛. □ 퐺 Multiplication formulas for the classes 풵푛 and 풵푛 follow immediately from Theorem 6.1 for homotopical and homological 푍푛-sets. Theorem 11.2 (Multiplication Formulas). Let 푛, 푚 ∈ 휔 ∪ ∞ and 푋 and 푌 be Tychonoﬀ spaces.

(1) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚, then 푋 × 푌 ∈ 풵푛+푚+1:

풵푛 × 풵푚 ⊂ 풵푛+푚+1. 퐺 퐺 (2) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚 for a coeﬃcient group 퐺, then 퐺 푋 × 푌 ∈ 풵푛+푚:

퐺 퐺 퐺 풵푛 × 풵푚 ⊂ 풵푛+푚.

푅 푅 (3) If 푋 ∈ 풵푛 and 푌 ∈ 풵푚 for a coeﬃcient group 푅 with 푅 휎(푅) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π}, then 푋 × 푌 ∈ 풵푛+푚+1:

푅 푅 푅 풵푛 × 풵푚 ⊂ 풵푛+푚+1.

퐺 Division and 푘-root formulas for the classes 풵푛 look as follows. For a class C of topological spaces, we denote by ∃∘C the class of topological spaces 푋 containing a non-empty open subspace 푈 ∈ C. Theorem 11.3 (Division Formulas). Let 푋 and 푌 be topological spaces. 퐹 퐹 (1) If 푋 × 푌 ∈ 풵푛+푚 for some ﬁeld 퐹 , then either 푋 ∈ 풵푛 or 퐹 푌 ∈ 풵푚: ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 71

풵퐹 푛+푚 = 풵퐹 . 퐹 푛 Top ∖ 풵푚 푅 (2) Assume that 푋 × 푌 ∈ 풵푛+푚 for a coefficient group 푅 with 휎(푅) ⊂ {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π} and either 휎(푅) is finite or 푋 is a metrizable separable Baire 푙푐푛-space. Then either 푅 푋 ∈ 풵푛 or else some non-empty open set 푈 ⊂ 푌 belongs ∪ 퐻 to the class 퐻∈푑(푅) 풵푚: 푅 풵푛+푚 ∪ 퐻 ⊂ ∃∘ 풵 . 푛 푅 퐻∈푑(푅) 푚 Br ∩ 푙푐 ∖ 풵푛 퐺 (3) Assume that 푋 ×푌 ∈ 풵푛+푚+1 for a coefficient group 퐺 and either 휎(퐺) is finite or 푋 is a metrizable separable Baire 푛 퐺 푙푐 -space. Then either 푋 ∈ 풵푛 or else some non-empty ∪ 퐹 open set 푈 ⊂ 푌 belongs to the class 퐹 ∈휑(퐺) 풵푚: 퐺 풵푛+푚+1 ∪ 퐹 ⊂ ∃∘ 풵 . 푛 퐺 퐹 ∈휑(퐺) 푚 Br ∩ 푙푐 ∖ 풵푛

퐹 Proof: (1) Assume that 푋 × 푌 ∈ 풵푛+푚 for some field 퐹 , but 퐹 퐹 퐹 푋∈ / 풵푛 and 푌∈ / 풵푚. Let 푈 be the interior of the set 푋 ∖ 풵푛 (푋) 퐹 and 푉 be the interior of the set 푌 ∖ 풵푚(푌 ). The product 푈 × 푉 is a non-empty open subset of 푋 × 푌 and thus contains some 퐹 - homological 푍푛+푚-point (푥, 푦) in 푋 × 푌 . By Lemma 8.1(1), either 퐹 퐹 푥 ∈ 풵푛 (푋) or 푦 ∈ 풵푚(푌 ). Both cases are not possible by the choice 퐹 퐹 of 푈 and 푉 . This contradiction shows that 푋 ∈ 풵푛 or 푌 ∈ 풵푚. 푅 (2) Assume that 푋 × 푌 ∈ 풵푛+푚 for a group 푅 with 휎(푅) ⊂ 푅 {ℚ, ℤ푝, 푅푝 : 푝 ∈ Π} but 푋∈ / 풵푛 . The latter means that 푋 contains 푅 a non-empty open set 푊 ⊂ 푋 disjoint with the set 풵푛 (푋) of 푅- homological 푍푛-points of 푋. It is necessary to find an open set ∪ 퐻 푈 ∈ 퐻∈푑(푅) 풵푚. Assume conversely that no such set 푈 exists. 퐻 This means that for every 퐻 ∈ 푑(퐺), the set 풵푚 (푌 ) is nowhere dense in 푌 . If 휎(푅) is finite, then so is the set ∪푑(푅) and we can find a non- 퐻 empty open set 푈 ⊂ 푋 disjoint with 퐻∈푑(푅) 풵푚 (푌 ). 72 T. BANAKH, R. CAUTY, AND A. KARASSEV

