Comultiplications on the Localized Spheres and Moore Spaces

mathematics

Article Comultiplications on the Localized Spheres and Moore Spaces

Dae-Woong Lee

Department of Mathematics, and Institute of Pure and Applied Mathematics, Jeonbuk National University, 567 Baekje-daero, Deokjin-gu, Jeonju-si, Jeollabuk-do 54896, Korea; [email protected]

Received: 2 December 2019; Accepted: 30 December 2019; Published: 5 January 2020

Abstract: Any nilpotent CW-space can be localized at primes in a similar way to the localization of a ring at a prime number. For a collection P of prime numbers which may be empty and a localization XP of a nilpotent CW-space X at P, we let |C(X)| and |C(XP )| be the cardinalities of the sets of all homotopy comultiplications on X and XP , respectively. In this paper, we show that if |C(X)| is ﬁnite, then |C(X)| ≥ |C(XP )|, and if |C(X)| is inﬁnite, then |C(X)| = |C(XP )|, where X is the k-fold wedge Wk ni sum i=1 S or Moore spaces M(G, n). Moreover, we provide examples to concretely determine the cardinality of homotopy comultiplications on the k-fold wedge sum of spheres, Moore spaces, and their localizations.

Keywords: comultiplications; localized spheres; basic Whitehead products; Hilton formula; Moore space

MSC: Primary 55Q20; Secondary 55P60, 55Q40, 55Q15, 20F18

1. Introduction Homotopy comultiplications, one of the Eckmann–Hilton dual notions of homotopy multiplications, play a fundamental role in classical and rational homotopy theories. One reason for this is that the set of pointed homotopy classes of base point preserving continuous maps from a co-H-space X to a space Y has a canonical binary operation with identity induced by the homotopy comultiplication on a co-H-space. The calculation of homotopy comultiplications is very complicated in that there are usually many homotopy comultiplications on a given co-H-space with many different homotopy properties. Localization theories in the pointed homotopy category of simply connected (or nilpotent) CW-complexes have been developed in algebraic or topological forms. Moreover, the study of homotopy comultiplications, binary operations for homotopy groups with coefﬁcients and the same n-type structures of co-H-spaces has been carried out by several authors; see [1–8]. For example, homotopy comultiplications on the wedge sum of circles were developed by using methods of group theory in [9]. The techniques of rational homotopy were applied to obtain some rational results for ﬁnite 1-connected co-H-spaces in [10,11]. The homotopy comultiplications on the wedge sum of two Moore spaces were investigated using homological algebra in [12]. The cardinality of homotopy comultiplications on a suspension has been determined in [13]. A topological transversality theorem for multivalued maps with continuous, compact selections was developed in [14]. From the equivariant homotopy theoretic point of view, an explicit expression of the behavior of the local cohomology spectral sequence graded on the representation ring was presented with many pictures in [15]. In this paper, we study the cardinality of the set of all homotopy comultiplications on a wedge of (localized) spheres or (localized) Moore spaces, and develop the properties of the homotopy

Mathematics 2020, 8, 86; doi:10.3390/math8010086 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 86 2 of 19 comultiplications of those CW-spaces. Our methods can be used to study homotopy comultiplications on a wedge of any number of spheres. The paper is organized as follows: In Section2, we describe the fundamental notions of homotopy comultiplications and introduce the localized version of Hilton’s formula. In Section3, we establish basic facts regarding the types of homotopy comultiplications on the k-fold wedge sum of localized spheres and deﬁne certain homotopy comultiplications on the k-fold wedge sum of localized spheres and Moore spaces. In Section4, we describe the main result of this paper and give an example to illustrate it. Convention. In this paper, we work in the pointed homotopy category C of connected nilpotent CW-complexes and homotopy classes of base point preserving continuous maps. We mostly use ‘=’ for the homotopy ‘'’ unless we emphasize the homotopy. We also do not distinguish notationally between a continuous map and its homotopy class.

2. Preliminaries

Let {Xγ|γ ∈ Γ} be a family of pointed topological spaces whose base points are xγ for each γ ∈ Γ. W The wedge sum γ∈Γ Xγ of {Xγ|γ ∈ Γ} is deﬁned as the quotient space

_ Xγ = ä Xγ/{xγ|γ ∈ Γ}, γ∈Γ γ∈Γ where äγ∈Γ Xγ is the topological sum of {Xγ|γ ∈ Γ}. For any abelian group G and a positive integer n ≥ 2, there exists a 1-connected CW-complex X such that ( G if i = n; Hi(X) = 0 if i 6= n. The homotopy type of X is uniquely determined up to homotopy and it is said to be a Moore space of type (G, n). We denote the Moore space of type (G, n) (or any CW-space which has the same homotopy type to it) by M(G, n). Localization plays a pivotal role in both algebra and algebraic topology, especially in homotopy theory. For a collection P of prime numbers which may be empty, a group G is called a P-local group if the map q : G → G given by q(g) = g + g + ... + g | {z } (q−times) c is bijective for all q ∈ P . A nilpotent CW-complex X is said to be P-local if πn(X) is P-local for all n ≥ 1. Any nilpotent CW-space can be localized at primes in the sense of Sullivan [16] and Bousﬁeld and Kan [17] in a similar way to the localization of a ring at a prime number. As usual, the localization of a nilpotent CW-space X at P is denoted as XP which is unique up to homotopy. A pair (X, ψ) consisting of a space X and a base point preserving continuous map ψ : X → X ∨ X 1 2 is said to be a co-Hopf space, or co-H-space for short, if π ψ ' 1X and π ψ ' 1X, where 1X is the identity map of X and π1, π2 : X ∨ X → X are the ﬁrst and second projections, respectively. In this case, the map ψ : X → X ∨ X is said to be a homotopy comultiplication, or comultiplication for short. Homotopy comultiplication is an Eckmann–Hilton dual notion of homotopy multiplication; see [18–20]. It can be seen that (X, ψ) is a co-H-space if jψ ' ∆ : X → X × X, where j : X ∨ X → X × X is the inclusion map and ∆ is the diagonal map. Mathematics 2020, 8, 86 3 of 19

