It is known that essential phantom mappings (of the second kind) between connected CW-complexes do exist. However, it appears that in the literature there are few explicit examples of such mappings. One usually finds descriptions of the domain and the codomain and an existence proof that the set of homotopy classes of mappings from the domain to the codomain is infinite. The purpose of the present paper is to describe some elementary examples of essential phantom mappings. The codomain is the n-sphere S^n, n ≥ 2, and the domain is the telescope Tn, associated with the sequence of copies of the canonical mapping f : S^n-1 → S^n-1 of odd degree p > 1. There are no essential phantom mappings whose codomain is the 1-sphere S^1

Sibe Mardešić

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RAD HAZU. MATEMATIČKE ZNANOSTIVol. 19 = 523 (2015): 143-149ELEMENTARY EXAMPLES OF ESSENTIAL PHANTOMMAPPINGSSibe MardešićAbstract. It is known that essential phantom mappings (of the sec-ond kind) between connected CW-complexes do exist. However, it appearsthat in the literature there are few explicit examples of such mappings. Oneusually finds descriptions of the domain and the codomain and an existenceproof that the set of homotopy classes of mappings from the domain to thecodomain is infinite. The purpose of the present paper is to describe someelementary examples of essential phantom mappings. The codomain is then-sphere Sn, n ≥ 2, and the domain is the telescope T n, associated withthe sequence of copies of the canonical mapping f : Sn−1 → Sn−1 of odddegree p > 1. There are no essential phantom mappings whose codomainis the 1-sphere S1.1. Introduction1.1. A mapping h : Z → W between connected CW-complexes withbasepoints ∗ is said to be a phantom mapping (of the second kind, see [9])provided the restriction h|C to any compact subset (equivalently, to any finitesubcomplex) C ⊆ Z is homotopic to a constant mapping. Since connectedCW-complexes are pathwise connected, there is no loss of generality in requir-ing that h|C ≃ ∗, for all C. Clearly, if a mapping h is homotopic to ∗, it isa phantom mapping. Therefore, of primary interest are phantom mappings,which are not homotopic to ∗, i.e., are essential mappings. It is known thatessential phantom mappings do exist [9].It appears that in the literature there are few explicit examples of essentialphantom mappings. One usually finds explicit descriptions of the domain andthe codomain and an existence proof that the set of homotopy classes ofmappings from the domain to the codomain is infinite.1.2. Professor Jaka Smrekar of the University of Ljubljana kindly com-municated to the author the following proof (here slightly modified) of theexistence of essential phantom mappings h : T n → Sn, n ≥ 2. Here Sn is2010 Mathematics Subject Classification. 55S37.Key words and phrases. Homotopy, essential mapping, phantom mapping, telescope.143144 S. MARDEŠIĆthe n-sphere and T n is the telescope, associated with the canonical mappingf : Sn−1 → Sn−1 of prime degree p > 1. A detailed description of T n is givenin Section 2. It is easy to show that every mapping h : T n → Sn is a phan-tom mapping (see Lemma 1 in Section 4). Therefore, in order to constructessential phantom mappings h : T n → Sn, it suffices to construct essentialmappings h : T n → Sn.By a generalization of the Hopf classification theorem, proved by C. H.Dowker (see [2], Theorem 7.5), there is a bijection between the set [Pn, Sn]of homotopy classes of mappings from an n-dimensional polyhedron Pn toSn and the n-th integral cohomology group Hn(Pn;Z) of that polyhedron.Since dim T n = n, we see that there is a bijection between the sets [T n, Sn]and Hn(T n;Z). Therefore, it suffices to show that Hn(T n;Z) 6= 0. Actually,the latter group is uncountable. Indeed, it is known that Hn(T n;Z) ≈ Ẑp/Z,where Ẑp denotes the group of p-adic integers and Z is naturally embeddedin Ẑp ([4], Example 3F.9). It is also known that the group Ẑp is uncountable([4], Example 3F.6). Since Z is countable, Ẑp/Z is also uncountable.1.3. The following explicit example of an essential phantom mappingh : Z → Sn, n ≥ 3, is described in [5] (pp. 83-84) and [9] (pp. 1211-1212). Let A and B be nonempty complementary sets of primes and letφA : Sn−1 → Sn−1A and φB : Sn−1 → Sn−1B be the respective localizationsof the (n − 1)-sphere Sn−1. Let λ : Sn−1 → Sn−1A ∨ Sn−1B be the compo-sition of two mappings. The first one Sn−1 → Sn−1 ∨ Sn−1 is the quo-tient mapping, obtained by collapsing to a point the equator of Sn−1. Thesecond one is φA ∨ φB : Sn−1 ∨ Sn−1 → Sn−1A ∨ Sn−1B . Let Z be the CW-complex obtained by attaching the n-cell Dn to Sn−1A ∨Sn−1B via the mappingλ : ∂Dn → Sn−1A ∨ Sn−1B . Then the quotient mapping h : Z → Sn, obtainedfrom Z by collapsing Sn−1A ∨Sn−1B to a point, is an essential phantom mappingh : Z → Sn.The present paper describes explicitly elementary examples of some essen-tial phantom mappings h : T n → Sn, n ≥ 2. There are no essential phantommappings, whose codomain is S1 [7].2. The telescope T n2.1. To fix notations we recall the definition of the telescope T n. Firstconsider the 1-sphere S1 = {ζ ∈ C : |ζ| = 1} with the basepoint ∗ = 1.The canonical mapping f : S1 → S1 of degree p > 1 is defined by f(ζ) = ζp,ζ ∈ S1. Note that f(∗) = ∗. In order to define the canonical mappingf : Sn−1 → Sn−1 of degree p > 1, for n > 2, one views the (n − 1)-sphereSn−1 as the (unreduced) (n − 2)-fold suspension Σn−2(S1) with basepointELEMENTARY EXAMPLES OF ESSENTIAL PHANTOM MAPPINGS 145∗ ∈ S1 ⊆ Sn−1. Then f : Sn−1 → Sn−1 is the (n − 2)-fold suspension off : S1 → S1. Note that also in this case f(∗) = ∗.The mapping cylinder M associated with f : Sn−1 → Sn−1 is obtainedfrom Sn−1 × [0, 1] by identifying the points (ζ1, 1) and (ζ2, 1), ζ ∈ Sn−1,whenever f(ζ1) = f(ζ2). The corresponding quotient mapping will be denotedby φ : Sn−1 × [0, 1] → M . The natural CW-structure of M consists of two0-cells, v− = φ(∗, 0) and v+ = φ(∗, 1), called the initial and the terminalbasepoints of M , of the arc A = φ(∗× [0, 1]) as the only 1-cell, called the spineof M , of two (n−1)-spheres B− = φ(Sn−1 ×0) and B+ = φ(Sn−1 ×1), calledthe initial and the terminal bases ofM and of the n-cell M = φ(S(n−1)×[0, 1]).Note that M is a connected 2-dimensional polyhedron.2.2. The telescope T n, associated with the sequence Sn−1f−→ Sn−1 f−→Sn−1 → . . . is obtained from the direct sum M1 ⊔ M2 ⊔ . . . of the sequenceM1,M2, . . . of copies of the mapping cylinder M of f : Sn−1 → Sn−1, byidentifying the terminal base of Mi with the initial base of Mi+1, i ∈ N. The(n− 1)-spheres obtained in this way will be denoted by Bi, i = 0, 1, . . .. Themapping cylinders Mi and their spines Ai are naturally embedded in T n. Thetwo endpoints of Ai are denoted by vi−1 and vi. Note that T n = M1 ∪M2 ∪. . .is a connected 2-dimensional polyhedron. It has a natural CW-structure,which consists of the points vi as 0-cells, of the arcs Ai as 1-cells, of the(n − 1)-spheres Bi as the (n − 1)-cells and of mapping cylinders Mi as then-cells.3. The mapping h : T n → Sn3.1. If in the cylinder Sn−1 × [0, 1] we collapse Sn−1 × 0 to a point v−and we collapse Sn−1 ×1 to a point v+, we obtain the (unreduced) suspensionΣSn−1 of Sn−1 with vertices v−, v+ and we obtain the corresponding quotientmapping ψ : Sn−1 × [0, 1] → ΣSn−1. Note that, whenever y ∈ M belongs tothe terminal base B+ of M , then φ−1(y) ⊆ Sn−1 × 1 and thus, ψ(φ−1(y)) =v+. If y ∈ M does not belong to B+, then φ−1(y) is a single point of Sn−1 ×[0, 1) and so is ψ(φ−1(y)). Therefore, there is a unique mapping χ : M →ΣSn−1 = Sn such that χφ = ψ. Clearly, χ is the quotient mapping, whichcollapses the initial base of M to v− and collapses the terminal base of M tov+.We now repeat the described procedure for every mapping cylinder Mi,i ∈ N, of the telescope T n, i.e., we collapse to a point vi each (n − 1)-sphereBi (including B0) and we consider the corresponding quotient mappings χi.We obtain a CW-complex T n∗, called the pinched telescope. We also obtaina quotient mapping χ : T n → T n∗ such that χ|Mi = χi. Note that χ mapsthe initial and the terminal bases Bi−1 and Bi of Mi to the vertices vi−1and vi of the n-sphere Sni = χi(Mi), respectively. The CW-structure of T∗n146 S. MARDEŠIĆconsists of the points v0, v1, . . . as 0-cells, of the arcs A1, A2, . . . as 1-cells andof the n-spheres Sni as n-cells, i ∈ N. Notice that T 2∗ looks like an infinitestring-of-beads.3.2. Our next goal is to define a mapping χ′ of T n∗ to the wedge of asequence of pointed n-spheres, S = Sn1 ∨ Sn2 ∨ . . .. In the proof we need thefollowing fact. Let v− and v+ be the vertices of the (unreduced) suspensionΣSn−1, let ∗ ∈ Sn−1 be the basepoint of Sn−1 and let A be the arc Σ(∗).Then the quotient space (ΣSn−1)/A is an n-sphere Sn. If χ′ : ΣSn−1 → Sn isthe corresponding quotient mapping and we denote the point A of (ΣSn−1)/Aby ∗, then χ′(∗) = ∗.The assertion is a very special case of well-known facts concerning decom-positions of manifolds. If X is a cellular subset of a closed n-manifold N , i.e.,X is the intersection of a sequence of n-disks Di ⊆ N , where Di+1 ⊆ IntDi,i ∈ N, then the decomposition GX of N , whose only nondegenerate elementis X , is a shrinkable upper semicontinuous decomposition ([1], 1, Proposition4 and 6, Proposition 2). Moreover, if G is a shrinkable upper semicontinuousdecomposition of a compact metric space N , then the corresponding quotientmapping π : N → N/G is a near-homeomorphism, i.e., it can be approximatedby homeomorphisms ([1], 5, Theorem 2) and thus, N/G is homeomorphic toN .In our case, A is a cellular subset of the n-sphere ΣSn−1 and thus,ΣSn−1/A is also an n-sphere. It is now clear that, by collapsing the unionof arcs A = A1 ∪ A2 ∪ . . . ⊆ T n∗ to a point ∗, one obtains the wedgeS = Sn1 ∨Sn2 ∨ . . . and a quotient mapping χ′ : T n∗ → S, χ′(∗) = ∗. Note thatthe natural CW-structure of S consists of a single 0-cell ∗, and of a sequenceof n-cells Sn1 , Sn2 , . . ..The next mapping considered is the folding map χ′′ : S → Sn. It mapseach summand Sni of S to Sn by the identity mapping. Note that χ′′(∗) = ∗.Finally, we define the desired mapping h : T n → Sn, by putting h = χ′′χ′χ.Note that