By Theorem 8.2(2), no point (푥, 푦) ∈ 푊 ×푈 is an 푅-homological 푍푛+푚-point in 푋 × 푌 . But this contradicts the density of the set 푅 풵푛+푚(푋 × 푌 ) in 푋 × 푌 . Next, we consider the case when 휎(푅) is inﬁnite and 푋 is a metrizable separable Baire 푙푐푛-space. By Theorem 9.2, for every 퐻 group 퐻 ∈ 휎(푅), the∩ set 풵푛 (푋) is of type 퐺훿 in 푋 and by The- 퐺 퐻 푅 orem 5.5, 풵푛 (푋) = 퐻∈휎(푅) 풵푛 (푋). Assuming that 풵푛 (푋) is not dense in the Baire space 푋, we conclude that for some group 퐻 퐻 ∈ 휎(푅), the set 풵푛 (푋) is not dense in 푋. Since 휎(퐻) = {퐻} is ﬁnite, we may apply the preceding case to conclude that some nonempty open set 푈 ⊂ 푌 belongs to the class

∪ 퐷 ∪ 퐷 풵푚 ⊂ 풵푚. 퐷∈푑(퐻) 퐷∈푑(퐺)

퐺 (3) Assume that 푋 × 푌 ∈ 풵푛+푚+1 for a coefficient group 퐺, but 퐺 푋∈ / 풵푛 . The latter means that 푋 contains a non-empty open set 퐺 푊 ⊂ 푋 disjoint with the set 풵푛 (푋) of 퐺-homological 푍푛-points of ∪ 퐹 푋. It is necessary to find an open set 푈 ∈ 퐹 ∈휑(퐺) 풵푚. Assume conversely that no such set 푈 exists. This means that for every 퐹 퐹 ∈ 휑(퐺), the set 풵푚(푌 ) is nowhere dense in 푌 . If 휎(퐺) is finite, then the set 휑(퐺) is also infinite,∪ and we can 퐹 find a non-empty open set 푈 ⊂ 푋 disjoint with 퐹 ∈휑(퐺) 풵푚(푌 ). By Theorem 8.2(1), no point (푥, 푦) ∈ 푊 × 푈 is a 퐺-homological 푍푛+푚+1-point in 푋 × 푌 . But this contradicts the density of the set 퐺 풵푛+푚+1(푋 × 푌 ) in 푋 × 푌 . The case of infinite 휎(퐺) can be reduced to the previous case by the same argument as in (2). □

Theorem 11.4 (푘-Root Formulas). Let 푛 ∈ 휔 ∪ {∞}, 푘 ∈ ℕ, 푋 be a topological space, and 퐺 be a coeﬃcient group.

퐹 푘 퐹 (1) 푋 ∈ 풵푛 for some ﬁeld 퐹 if and only if 푋 ∈ 풵푛푘: √ 푘 퐹 퐹 풵푛푘 = 풵푛 .