n n n n We now consider the k-fold wedge sum of spheres X := S 1 ∨ S 2 ∨ · · · ∨ S k . Let ξi : S i → X, i = 1, 2, ... , k be the canonical inclusion maps. We now construct the so-called basic Whitehead products (see also [21–24]) by using the homotopy classes of those canonical inclusion maps by induction on n. We denote the basic Whitehead products of weight 1 by ξ1, ξ2, ... , ξk that are ordered by the canonical inclusion maps in the way so that ξ1 < ξ2 < ··· < ξk. Assume that the basic Whitehead products of weight less than or equal to n − 1 have been already deﬁned and ordered so that if r < s < n, all basic Whitehead products of weight r are less than all of the basic Whitehead products of weight s. Then a basic Whitehead product of weight n is a basic Whitehead product [a, b], where a and b are basic Whitehead products of weights p and q, respectively, with p + q = n and a < b. If b is a basic Whitehead product [c, d] consisting of basic Whitehead products c and d, then the condition c ≤ a is necessary. Then the basic Whitehead products of weight n are greater than any basic Whitehead product of weight less than n and are ordered among themselves arbitrarily. For a basic Whitehead product of weight n, we associate to it a string of canonical inclusion maps ξ1, ξ2, ... , ξk that appear in the basic Whitehead products. The height hs of the basic Whitehead product ws is deﬁned as hs = ∑ ti(ni − 1), i where ti is the number of the canonical inclusion map ξi that appears in the basic Whitehead product ws for s = 1, 2, 3, . . . . We are interested in the cardinality of the set of homotopy comultiplications on a co-H-space. In the case of a wedge of two spheres, we obtain the following.

Lemma 1. The cardinality of homotopy comultiplications of Sn ∨ Sm is

∞ hs+1 ∏ |πm(S )|, s=3 where 2 ≤ n < m and |X| denotes the cardinality of a set X.

Proof. See ([22], Corollary 2.9) and ([24], Corollary 2.15) for the more general case.

One of the most important theorems in algebraic topology is the Hilton’s formula [25], which is described as follows.

Theorem 1. Let w1, w2,..., ws,... be the basic Whitehead products of the k-fold wedge sum of spheres

X = Sn1 ∨ Sn2 ∨ · · · ∨ Snk .

Then for all m, we have ∞ ∼ M hs+1 πm(X) = πm(S ). s=1 Here,

• hs is the height of ws; and L∞ hs+1 • the isomorphism θ : s=1 πm(S ) → πm(X) is given by

hs+1 hs+1 θ|πm(S ) = ws∗ : πm(S ) → πm(X).

The direct sum is ﬁnite for each positive integer m since the height hs goes to inﬁnity as s → ∞. Mathematics 2020, 8, 86 4 of 19

3. Comultiplications on a Wedge of (Localized) Spheres and Moore Spaces In this section, we investigate the cardinality of the set of all homotopy comultiplications on the k-fold wedge sum of localized spheres and Moore spaces. In particular, we are interested in determining inequalities and relationships between the cardinalities of the sets of homotopy comultiplications on the wedge sum of spheres, Moore spaces and their localizations.

3.1. The Localized Version of Hilton’s Formula

Let P be a collection of prime numbers and let GP denote the P-localization of a nilpotent group G. It is well known that if h : G → H is any homomorphism in the category of nilpotent groups, then we have a unique map hP : GP → HP such that the diagram

h G / H

e e

hP GP / HP is strictly commutative, where e : G → GP is a P-localizing map; similarly, for the nilpotent CW-spaces.

Lemma 2. Let ψ : X → X ∨ X be a comultiplication on a nilpotent CW-space X. Then the localization ψP : XP → XP ∨ XP at a set P of primes is a comultiplication.

Proof. Since the localization of a nilpotent CW-space has the functorial property, all the faces including the bottom triangle colored in blue of the following triangular prism are strictly commutative. Here (in Figure1),

• π1 : X ∨ X → X is the ﬁrst projection; 1 1 • πP : XP ∨ XP → XP is the ﬁrst projection (obtained by π by taking the P-localization); • 1Y is the identity map of Y; and • e : X → XP is a P-localizing map.

Similarly, we obtain the same result for the second projection π2 : X ∨ X → X. This implies that if ψ : X → X ∨ X is a comultiplication on X, then its localization ψP : XP → XP ∨ XP also has the comultiplication structure on the localization XP of X.

ψ X - X ∨ X PP PP P 1 1X PPPq π X ) e e ∨ e

? e ? ψP - XP XP ∨ XP PP PP P 1 1XP PPq ? ) πP XP

Figure 1. The triangular prism.

We now consider the localized version of the Hilton’s formula described in Theorem1 of Section2 as follows. Mathematics 2020, 8, 86 5 of 19

n n n Theorem 2. Let XP be the localization of the k-fold wedge sum of spheres X := S 1 ∨ S 2 ∨ · · · ∨ S k . Then we have ∞ ∼ M hs+1 πm(XP ) = πm(SP ) s=1 for all m.

Proof. We note that localization preserves the Cartesian product, the smash product and the wedge sum in the pointed homotopy category of pointed nilpotent CW-complexes; that is,

• (X × Y)P ' XP × YP ; • (X ∧ Y)P ' XP ∧ YP ; and • (X ∨ Y)P ' XP ∨ YP .

Since the localizations of nilpotent CW-complexes are unique up to homotopy equivalence and the localizations of nilpotent groups commute with ﬁnite direct sums, we have

∼ πm(XP ) = πm(X) ⊗ ZP ∞ ∼ M hs+1 = πm(S ) ⊗ ZP s=1 ∞ ∼ M hs+1 = πm(S ) ⊗ ZP s=1 ∞ ∼ M hs+1 = πm(SP ), s=1 as required.