h(∗) = ∗.Theorem 1. If p > 1 is an odd integer, the mapping h : T n → Sn, n ≥ 2,is an essential phantom mapping.4. Proof of Theorem 14.1. We first prove the following lemma.Lemma 1. Every mapping k : T n → Sn from the telescope T n to then-sphere Sn, n ≥ 2, is a phantom mapping.Proof. It is well known that there is a strong deformation retraction r ofM to its terminal base B+. Denote by ri : Mi → Bi, i ∈ N, the mappingswhich correspond to r. For i ≤ n, put Tin = Mi ∪ . . . ∪ Mn and note thatELEMENTARY EXAMPLES OF ESSENTIAL PHANTOM MAPPINGS 147Tin = Mi∪Ti+1n, i < n. Also consider the deformation retractions rin : Tin →Ti+1n, where rin|Mi = ri and rin|Ti+1n is the identity mapping. Note thatTnn = Mn and put rnn = rn : Mn → Bn. Clearly, the composition rn1 =rnn . . . r2nr1n : T1n → Bn is a strong deformation retraction. In particular,the inclusion in : Bn → T1n has the property that inrn1 is homotopic to theidentity mapping on T1n. Therefore, k|T1n ≃ kinrn1 = (k|Bn)rn1 . Now notethat every mapping of Sn−1 to Sn is homotopic to the constant mapping ∗.Since Bn is an (n − 1)-sphere, k|Bn ≃ ∗. However, this implies that alsok|T1n ≃ ∗. Now consider a compact subset C of T n. Since T11 ⊆ T12 . . . andT11 ∪ T12 . . . = T n, one concludes that there is an n ∈ N such that C ⊆ T1n.Consequently, k|T1n ≃ ∗ implies k|C ≃ ∗ and k is indeed a phantom mapping.4.2. In view of Lemma 1, the proof of Theorem 1 will be completed if weprove the following lemma.Lemma 2. If p > 1 is an odd integer, then the mapping h : T n → Sn,n ≥ 2, is essential, i.e., it is not homotopic to the constant mapping ∗.Proof. In proving the assertion, we will use cellular cohomology groupsof CW-complexes with coefficients in Z (see e.g., [6], [10], [4], [3]). The map-ping h : T n → Sn induces a homomorphism hn∗ : Hn(Sn;Z) → Hn(T n;Z).Since homotopic cellular mappings induce equal homomorphisms of cellularcohomology groups (see e.g., [3], Theorem 12.1.9 or [4], p. 201) and the ho-momorphism H2(Yµ;Z) → H2(Tiµ ;Z), induced by the constant mapping ∗,equals 0, Lemma 2 will be proved if we show that hn∗ 6= 0.Recall that Sn has a CW-structure, which consists of a single 0-cell ∗ anda single n-cell Sn. Let a be the cellular n-cochain of Sn, which assumes an oddvalue α at the n-cell Sn. Since in Sn there are no (n+ 1)-cells, a is a cocycle.The folding mapping χ′′ associates with a the n-cocycle χ′′n(a), which oneach n-cell Sni of the cellular chain complex of S assumes the same valueα. Furthermore, the mapping χ′ associates with the n-cocycle χ′′n(a) then-cocycle χ′nχ′′n(a), which on each n-cell ΣS1i of the cellular chain complexof T n∗ assumes the same value α. Finally, the mapping χ associates with then-cocycle χ′nχ′′n(a) the n-cocycle h∗(a) = χ∗χ′nχ′n(a), which on each n-cellMi of the cellular chain complex of T n assumes the same value α. Therefore,to complete the proof of Lemma 2, it suffices to show that hn(a) is not thecoboundary of some (n− 1)-cochain of the cellular chain complex of T n.4.3. The latter assertion is an immediate consequence of the next lemma.Lemma 3. Let p > 1 be an odd integer and let a be the n-cochain of thecellular chain complex of T n, which on each n-cell Mi, i ∈ N, of T n assumesthe same odd value α ∈ Z, i.e., a(Mi) = α, for all i ∈ N. Then a is ann-cocycle, which is not a coboundary.148 S. MARDEŠIĆProof. Consider the telescope T n and assume that there is an (n − 1)-cochain