(2) If 푋 is a metrizable separable Baire 푙푐푛-space and 푋푘 ∈ 퐺 퐺 풵푛푘+푘, then 푋 ∈ 풵푛 : ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 73 √ 푛 푘 퐺 퐺 Br ∩ 푙푐 ∩ 풵푛푘+푘 ⊂ 풵푛 .

(3) If 푋 is a metrizable separable Baire 푙푐푛푘+푘−1-space and 푘 퐺 퐺 푋 ∈ 풵푛푘+푘−1, then 푋 ∈ 풵푛 : √ 푘푛+푘−1 푘 퐺 퐺 Br ∩ 푙푐 ∩ 풵푛푘+푘−1 = 풵푛 .

Proof: (1) Can be derived by induction from Theorem 11.2(2) and Theorem 11.3(1). (2) Assume that 푋 is a metrizable separable Baire 푙푐푛-space with 푘 퐺 푋 ∈ 풵푘푛+푘. It follows from Theorem 9.2 that for every group 퐻 퐺 ∩퐻 ∈ 휎(퐺), the set 풵푛 (푋) is of type 퐺훿 in 푋. Since 풵푛 (푋) = 퐻 퐺 퐻∈휎(퐺) 풵푛 (푋), the density of 풵푛 (푋) in the Baire space 푋 will 퐻 follow as soon as we prove the density of 풵푛 (푋) in 푋 for each group 퐻 ∈ 휎(퐺). 푘 퐺 퐻 If 퐻 is a field, then the inclusion 푋 ∈ 풵푘푛+푘 ⊂ 풵푘푛+푘 combined 퐻 퐻 with the first item implies that 푋 ∈ 풵푛+1 ⊂ 풵푛 , which means that 퐻 the set 풵푛 (푋) is dense in 푋. 푘 If 퐻 = 푅푝 for some 푝, then Theorem 5.6(1) implies that 푋 ∈ 푅푝 ℚ ℤ푝 ℚ ℤ푝 풵푛푘+푘 ⊂ 풵푛푘+푘 ∩풵푛푘+푘 and the first item yields 푋 ∈ 풵푛+1 ∩풵푛+1. Applying Theorem 5.6(2) and (4), we get

ℚ ℤ푝 ℚ ℚ푝 푅푝 퐻 푋 ∈ 풵푛+1 ∩ 풵푛+1 ⊂ 풵푛 ∩ 풵푛+1 ⊂ 풵푛 = 풵푛 . 푘 If 퐻 = ℚ푝 for some 푝, then Theorem 5.6(3) implies that 푋 ∈ ℚ푝 ℤ푝 ℤ푝 풵푛푘+푘 ⊂ 풵푛푘+푘−1 ⊂ 풵푛푘 and the first item combined with Theo- ℤ푝 ℚ푝 퐻 rem 5.6(2) implies 푋 ∈ 풵푛 ⊂ 풵푛 = 풵푛 . (3) Assume that 푋 is a metrizable separable Baire 푙푐푛푘+푘−1- 푘 퐺 space and 푋 ∈ 풵푛푘+푘−1. Arguing as in the preceding case, we can reduce the problem to the case 퐺 = 푅푝 for some prime number 푝. 푅푝 푘 푘 By Theorem 9.2, 풵푘푛+푘−1(푋 ) is a dense 퐺훿-set in 푋 . Assuming 푅푝 that 풵푛 (푋) is not dense in 푋, find a non-empty open set 푈 ⊂ 푋 푅푝 푅푝 ℚ ℚ disjoint with 풵푛 (푋). Since 풵푛푘+푘−1 ⊂ 풵푛푘+푘−1 ⊂ 풵푛푘, the first ℚ ℚ item implies that 푋 ∈ 풵푛 . Then 퐷 = 푈 ∩ 풵푛 (푋) is a dense 퐺훿-set 푘 푘 in 푈 by Theorem 9.2 and hence, 퐷 is a dense 퐺훿 set in 푈 . Since 74 T. BANAKH, R. CAUTY, AND A. KARASSEV