3.2. The 2-Fold Wedge of (Localized) Spheres We begin with the 2-fold wedge sum of (localized) spheres to illustrate our methods. The results will be applied to the k-fold wedge sum of (localized) spheres for k ≥ 3. Notation. Throughout this (sub)section, we will make use of the following notations.

• X := Sn ∨ Sm is the wedge sum of spheres, where 2 ≤ n < m. • k1 : Sn → X and k2 : Sm → X are the inclusion maps. • i1, i2 : Sn → Sn ∨ Sn are the first and second inclusion maps, respectively. • p1, p2 : Sn ∨ Sn → Sn are the first and second projections, respectively. • ι1, ι2 : X → X ∨ X are the first and second inclusion maps, respectively. • π1, π2 : X ∨ X → X are the first and second projections, respectively. • P is a collection of prime numbers. • fP : AP → BP is the P-localization of a map f : A → B, where A and B are object classes in the pointed homotopy category C of connected nilpotent CW-spaces. • C(X) ⊆ [X; X ∨ X] is the set of all homotopy classes of comultiplications on a nilpotent CW-space k ni X, e.g., X := ∨i=1S , Moore spaces, or their localizations.

n m Lemma 3. Let XP = SP ∨ SP be a wedge sum of localized spheres with 2 ≤ n < m. Then any comultiplication ψ : XP → XP ∨ XP has the following type ( ψk1 = ι1 k1 + ι2 k1 P P P P P (1) 2 1 2 2 2 ψkP = ιP kP + ιP kP + Q

m 1 2 for some homotopy class Q : SP → XP ∨ XP satisfying πP Q = 0 = πP Q, where the additions are the homotopy additions in the homotopy groups. Mathematics 2020, 8, 86 6 of 19

We note that if Y is a CW-complex which is ﬁltered by its n-skeletons Yn for n ≥ 0, then its n n+1 n n+1 n localization YP of Y at P is ﬁltered by YP . Moreover, the two CW-spaces YP /YP and (Y /Y )P n+1 have the same homotopy type, and each is a wedge sum of the localized spheres SP .

Proof. We observe that XP is a localized CW-complex and thus there is a cellular map ([19] p. 77)

0 ψ : XP → XP ∨ XP such that 0 ψ ' ψ : XP → XP ∨ XP . Therefore, we have 0 n n ψ (XP ) ⊂ (XP ∨ XP ) . n n n We note that XP = SP and that the localized sphere SP has a CW-decomposition that consists of one localized zero cell and one localized n-cell. Therefore, we obtain

n n n (XP ∨ XP ) = SP ∨ SP

0 n n n and observe that ψ gives a base point preserving continuous map η : SP → SP ∨ SP such that the following diagram η n n n I n n SP / SP ∨ SP / SP × SP (2)

1 1 1 1 1 kP kP ∨kP kP ×kP ψ=∼ψ0 J XP / XP ∨ XP / XP × XP is commutative up to homotopy, where I and J are the inclusion maps. We also note that the following triangle ψ XP / XP ∨ XP (3)

J ∆ ' XP × XP is homotopy commutative, where ∆ : XP → XP × XP is the diagonal map. From the homotopy commutative diagrams (2) and (3), we obtain

1 1 1 1 1 1 ¯ (kP × kP )Iη ' JψkP ' ∆kP = (kP × kP )∆, (4)

¯ n n n where ∆ : SP → SP × SP is the diagonal map. Since

1 1 n n n n (kP × kP )∗ : [SP ; SP × SP ] → [SP ; XP × XP ] is a monomorphism of homotopy groups, from (4) we have

Iη ' ∆¯ ; that is, the following triangle η n n n SP / SP ∨ SP (5)

I ∆¯ & n n SP × SP ¯ n n n is also commutative up to homotopy, where ∆ : SP → SP × SP is the diagonal map. This implies that n n n n the map η : SP → SP ∨ SP is also a comultiplication. Since the localized sphere SP is 1-connected for Mathematics 2020, 8, 86 7 of 19 n ≥ 2, we see that the comultiplication η is unique up to homotopy (see ([1] Proposition 3.1)) and has the form 1 2 η = iP +η iP , where +η is the addition induced by the comultiplication η. By the uniqueness of the comultiplication

n n n η : SP → SP ∨ SP as the standard (or suspension) comultiplication, we have

1 2 1 2 iP +η iP = iP + iP , where + is the homotopy addition in the homotopy group

n ∼ n ∼ [SP ; XP ∨ XP ] = [S ; XP ∨ XP ] = πn(XP ∨ XP ).

Hence 1 1 1 1 1 1 2 1 1 2 1 ψkP = (kP ∨ kP )η = (kP ∨ kP )(iP +η iP ) = ιP kP + ιP kP . n We now consider the homotopy class Q in the homotopy group [SP ; XP ∨ XP ] as an abelian group given by 2 1 2 2 2 Q = ψkP − ιP kP − ιP kP ; that is, 2 1 2 2 2 ψkP = ιP kP + ιP kP + Q. Then we obtain

1 1 2 1 1 2 1 2 2 2 2 πP Q = πP ψkP − πP ιP kP − πP ιP kP = kP − kP − 0 = 0 and 2 2 2 2 1 2 2 2 2 2 2 πP Q = πP ψkP − πP ιP kP − πP ιP kP = kP − 0 − kP = 0, as required.

m m Lemma 4. If Q : SP → XP ∨ XP is any homotopy class in [SP ; XP ∨ XP ] such that

1 2 πP Q = 0 = πP Q, then the map ψ : XP → XP ∨ XP deﬁned in Equation (1) is a comultiplication.