b such that δb = a. Since the n-cells of T n are the mapping cylindersMi, we must have (δb)(Mi) = a(Mi) = α. Note that, for n > 2, the (n − 1)-cochain b is a function on the set {Bi|i ∈ N}. Put b(Bi) = βi. If n = 2,the set of 1-cells also includes the arcs Ai, i ∈ N. By definition, (δb)(Mi) =b(∂Mi) and thus, b(∂Mi) = α. Since ∂Mi = pBi − Bi−1, one concludes thatb(∂Mi) = b(pBi −Bi−1) = pβi − βi−1. Consequently,(4.1) pβi − βi−1 = α, i ∈ N.Subtracting pairs of consecutive equalities (4.1), one sees that(4.2)β1 −β0 = p(β2 −β1), β2 −β1 = p(β3 −β2), . . . , βi−βi−1 = p(βi+1 −βi) = . . . .The equalities (4.2) show that(4.3) β1 − β0 = p(β2 − β1) = p2(β3 − β2) = . . . = pi(βi+1 − βi) = . . . .Since p > 1, formula (4.3) shows that the integer β1 −β0 has arbitrarily largedivisors pi. This is possible only when β1 −β0 = 0, i.e., β0 = β1. In that case,for i = 1, (4.1) becomes α = pβ1 −β0 = (p−1)β0. Since it was assumed that pis odd, p− 1 is even and so is α = (p− 1)β0. However, this is in contradictionwith the assumption that α is an odd number. Acknowledgements.The author wishes to express his thanks to Professor Jaka Smrekar of theUniversity of Ljubljana, for sharing information on phantom mappings andmaking his Master’s thesis available to the author [11].References[1] R. J. Daverman, Decompositions of manifolds, Academic Press, New York, 1986.[2] C. H. Dowker, Mapping theorems for noncompact spaces, Amer. J. Math. 69 (1947),200–242.[3] R. Geoghegan, Topological methods in group theory, Springer Science + BusinessMedia, New York, 2008.[4] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.[5] P. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces,North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co.,Inc., New York, 1975.[6] A. T. Lundell and S. Weingram, The topology of CW-complexes, Van Nostrand, NewYork, 1969.[7] S. Mardešić, There are no phantom pairs of mappings to 1-dimensional CW -complexes, Bull. Polish Acad. Sci. Math. 55 (2007), 365–371.[8] S. Mardešić and J. Segal, Shape theory, North-Holland Publishing Co., Amsterdam,1982.[9] C. A. McGibbon, Phantom maps, Chapter 25 of Handbook of algebraic topology (ed.I. M. James), North-Holland, Amsterdam, 1995, pp. 1209–1257.[10] J. R. Munkres, Elements of algebraic topology, Addison-Wesley Publ. Comp., MenloPark, CA, 1984.ELEMENTARY EXAMPLES OF ESSENTIAL PHANTOM MAPPINGS 149[11] J. Smrekar, On phantom mappings, Master’s thesis, University of Ljubljana, Ljubljana,2002 (in Slovenian).Elementarni primjeri bitnih fantomskih preslikavanjaSibe MardešićSažetak. Poznato je da bitna fantomska preslikavanja(druge vrste) izmedu povezanih CW-kompleksa postoje. U lite-raturi su rijetki eksplicitni primjeri takvih preslikavanja. Običnoje dana domena i kodomena preslikavanja te egzistencijski dokazda je skup klasa homotopije iz domene u kodomenu beskonačan.U ovom radu opisuju se neki elementarni primjeri bitnih fantom-skih preslikavanja. Kodomena je n-sfera Sn, n ≥ 2, a domenaje teleskop T n pridružen nizu kopija kanonskoga preslikavanjaf : Sn−1 → Sn−1, neparnoga stupnja p > 1. Nema bitnih fan-tomskih preslikavanja čija je kodomena 1-sfera S1.Sibe MardešićDepartment of Mathematics, University of Zagreb, Bijenička cesta 30,10 002 Zagreb, P.O. Box 335, CroatiaE-mail: smardes@math.hrReceived: 13.9.2014.150

Elementary examples of essential phantom mappings

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