푘 푘 푅푝 푘 푋 is a Baire space, there is a point ⃗푥 ∈ 퐷 ∩ 풵푛푘+푘−1(푋 ) which can be written as ⃗푥 = (푥1, . . . , 푥푘). Each point 푥푖 is a ℚ-homological but not an 푅푝-homological 푍푛-point in 푋. This means that for some 푛푖 ≤ 푛 the group

퐻푛푖 (푋, 푋 ∖{푥푖}; 푅푝) = 퐻푛푖 (푋, 푋 ∖{푥푖})⊗푅푝 is not trivial. Since 푥푖 is a ℚ-homological 푍푛-set in 푋, the group 퐻푛푖 (푋, 푋 ∖ {푥푖}) cannot contain an element of inﬁnite order. Since 퐻푛푖 (푋, 푋∖{푥푖})⊗푅푝∕=0, the group 퐻푛푖 (푋, 푋 ∖ {푥푖}) contains an element of order 푝. ∑푖 Let 푚푖 = 푖 − 1 + 푗=1 푛푗 for 푖 ≤ 푘. The torsion product

퐻푛1 (푋, 푋 ∖ {푥1}) ∗ 퐻푛2 (푋, 푋 ∖ {푥2}) contains an element of order 2 2 푝 and so does the group 퐻푚2 (푋 , 푋 ∖ {(푥1, 푥2)}) by the Künneth formula. Now by induction, we can show that for every 푖 ≤ 푘, the 푖 푖 homology group 퐻푚푖 (푋 , 푋 ∖ {(푥1, . . . , 푥푖)}) contains an element of order 푝. For 푖 = 푘, we get that the group 퐻 (푋푘, 푋푘 ∖ {⃗푥}) 푚푘 contains an element of order 푝 and hence, the tensor product 퐻 (푋푘, 푋푘 ∖ {⃗푥}) ⊗ 푅 = 퐻 (푋푘, 푋푘 ∖ {⃗푥}; 푅 ) 푚푘 푝 푚푘 푝 is not trivial, which is not possible because 푚푘 ≤ 푛푘 + 푘 − 1 and ⃗푥 푘 is an 푅푝-homological 푍푛푘+푘−1-point in 푋 . □ Remark 11.5. A dendrite 퐷 with a dense set of end-points is an example of a space in which 푍∞-points form a dense 퐺훿-set. Being large in the sense of a Baire category, the set of 푍∞-points of 퐷 is small in a geometric sense: It is locally ∞-negligible in 퐷. On the other hand, W̷lodzimierz Kuperberg [29] has constructed a finite polyhedron 푃 in which the set of all 푍∞-points fails to be locally ∞- negligible. Yet, according to I. Namioka [32], the set 푍 of 푍∞-points in a finite-dimensional LC∞-space 푋 is small in a homological sense: 퐻푘(푈, 푈 ∖푍) = 0 for any open set 푈 ⊂ 푋 and any 푘 ∈ 휔. This result of Namioka is similar in spirit to our Theorem 4.4 asserting that a closed trt-dimensional subspace 퐴 ⊂ 푋 consisting of 퐺-homological 푍∞-points of 푋 is a 퐺-homological 푍∞-set in 푋.

12. 푍푛-points and dimension In this section we study the dimension properties of spaces all of whose points are homological 푍푛-points. First, we note a simple corollary of Theorem 4.3. Theorem 12.1. If a (separable metrizable) topological space 푋 ∈ 퐺 ∪퐺풵푛 , then trind(푋) ≥ trt(푋) ≥ 1+푛 (and hence dim(푋) ≥ 1+푛). ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 75

A similar lower bound holds also for the cohomological and extension dimensions. Given a space 푋 and a CW-complex 퐿, we write e-dim푋 ≤ 퐿 if any map 푓 : 퐴 → 퐿 deﬁned on a closed subset 퐴 ⊂ 푋 extends to a map 푓¯ : 푋 → 퐿. The extension dimension (e-dim) generalizes both the usual covering dimension dim and the cohomological dimension dim퐺 because ∙ dim 푋 ≤ 푛 if and only if e-dim푋 ≤ 푆푛 and ∙ dim퐺 푋 ≤ 푛 if and only if e-dim푋 ≤ 퐾(퐺, 푛), where 퐾(퐺, 푛) is an Eilenberg-MacLane complex, i.e., a CW-complex 퐾 with a unique non-trivial homotopy group 휋푛(퐾) = 퐺. The main (and technically most diﬃcult) result of this section follows.