Proof. We have

1 1 1 1 1 2 1 1 1 • πP ψkP = πP (ιP kP + ιP kP ) = kP + 0 = kP ; 1 2 1 1 2 2 2 2 1 2 2 • πP ψkP = πP (ιP kP + ιP kP + Q) = kP + 0 + πP Q = kP + 0 + 0 = kP ; 2 1 2 1 1 2 1 1 1 • πP ψkP = πP (ιP kP + ιP kP ) = 0 + kP = kP ; and 2 2 2 1 2 2 2 2 2 2 2 • πP ψkP = πP (ιP kP + ιP kP + Q) = 0 + kP + πP Q = 0 + kP + 0 = kP ; 1 2 that is, πP ψ = 1XP and πP ψ = 1XP . This implies that ψ is a comultiplication on XP . Using the results above, we can deﬁne a comultiplication on a wedge of localized spheres as follows.

m m Deﬁnition 1. Let Q : SP → XP ∨ XP be any element of [SP ; XP ∨ XP ] satisfying

1 2 πP Q = 0 = πP Q. Mathematics 2020, 8, 86 8 of 19

We deﬁne a comultiplication ψQ : XP → XP ∨ XP by

( 1 1 1 2 1 ψQkP = ιP kP + ιP kP 2 1 2 2 2 ψQkP = ιP kP + ιP kP + Q.

The homotopy class Q is called a perturbation of the comultiplication ψQ.

2n−1 2n−1 We note that if e : S → SP is the P-localizing map, then the homomorphism

∼ ∗ 2n−1 n n = 2n−1 n n e : [SP ; SP ∨ SP ] / [S ; SP ∨ SP ] (6) induced by e is an isomorphism of homotopy groups. Convention. For our convenience in notation, using the isomorphism of homotopy groups in (6), we make use of the same notations of the homotopy classes on both sides of the isomorphism; that is,

∗ 2n−1 n n γ 7→ e (γ) = γ ◦ e ←→ γ ∈ [SP ; SP ∨ SP ]

2n−1 n n for any homotopy class γ in the homotopy group [SP ; SP ∨ SP ]. In particular,

1 2 2n−1 n n γ = [iP , iP ] ∈ [SP ; SP ∨ SP ] as the generalized Whitehead product in the sense of Arkowitz [26]; see below.

− Lemma 5. Let [i1, i2] : S2n 1 → Sn ∨ Sn be the Whitehead product. Then we have

1 2 1 2 [i , i ]P = [iP , iP ].

Proof. The functorial property of the localization shows that the following diagram

− a i1∨i2 ∇ S2n 1 / Sn ∨ Sn / Sn ∨ Sn ∨ Sn ∨ Sn / Sn ∨ Sn (7)

e e e e i1 ∨i2 ∇ 2n−1 aP n n P P n n n n P n n SP / SP ∨ SP / SP ∨ SP ∨ SP ∨ SP / SP ∨ SP

2n−1 n n is commutative up to homotopy, where a : S → S ∨ S is the attaching map and e : Y → YP is a P-localizing map. Since 2n−1 n n ∼ 2n−1 n n [S ; SP ∨ SP ] = [SP ; SP ∨ SP ], the commutative diagram in (7) shows the proof.

m 2n−1 Example 1. Let α : SP → SP be any homotopy class. We deﬁne a homotopy class Q in the homotopy group m ∼ m πm(XP ∨ XP ) = [S ; XP ∨ XP ] = [SP ; XP ∨ XP ] as the composition

[i1 i2] k1 ∨k1 m α 2n−1 , P n n P P SP / SP / SP ∨ SP / XP ∨ XP of maps; that is, 1 1 1 2 1 1 2 1 Q = (kP ∨ kP )[i , i ]P (α) = [ιP kP , ιP kP ]α, Mathematics 2020, 8, 86 9 of 19

1 2 1 2 where [i , i ]P is the localization of the Whitehead product [i , i ]. Then the map ψQ : XP → XP ∨ XP deﬁned by Deﬁnition1 is a comultiplication on XP ∨ XP whose perturbation is Q.

2 m Let qP : XP → SP be the projection. Then the coﬁbration

k1 ∨k1 q2 ∨q2 n n P P P P m m SP ∨ SP / XP ∨ XP / SP ∨ SP , asserts that if Q ∈ πm(XP ∨ XP ) is any homotopy class satisfying

1 2 πP Q = 0 = πP Q,

n n 1 1 then there exists a unique homotopy class W ∈ πm(SP ∨ SP ) such that Q = (kP ∨ kP )W and

1 2 pP W = 0 = pP W,

1 2 n n n 1 2 where pP , pP : SP ∨ SP → SP are the ﬁrst and second projections induced by p and p , respectively. To investigate the fundamental property of homotopy comultiplications on a wedge of localized spheres, we need to study the perturbations of all homotopy comultiplications. This naturally raises the following question: How can we construct the perturbation of a comultiplication on the wedge of localized spheres? The following lemma gives an answer to this query. 1 2 We denote the (generalized) basic Whitehead products of iP and iP by w1, w2, ... , ws, ... and the height of ws is denoted by hs for each s = 1, 2, 3, . . .. Then we have the following.

m Lemma 6. A perturbation Q : SP → XP ∨ XP of any comultiplication ψQ : XP → XP ∨ XP can be expressed uniquely as in Deﬁnition1 as follows:

∞ 1 1 Q = (kP ∨ kP )∗( ∑ wsas), s=3 where ws is the sth (generalized) basic Whitehead product localized at P and as is any homotopy class in the homotopy group m hs+1 ∼ m hs+1 [SP ; SP ] = [S ; SP ] for s = 3, 4, 5, . . . .

Proof. By using Lemmas3–5, we see that every comultiplication ψQ : XP → XP ∨ XP has the type of Deﬁnition1 satisfying 1 2 πP Q = 0 = πP Q. Using Theorem2, we obtain

∞ n n ∼ M hs+1 πm(SP ∨ SP ) = πm(SP ) s=1 ∞ n n M hs+1 = πm(SP ) ⊕ πm(SP ) ⊕ πm(SP ). s=3

m Moreover, every homotopy class Q in the homotopy group [SP ; XP ∨ XP ] can be uniquely expressed as the following type:

∞ 1 1 1 2 m ∼ (kP ∨ kP )∗(iP a1 + iP a2 + ∑ wsas) ∈ [SP ; XP ∨ XP ] = πm(XP ∨ XP ). s=3

Here, Mathematics 2020, 8, 86 10 of 19

m n ∼ n • a1 and a2 are any homotopy classes in [SP ; SP ] = πm(SP ); m hs+1 ∼ hs+1 • as is any homotopy class in [SP ; SP ] = πm(SP ) for s = 3, 4, 5, . . .; and • ws is the (generalized) basic Whitehead product for s = 3, 4, 5 . . ..

1 We recall that all of the (generalized) basic Whitehead products ωs consists of at least one iP 2 1 2 1 1 2 2 1 2 and at least one iP , for example, ω3 = [iP , iP ], ω4 = [iP , [iP , iP ]], ω5 = [iP , [iP , iP ]] and so on. We now have ∞ 1 1 1 1 πP ◦ Q = πP (kP ∨ kP )∗( ∑ wsas) s=1 ∞ 1 1 1 1 2 = πP (kP ∨ kP )∗(iP a1 + iP a2 + ∑ wsas) (8) s=3 = a1 + 0∗(a2) + 0 = a1 and ∞ 2 2 1 1 πP ◦ Q = πP (kP ∨ kP )∗( ∑ wsas) s=3 ∞ 2 1 1 1 2 = πP (kP ∨ kP )∗(iP a1 + iP a2 + ∑ wsas) (9) s=3 = 0∗(a1) + a2 + 0 = a2, where 0∗ is the trivial homomorphism between homotopy groups and 0 is the trivial homotopy class. 1 2 From the fact that πP Q = 0 = πP Q, the homotopy classes a1 and a2 in (8) and (9) should be zero, as required.

For the 2-fold wedge sum of localized spheres, we have

n m Theorem 3. The cardinality of the set of comultiplications on SP ∨ SP is

∞ hs+1 ∏ |πm(SP )|, s=3 where 2 ≤ n < m and P is a collection of prime numbers.

Proof. By Lemma6, we have the proof.

Let |C(XP )| be the cardinality of the set C(XP ) of comultiplications on XP .

Example 2. If P = {2, 3}, then we have

4 7 (1) |C(SP ∨ SP )| = ∞; 4 8 (2) |C(SP ∨ SP )| = 2; and 4 9 (3) |C(SP ∨ SP )| = 2. Indeed, by Theorem3, we have

4 8 7 10 |C(SP ∨ SP )| = |π8(SP )| × |π8(SP )| × · · · = |Z2 ⊗ ZP | × |{e} ⊗ ZP | × · · · , = 2 × 1 × 1 × · · · × 1 × · · · = 2 where {e} is the trivial group, and similarly for the other cases. Mathematics 2020, 8, 86 11 of 19

3.3. The k-Fold Wedge Sum of (Localized) Spheres

We now consider the k-fold wedge sum of 1-connected spheres {Sni |i = 1, 2, ... , k} and their ni localized spheres {SP |i = 1, 2, . . . , k} as a generalization of the statements above; that is,

X = Sn1 ∨ Sn2 ∨ · · · ∨ Snk and n1 n2 nk XP = SP ∨ SP ∨ · · · ∨ SP for 2 ≤ n1 < n2 < ··· < nk. Notation. We make use of the following notations in the rest of this paper.

Wk ni • X := i=1 S for 2 ≤ n1 < n2 < ··· < nk. 1 2 ni • αi , αi : S → X ∨ X are the ith and (k + i)th inclusions, respectively, for i = 1, 2, . . . , k − 1. • ζi : Sni → X is the ith inclusion for i = 1, 2, . . . , k. • ι1, ι2 : X → X ∨ X are the ﬁrst and second inclusions, respectively. • π1, π2 : X ∨ X → X are the ﬁrst and second projections, respectively. † • Wj is the set of all (generalized) basic Whitehead products localized at P containing the homotopy 1 classes from αi for i = 1, 2, . . . , j − 1 and j = 2, 3, . . . , k. ] • Wj is the set of all (generalized) basic Whitehead products localized at P containing the homotopy 2 classes from αi for i = 1, 2, . . . , j − 1 and j = 2, 3, . . . , k. ∗ • Wj is the set of all (generalized) basic Whitehead products localized at P containing as a factor at 1 2 least one of the homotopy classes from αi and at least one of the homotopy classes from αi for i = 1, 2, . . . , j − 1 and j = 2, 3, . . . , k.

n m We note that if k = 2, n1 = n and n2 = m; that is, X = S ∨ S , then we have

1. ζ1 = k1 : Sn ,→ Sn ∨ Sm; 2. ζ2 = k2 : Sm ,→ Sn ∨ Sm; 1 1 1 1 1 1 1 n i n n k ∨k 3. α1 = (k ∨ k ) ◦ i : S / S ∨ S / X ∨ X ; and 2 1 1 2 1 1 2 n i n n k ∨k 4. α1 = (k ∨ k ) ◦ i : S / S ∨ S / X ∨ X in the notations above. To develop the basic Whitehead products in the k-fold wedge sum of spheres localized at P, we order the basic Whitehead products of weight 1 as follows:

1 2 • αi < αj for i, j = 1, 2, . . . , k − 1; and • ι1ζi < ι2ζj for i, j = 1, 2, . . . , k.

nj nj Deﬁnition 2. Let Qj : SP → XP ∨ XP be any element of [SP ; XP ∨ XP ] satisfying

1 2 πP Qj = 0 = πP Qj

1 2 for each j = 2, 3, ... , k, where πP , πP : XP ∨ XP → XP are the ﬁrst and second projections, respectively. We deﬁne a comultiplication ψ = ψ : X → X ∨ X (Q2,Q3,...,Qk) P P P Mathematics 2020, 8, 86 12 of 19 by