ℤ 푛 Theorem 12.2. If 푋 ∈ 풵푛 is a locally compact 푙푐 -space, then dim퐺 푋 ≥ 푛 + 1 for any non-trivial abelian group 퐺.

Proof: Assume that dim퐺 푋 = 푛 < ∞ for some abelian group 퐺. By [30, Theorem 2] (or [20, Theorem 1.8]), the space 푋 contains a point 푥 having an open neighborhood 푈 ⊂ 푋 with compact closure such that for any smaller neighborhood 푉 ⊂ 푈 of 푥, the homomorphism in the relative Cechˇ cohomology groups ˇ 푛 ¯ ˇ 푛 ¯ 푖푉,푈 : 퐻 (푋, 푋 ∖ 푉 ; 퐺) → 퐻 (푋, 푋 ∖ 푈; 퐺), induced by the inclusion (푋, 푋 ∖ 푈¯) ⊂ (푋, 푋 ∖ 푉¯ ), is non-trivial. The complete regularity of the locally compact space 푋 allows us to ﬁnd a compact 퐺훿-set 퐾1 ⊂ 푈 containing 푥 in its interior. It is well known (see [33, Ch. 6, Sec. 9]) that in paracompact 푙푐푛-spaces, Cechˇ cohomology coincides with singular cohomology. Singular cohomology relates to singular homology via the following exact sequence, see [25, Sec. 3.1]: 푛 0 → Ext(퐻푛−1(푋, 퐴); 퐺) → 퐻 (푋, 퐴; 퐺) → Hom(퐻푛(푋, 퐴); 퐺) → 0. This sequence will be applied to the pairs

(푋, 푋 ∖ 퐾1) ⊂ (푋, 푋 ∖ 퐾2) ⊂ (푋, 푋 ∖ 퐾3), where 퐾3 ⊂ 퐾2 ⊂ 퐾1 are compact 퐺훿-neighborhoods of 푥 so small that the inclusion homomorphisms

퐻푘(푋, 푋 ∖ 퐾1) → 퐻푘(푋, 푋 ∖ 퐾2) → 퐻푘(푋, 푋 ∖ 퐾3) are trivial for all 푘 ≤ 푛 (the existence of such neighborhoods 퐾2 and 퐾3 follows from Lemma 1.6). 76 T. BANAKH, R. CAUTY, AND A. KARASSEV

These trivial homomorphisms induce trivial homomorphisms

푒2,1 : Ext(퐻푛−1(푋, 푋 ∖ 퐾2); 퐺) → Ext(퐻푛−1(푋, 푋 ∖ 퐾1); 퐺) and

ℎ3,2 : Hom(퐻푛(푋, 푋 ∖ 퐾3); 퐺) → Hom(퐻푛(푋, 푋 ∖ 퐾2); 퐺). Now consider the commutative diagram / 푛 / Ext(퐻푛−1(푋, 푋∖퐾3);퐺) 퐻 (푋, 푋∖퐾3;퐺) Hom(퐻푛(푋, 푋∖퐾3);퐺)

푖3,2 ℎ3,2 / 푛 / Ext(퐻푛−1(푋, 푋∖퐾2);퐺) 퐻 (푋, 푋∖퐾2;퐺) Hom(퐻푛(푋, 푋∖퐾2);퐺)

푒2,1 푖2,1 / 푛 / Ext(퐻푛−1(푋, 푋∖퐾1);퐺) 퐻 (푋, 푋∖퐾1;퐺) Hom(퐻푛(푋, 푋∖퐾1);퐺).