 1 1 1 2 1 ψQζ = ι ζ + ι ζ  P P P P P  ψ ζ2 = ι1 ζ2 + ι2 ζ2 + Q  Q P P P P P 2  .  . j j j (10) ψ ζ = ι1 ζ + ι2 ζ + Q  Q P P P P P j  .  .  .  k 1 k 2 k ψQζP = ιP ζP + ιP ζP + Qk.

n Here, we call Q ∈ [ j ; X ∨ X ] the jth perturbation of ψ = ψ for j = 2, 3, . . . , k. j SP P P (Q2,Q3,...,Qk)

n We need to investigate the jth perturbation Q ∈ [ j ; X ∨ X ] of ψ : X → X ∨ X j SP P P (Q2,Q3,...,Qk) P P P as a generalization of Lemma6 more concretely.

ni Lemma 7. The jth perturbation Qj ∈ [SP ; XP ∨ XP ] of any comultiplication ψ : XP → XP ∨ XP can be uniquely expressed as in Deﬁnition2 by

Qj = ∑ wj ◦ cj. (11) ∗ wj∈Wj for j = 2, 3, . . . , k. Here,

+ hwj 1 • wj : SP → XP ∨ XP is the jth (generalized) basic Whitehead product localized at P; and n hw +1 n hw +1 hw +1 j j j j ∼ nj j • cj : SP → SP is any homotopy class in the homotopy group [SP ; SP ] = [S ; SP ], where hwj ∗ is the height of the (generalized) basic Whitehead product wj ∈ Wj for each j = 2, 3, . . . , k.

Proof. The localized version of the Hilton’s formula says that the jth perturbation Qj of ψ : XP → XP ∨ XP can be uniquely written by

j−1 j−1 1 i 2 i Qj = ∑(αi )P ◦ βj + ∑(αi )P ◦ γj + ∑ uj ◦ aj + ∑ vj ◦ bj + ∑ wj ◦ cj (12) ] ∗ i=1 i=1 v ∈W† wj∈W j j vj∈Wj j

Here, n • i i [ j ni ] ∼ ( ni ) = − βj and γj are any homotopy classes in SP ; SP = πnj SP for i 1, 2, . . . , j 1; h +1 h +1 nj uj ∼ uj • aj is any homotopy class in [SP ; SP ] = πnj (SP ) for j = 2, 3, . . . k; h +1 h +1 nj vj ∼ vj • bj is any homotopy class in [SP ; SP ] = πnj (SP ) for j = 2, 3, . . . k; h +1 h +1 nj wj ∼ wj • cj is any homotopy class in [SP ; SP ] = πnj (SP ) for j = 2, 3, . . . k; and • uj, vj and wj are the (generalized) basic Whitehead product localized at P for j = 2, 3, . . . k.

i i We show that the homotopy classes βj, γj, aj and bj are all zero. By taking the ﬁrst and second 1 2 projections πP , πP : XP ∨ XP → XP to the perturbation Qj of any comultiplication ψ : XP → XP ∨ XP , we have

1 2 πP ◦ Qj = 0 = πP ◦ Qj (13) Mathematics 2020, 8, 86 13 of 19 for each j = 2, 3, . . . , k; that is,

1 0 = πP ◦ Qj j−1 j−1 1 1 i 1 2 i 1 1 = ∑πP ◦ (αi )P ◦ βj + ∑πP ◦ (αi )P ◦ γj + ∑ πP ◦ uj ◦ aj + ∑ πP ◦ vj ◦ bj i=1 i=1 u ∈W† ] (14) j j vj∈Wj 1 + ∑ πP ◦ wj ◦ cj ∗ wj∈Wj

2 and similarly, for the second projection πP ; see below. Since the map

(α1) π1 1 1 nj i P P πP ◦ (αi )P : SP / XP ∨ XP / XP (15) in (14) is the inclusion map, the localized version of the Hilton’s formula asserts that the homotopy i class βj should be zero for each i = 1, 2, ... , j − 1 and j = 2, 3, ... , k. The choice of the (generalized) † basic Whitehead products localized at P in Wj says that all the homotopy classes aj should be zero. We note that the maps

(α2) π1 1 2 nj i P P πP ◦ (αi )P : SP / XP ∨ XP / XP , (16)

h +1 v π1 1 vj j P πP ◦ vj : SP / XP ∨ XP / XP (17) and

h +1 w π1 1 wj j P πP ◦ wj : SP / XP ∨ XP / XP (18)

2 are all zero by the choice of (αi )P , vj and wj for each i = 1, 2, . . . , j − 1 and j = 2, 3, . . . , k. 2 Similarly, by taking the second projection πP : XP ∨ XP → XP to the perturbation Qj again, we see that the map

(α2) π2 2 2 nj i P P πP ◦ (αi )P : SP / XP ∨ XP / XP (19)

i from (12) is the inclusion map so that γj = 0 for each i = 1, 2, ... , j − 1 and j = 2, 3, ... , k. The choice of ] the (generalized) basic Whitehead products localized at P in Wj says that all the homotopy classes bj should be zero. Moreover, the localized version of the Hilton’s formula asserts that the homotopy class 2 1 2 2 1 πP ◦ (αi )P , πP ◦ uj, and πP ◦ wj are all zero by the choice of (αi )P , uj and wj for each i = 1, 2, ... , j − 1 and j = 2, 3, . . . , k, as required.

For the k-fold wedge sum of localized spheres, we have the following.

n1 n2 nk Theorem 4. The cardinality of the set of comultiplications on XP := SP ∨ SP ∨ · · · ∨ SP is

h +1 h +1 h +1 | ( w2 )| × | ( w3 )| × · · · × | ( wk )| ∏ πn2 SP ∏ πn3 SP ∏ πnk SP , ∗ ∗ ∗ w2∈W2 w3∈W3 wk∈Wk where 2 ≤ n1 < n2 < ··· < nk and P is a collection of prime numbers.

Proof. By Lemma7, we complete the proof. Mathematics 2020, 8, 86 14 of 19

Example 3. If P = {2, 3, 5}, then we have

8 12 26 (1) |C(SP ∨ SP ∨ SP )| = ∞; and 8 12 27 (2) |C(SP ∨ SP ∨ SP )| = 512.

3.4. The (Localized) Moore Spaces A comultiplication ϕ : Y → Y ∨ Y is said to be associative if

(ϕ ∨ 1) ◦ ϕ ' (1 ∨ ϕ) ◦ ϕ : Y → Y ∨ Y ∨ Y, where 1 : Y → Y is the identity map. If ϕ is associative, we call (Y, ϕ) an associative co-H-space or co-H-group. It is well known that all Moore spaces M(G, n), n ≥ 2, are co-H-groups, where G is an abelian group. In general, the cardinality of the set C(Y) ⊆ [Y; Y ∨ Y] of all homotopy classes of comultiplications on Y is extremely difﬁcult to detect. As a particular case, it was shown in [27] that the set of comultiplications C(M(G, 2)) is in one-one correspondence with the group Ext(G, G ⊗ G), and that if n ≥ 3, then C(M(G, n)) has one element as the standard comultiplication, where G is an abelian group. We note that this result is also valid for localizations.

4. Main Result and Its Illustration

Let P be a collection of prime numbers which may be empty and let XP be the localization of the k-fold wedge sum of spheres

k _ X := Sni = Sn1 ∨ Sn2 ∨ · · · ∨ Snk i=1 for 2 ≤ n1 < n2 < ··· < nk or a Moore space M(G, n), where G is an abelian group and n ≥ 2. Then, by using the results in Sections2 and3, we have the following theorem.

Theorem 5. Let |C(X)| and |C(XP )| be the cardinalities of the sets of all the homotopy comultiplications on X and XP , respectively.

(1) If |C(X)| is ﬁnite, then |C(X)| ≥ |C(XP )|. (2) If |C(X)| is inﬁnite, then |C(X)| = |C(XP )|.

Proof. We prove the theorem in the case of the k-fold wedge sum of spheres and its localization. + hwj 1 We ﬁrst note that if the homotopy group πnj (S ) contains a q-torsion subgroup Tq with (p, q) = 1 h +1 ∈ ∗ = ( wj ) for wj Wj , j 2, 3, ... , k, then all of the q-torsion subgroups of πnj S could entirely vanish in hw +1 ( j ) ∈ ∗ = ( ) = ∈ P πnj SP for wj Wj , j 2, 3, . . . , k, where q is a prime number with p, q 1 for all p . + hwj 1 (1) If |C(X)| is ﬁnite, then all of the homotopy groups πnj (S ), j = 2, 3, ... , k must be the torsion abelian groups Twj , j = 2, 3, . . . , k, where Twj is the torsion subgroup of

k k h +1 ( wj ) = ∏ ∏ πnj S ∏ ∏ Twj . (20) = ∗ = ∗ j 2 wj∈Wj j 2 wj∈Wj

We also note that ∼ Tw = j ⊕ j ⊕ · · · ⊕ j , (21) j Zp1 1 Zp2 2 Zpl l Mathematics 2020, 8, 86 15 of 19

where p1, p2, ... , pl are (not necessarily distinct) primes and j1, j2, ... , jl are (not necessarily distinct) positive integers. We thus have

k h +1 | ( )| = | ( wj )| C X ∏ ∏ πnj S (by Lemma 1) = ∗ j 2 wj∈Wj k = | | ∏ ∏ Twj = ∗ j 2 wj∈Wj k (22) ≥ | ⊗ | ∏ ∏ Twj ZP = ∗ j 2 wj∈Wj k hw +1 = | ( j )| ∏ ∏ πnj SP (by Theorem 4) = ∗ j 2 wj∈Wj = |C(XP )|.

We note that equality of the inequality in (22) holds only in the case that the collection P of primes contains all of the prime numbers expressed in the torsion subgroups Twj , j = 2, 3, . . . , k. + hwj 1 (2) If |C(X)| is infinite, then at least one of the homotopy groups πnj (S ), j = 2, 3, ... , k contains an infinite cyclic group, say in dimension ns = hws + 1 or ns = 2(hws + 1) − 1, where hws + 1 is an even integer for some s = 2, 3, ... , k. Moreover, it can be seen that the numbers of nonzero + hw +1 hwj 1 j homotopy groups πnj (S ) or πnj (SP ), j = 2, 3, ... , k are always finite because the height hwj goes to infinity. hws +1 Let Fws be a free abelian group of rank 1 and let Tws be a torsion subgroup of πns (S ) for some s = 2, 3, . . . , k. Then we have

k h +1 | ( )| = | ( wj )| C X ∏ ∏ πnj S = ∗ j 2 wj∈Wj h +1 = | ⊕ | × | ( wj )| (23) Fws Tws ∏ ∏ πnj S ∗ j6=s wj∈Wj ≈ |Z|.

Here, ≈ means the same cardinality as sets and ( = + ∼ Z for ns hws 1 Fws ⊕ Tws = Z ⊕ Tws for ns = 2(hws + 1) − 1 and hws + 1 even.