The exactness of rows of the diagram and the triviality of the homomorphisms 푒2,1 and ℎ3,2 imply the triviality of the homomorphism 푛 푛 푖3,1 = 푖2,1 ∘ 푖3,2 : 퐻 (푋, 푋 ∖ 퐾3; 퐺) → 퐻 (푋, 푋 ∖ 퐾1; 퐺). The local compactness of 푋 allows us to ﬁnd an open 휎-compact subset 푊 ⊂ 푋 containing the compact set 퐾1. Since the sets 퐾푖, 푖 ∈ {1, 3}, are of type 퐺훿 in 푋, the spaces 푊 ∖ 퐾푖 are 휎-compact and thus paracompact. The excision axiom for singular cohomology (see [33, Ch. 5, 푛 푛 Sec. 4, 14]) implies that 퐻 (푋, 푋 ∖ 퐾푖; 퐺) = 퐻 (푊, 푊 ∖ 퐾푖; 퐺) for 푖 ∈ {1, 3}. This observation implies that the inclusion homomor- 푛 푛 phism 푖3,1 : 퐻 (푊, 푊 ∖퐾3; 퐺) → 퐻 (푊, 푊 ∖퐾1; 퐺) is trivial. Since 푛 푊 and 푊 ∖ 퐾푖, 푖 ∈ {1, 3}, are paracompact 푙푐 -spaces, the singu- 푛 ˇ lar cohomology group 퐻 (푊, 푊 ∖ 퐾푖; 퐺) coincides with the Cech ˇ 푛 cohomology group 퐻 (푊, 푊 ∖ 퐾푖; 퐺). Consequently, the inclusion ˇ 푛 ˇ 푛 homomorphism 퐻 (푊, 푊 ∖ 퐾3; 퐺) → 퐻 (푊, 푊 ∖ 퐾1; 퐺) is trivial. ˇ ˇ 푛 By the excision axiom for Cech cohomology, 퐻 (푊, 푊 ∖ 퐾푖; 퐺) = ˇ 푛 퐻 (푋, 푋 ∖ 퐾푖; 퐺) for 푖 ∈ {1, 3}. Consequently, the inclusion homo- ˇ 푛 ˇ 푛 morphism 퐻 (푋, 푋 ∖퐾3; 퐺) → 퐻 (푋, 푋 ∖퐾1; 퐺) is trivial and so is the inclusion homomorphism 퐻ˇ 푛(푋, 푋 ∖푉¯ ; 퐺) → 퐻ˇ 푛(푋, 푋 ∖푈¯; 퐺), where 푉 is the interior of 퐾3. But this contradicts the choice of the neighborhood 푈. □ Theorem 12.2 will help us to evaluate the extension dimension of a locally compact LC푛-space all of whose points are homological 푍푛-points. ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 77

푛 ℤ Theorem 12.3. If a metrizable locally compact LC -space 푋 ∈ 풵푛 has e-dim(푋) ≤ 퐿 for some CW-complex 퐿, then 휋푘(퐿) = 0 for all 푘 ≤ 푛. Proof: Separately we shall consider the cases of 푛 = 0, 푛 = 1, and 푛 ≥ 2. 푛 = 0: It suffices to check that the CW-complex 퐿 is connected. 0 Since each point of the LC -space 푋 is a homological 푍0-point, 푋 contains no isolated point and thus, 푋 contains an arc 퐽 connecting two distinct points 푎, 푏 ∈ 푋. Assuming that the complex 퐿 is disconnected, consider any map 푓 : {푎, 푏} → 퐿 sending the points 푎 and 푏 to different components of 퐿. Because of the connectedness of 퐽 ⊂ 푋, the map 푓 does not extend to 푋, which contradicts e-dim푋 ≤ 퐿. 푛 = 1: We should prove the simple-connectedness of 퐿. Theo- rem 12.1 implies that dim(푋) > 1. Then there is a point 푥 ∈ 푋 whose any neighborhood 푈 ⊂ 푋 has dimension dim 푈 > 1. Since 푋 is an LC1-space, the point 푥 has a closed neighborhood 푁 such that any map 푓 : ∂퐼2 → 푁 is null-homotopic in 푋. Moreover, we can assume that 푁 is a Peano continuum. Since dim 푁 > 1, the continuum 푁 is not a dendrite and, consequently, contains a simple closed curve 푆 ⊂ 푁. Assuming that the CW-complex 퐿 is not simply-connected, we can find a map 푓 : 푆 → 퐿 that is not homotopic to a constant map. Then the map 푓 cannot be extended over 푋 since the identity map of 푆 is null-homotopic in 푋. But this contradicts e-dim푋 ≤ 퐿.