By Theorem4 or by tensoring ZP on the homotopy groups described in (23), we see that if ns = hws + 1 for some s, then

k hw +1 | ( )| = | ( j )| C XP ∏ ∏ πnj SP = ∗ j 2 wj∈Wj hw +1 = |( ⊕ ) ⊗ | × | ( j )| Fws Tws ZP ∏ ∏ πnj SP ∗ (24) j6=s wj∈Wj hw +1 = | ⊗ | × | ( j )| Z ZP ∏ ∏ πnj SP ∗ j6=s wj∈Wj ≈ |ZP |

and if hws + 1 is even and ns = 2(hws + 1) − 1 for some s, then Mathematics 2020, 8, 86 16 of 19

k hw +1 | ( )| = | ( j )| C XP ∏ ∏ πnj SP = ∗ j 2 wj∈Wj hw +1 = |( ⊕ ) ⊗ | × | ( j )| Fws Tws ZP ∏ ∏ πnj SP ∗ (25) j6=s wj∈Wj hw +1 = |( ⊕ ) ⊗ | × | ( j )| Z Fws ZP ∏ ∏ πnj SP ∗ j6=s wj∈Wj ≈ |ZP |

We note that some ﬁnite groups in (24) and (25) will vanish for some torsion groups, each of whose orders do not belong to the collection P of prime numbers. Moreover, because P is a collection of primes and the free abelian group Z is a subgroup of ZP as a subring of the ring of rational numbers Q; that is, Z ⊆ ZP ⊆ Q, their cardinalities are the same as countably inﬁnite sets. The proof in the case of a Moore space M(G, n) and its localization runs through the similar way to the case of the k-fold wedge sum of spheres and its localization.

Remark 1. We note that if the collection P of prime numbers is the empty collection, then the above statement Wk ni is also true as a rationalization XQ of a 1-connected CW-complex X := i=1 S for 2 ≤ n1 < n2 < ··· < nk.

We ﬁnally give an example to illustrate our results.

Example 4. The ﬁnite (resp. inﬁnite) cases are described in Tables1–3 (resp. Tables4 and5). In the tables, φ denotes the empty collection of primes and ℵ0 is aleph-zero.

Table 1. The ﬁnite case.

n m X = S ∨ S P |C(X)| |C(XP )| S3 ∨ S6 {2, 5} 2 2 S3 ∨ S6 {3, 5} 2 1 S8 ∨ S26 {11, 13} 504 1 S3 ∨ S6 φ 2 1

Table 2. The ﬁnite case for k ≥ 3.

Wk ni X = i=1 S P |C(X)| |C(XP )| S4 ∨ S6 ∨ S8 {3, 5} 2 1 S5 ∨ S6 ∨ S7 {3, 5} 1 1 S5 ∨ S7 ∨ S10 {3, 5} 2 1 S7 ∨ S9 ∨ S11 ∨ S22 {3, 5} 19, 568, 944, 742, 400 1

Table 3. The ﬁnite case: Moore spaces.

Y = M(G, n) P |C(Y)| |C(YP )| M(Z, 2) {2, 5} 1 1 M(Z3, 2) {2, 5} 3 1 M(Q, 2) {3, 5} 1 1 M(G, n), n ≥ 3 φ 1 1 Mathematics 2020, 8, 86 17 of 19

Table 4. The inﬁnite case.

n m X = S ∨ S P |C(X)| |C(XP )| 4 7 S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 5 9 S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 10 19 S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 4 7 S ∨ S φ |Z| = ℵ0 |Q| = ℵ0

Table 5. The inﬁnite case for k ≥ 3.

Wk ni X = i=1 S P |C(X)| |C(XP )| 3 4 5 S ∨ S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 4 6 12 S ∨ S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 7 12 24 S ∨ S ∨ S {2, 5} |Z| = ℵ0 |ZP | = ℵ0 6 8 20 S ∨ S ∨ S φ |Z| = ℵ0 |Q| = ℵ0

Proof. To prove the example, we use the previous results in Sections2 and3, and prove the ﬁrst two 5 cases to illustrate the method. We note that π6(S ) is isomorphic to the cyclic group of order 2; that is, 5 ∼ 3 6 π6(S ) = Z2; see [28]. If X = S ∨ S and P = {2, 5}, then we have

5 7 7 |C(X)| = |π6(S )| × |π6(S )| × |π6(S )| 9 9 9 11 ×|π6(S )| × |π6(S )| × |π6(S )| × |π6(S )| × · · · = 2 × 1 × 1 × 1 × 1 × 1 × 1 × · · · = 2 and

5 7 7 |C(XP )| = |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | 9 9 9 11 ×|π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × · · · = |Z2 ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | , ×|{e} ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | × · · · = 2 × 1 × 1 × 1 × 1 × 1 × 1 × · · · = 2 where {e} is the trivial group. If X = S3 ∨ S6 and P = {3, 5}, then we get

5 7 7 |C(XP )| = |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | 9 9 9 11 ×|π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × |π6(S ) ⊗ ZP | × · · · = |Z2 ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | . ×|{e} ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | × |{e} ⊗ ZP | × · · · = 1 × 1 × 1 × 1 × 1 × 1 × 1 × · · · = 1

Therefore, if X = S3 ∨ S6, then we have ( |C(X)| = 2 = |C(XP )| for P = {2, 5} , |C(X)| = 2 > 1 = |C(XP )| for P = {3, 5} as required.

5. Conclusions and Further Prospects Co-H-spaces, also called spaces with a comultiplication, and localization theory play a pivotal role in (equivariant) homotopy theory. In general, it turns out that the computation of the cardinality of the Mathematics 2020, 8, 86 18 of 19 set of homotopy comultiplications is very difﬁcult. In this paper, we have investigated the inequalities and relationships between the cardinalities of the sets of homotopy comultiplications on the wedge of localized spheres and localized Moore spaces. We do hope that our methods can be used to study homotopy comultiplications on the wedge sum of any number localized spheres and localized Moore spaces. We also hope that the results in this paper can be applied to the notions of algebra comultiplications on the (localized) algebraic objects such as (free) Lie algebras by considering the coproduct of Lie algebras or cohomology modules over ZP .

Funding: This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2018R1A2B6004407). Acknowledgments: The author is grateful to the anonymous referees for a careful reading and many helpful suggestions that improved the quality of this paper. Conﬂicts of Interest: The author declares no conﬂict of interest.

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