푛 ≥ 2: Suppose that 휋푘(퐿) ∕= 0 for some 푘 ≤ 푛. We can assume that 푘 is the smallest number with 휋푘(퐿) ∕= 0. The simple connectedness of 퐿 implies that 푘 > 1. Applying the Hurewicz isomorphism theorem, we conclude that 퐻푘(퐿) = 휋푘(퐿) ∕= 0. Since e-dim푋 ≤ 퐿, we may apply [21, Theorem 7.14] to conclude that dim 푋 ≤ 푛. But this contradicts Theorem 12.2. □ 퐻푛(퐿)

13. Dimension of spaces all of whose points are 푍∞-points In this section we study the dimension properties of spaces all of whose points are homological 푍∞. We shall show that locally compact ANR’s with this property are infinite-dimensional in a 78 T. BANAKH, R. CAUTY, AND A. KARASSEV rather strong sense: They cannot be 퐶-spaces and have infinite cohomological dimension. We recall that a topological space 푋 is defined to be a 퐶-space if for any sequence {풱푛 : 푛 ∈ 휔} of open covers of 푋 there exists a sequence {풰푛 : 푛 ∈ 휔} of disjoint∪ families of open sets in 푋 such that each 풰푛 refines 풱푛 and {풰푛 : 푛 ∈ 휔} is a cover of 푋. By [23, 6.3.8], each metrizable countable-dimensional space is a 퐶-space.

ℤ 0 Theorem 13.1. If 푋 ∈ 풵∞ is a locally compact metrizable LC - space, then (1) 푋 is neither trt-dimensional nor countable-dimensional; ∞ (2) if 푋 is an 푙푐 -space, then dim퐺 푋 = ∞ for any abelian group 퐺; (3) if 푋 is an LC∞-space, then e-dim푋 ∕≤ 퐿 for any non- contractible CW-complex 퐿; (4) if 푋 is locally contractible, then 푋 fails to be a 퐶-space. Proof: (1)–(3) Follow from Theorem 4.4, Theorem 12.2, and The- orem 12.3, respectively. (4) First, we prove the fourth item under a stronger assumption that all points of 푋 are homotopical 푍∞-points. Assume that 푋 is a 퐶-space. By a theorem of John H. Gresham [24], the 퐶-space 푋, being locally contractible, is an ANR. Then the product 푋 × 푄 is a Hilbert cube manifold according to the ANR theorem of Edwards, see [15, Theorem 44.1]. The product 푋 × 푄, being a 푄-manifold, contains an open subset 푈 ⊂ 푋 × 푄 whose closure 푈 is homeomorphic to the Hilbert cube 푄 and whose boundary ∂푈 ¯ is a 푍∞-set in 푈. Now consider the multivalued map Φ : 푋 → 푈, assigning to each point 푥 ∈ 푋 the set Φ(푥) = 푈∖({푥}×푄). Our goal is to show that the map Φ has a continuous selection 푠 : 푋 → 푈. Assuming that this is done, consider the map 푠∘pr : 푈 → 푈, where pr : 푈 → 푋 stands for the natural projection of 푈 ⊂ 푋 × 푄 onto the ﬁrst factor. This map is continuous and has no ﬁxed point, which is a contradiction. So, it remains to construct the continuous selection 푠 of the multivalued map Φ. The existence of such a selection will follow from selection theorem of V. V. Uspenskij [39] as soon as we verify that ∙ for each 푥 ∈ 푋, the complement 푈 ∖ Φ(푥) = 푈 ∩ ({푥} × 푄) is a 푍∞-set in 푈; ON HOMOTOPICAL AND HOMOLOGICAL 푍푛-SETS 79

∙ for any compact set 퐾 ⊂ 푈¯, the set {푥 ∈ 푋 : Φ(푥) ⊃ 퐾} is open in 퐾.

The first condition holds since each point of 푋 is a 푍∞-point in 푋 and the boundary ∂푈 is a 푍∞-set in 푈. The second condition holds because {푥 ∈ 푋 : 퐹 (푥) ⊃ 퐾} = 푋 ∖ pr(퐾). This completes the proof of the special case when all points of 푋 are 푍∞-points. Now assume that all points of 푋 are merely homological 푍∞- points in 푋. Then the points of the product 푋 × [0, 1] are (homotopical) 푍∞-points by Corollary 8.4. Now the preceding discussion implies that 푋 × [0, 1] fails to be a 퐶-space. Taking into account that the product of a metrizable 퐶-space with the interval is a 퐶- space [1, Theorem 2.2.3], we conclude that 푋 is not a 퐶-space. □ Remark 13.2. The last item of Theorem 13.1 is true in a bit stronger form: Each locally compact locally contractible space 푋 ∈ 퐺 ∪퐺풵∞ fails to be a 퐶-space; see [5]. However, the proof of this stronger result requires non-elementary tools like a homological version of Uspenskij’s selection theorem combined with a homological version of the Brouwer fixed point theorem. On the other hand, this stronger result would follow from Theorem 4.4 if each compact C-space were trt-dimensional. However, we are not sure that this is true. Remark 13.3. In light of Theorem 13.1, it is interesting to mention that a compact AR all of whose points are 푍∞-points need not be homeomorphic to the Hilbert cube. A suitable counterexample was constructed in [17, Corollary 9.3]. Problem 13.4. Let 푋 be a compact absolute retract all of whose points are 푍∞-points. Is 푋 strongly infinite-dimensional? Is 푋 × 퐼 homeomorphic to the Hilbert cube? In this respect let us mention the following characterization of Hilbert cube manifolds [7] which can be deduced from Theorem 4.3 and the homological characterization of Q-manifolds due to Robert J. Daverman and John J. Walsh [17]; see also [9]. Theorem 13.5. A locally compact ANR-space 푋 is a Hilbert cube manifold if and only if (1) 푋 has the disjoint disks property; (2) each point of 푋 is a homological 푍∞-point; 80 T. BANAKH, R. CAUTY, AND A. KARASSEV

(3) each map 푓 : 퐾 → 푋 of a compact polyhedron can be approximated by a map with trt-dimensional image. We recall that a space 푋 has the disjoint disks property if any two maps 푓, 푔 : 퐼2 → 푋 can be approximated by maps with disjoint images. Theorem 4.4 implies that each closed trt-dimensional subspace of the Hilbert cube 푄 is a homological 푍∞-set in 푄. By [2, Proposition 4.7], each compact trt-dimensional space is a 퐶-space.

Question 13.6. Is a closed subset 퐴 ⊂ 푄 a homological 푍∞-set in 푄 if 퐴 is weakly inﬁnite-dimensional or a 퐶-space?

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(Banakh) Ivan Franko National University of Lviv (Ukraine), Uniwersytet Humanistyczno-Przyrodniczy, (Kielce, Poland), Nipissing University, (North Bay, Ontario, Canada) E-mail address: [email protected]; [email protected]

(Cauty) Universite´ Paris VI (France) E-mail address: [email protected]

(Karassev) Nipissing University (North Bay, Ontario, Canada) E-mail address: [email protected]