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On products in the coarse shape and strong coarse shape

categories

Tayyebe Nasri, Behrooz Mashayekhy∗, Hanieh Mirebrahimi

Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,

Ferdowsi University of Mashhad,

P.O.Box 1159-91775, Mashhad, Iran.

Abstract

The paper is devoted to introduce a new category, called strong coarse shape cat-egory for topological spaces, SSh∗(Top), that is coarser than strong shape cate-gory SSh(Top). If the Cartesian product of two spaces X and Y admits a strongHPol∗-expansion (HPol∗-expansion), which is the Cartesian product of strong HPol∗-expansions (HPol∗-expansions) of these spaces, then X×Y is a product in the strongcoarse shape category (coarse shape category). As a consequence, the Cartesianproduct of two compact Hausdorff spaces is a product in the coarse shape and strongcoarse shape categories.

Keywords: Coarse shape category, Shape category, Inverse limit.2010 MSC: 55P55, 54C56, 54B10, 55Q07.

1. Introduction

F. W. Bauer [1] was the first to define and study strong shape for arbitrary spaces.Recently, N. Ugleic and N. Bilan [3] have extended the shape theory by constructinga coarse shape category, denoted by Sh∗, whose objects are all topological spaces. Itsisomorphisms classify topological spaces strictly coarser than the shape does. Theshape category Sh can be considered as a subcategory of Sh∗. The shape and coarseshape coincide on the class of spaces having homotopy type of polyhedra.

In this paper, we take inspiration from these two categories and introduce inSection 2 the strong coarse shape category SSh∗(Top), that is coarser than strong

∗Corresponding authorEmail addresses: tnasri72@yahoo.com (Tayyebe Nasri), bmashf@um.ac.ir

(Behrooz Mashayekhy), h−mirebrahimi@um.ac.ir (Hanieh Mirebrahimi)

Preprint submitted to December 11, 2013

shape category SSh(Top).Keesling in 1974 [5] proved that if X and Y are compact Hausdorff spaces, then

X × Y is a product in the ordinary shape category. Also, Mardesic in 2004 [6]investigated on the product in the strong shape and showed that if X × Y admits astrong expansion, then X×Y is a product in the strong shape category. In particularthe Cartesian product of two compact Hausdorff spaces is a product in strong shape.In Section 3, we study the existence of product in the coarse shape and strongcoarse shape categories. We prove that if the Cartesian product of two spaces Xand Y admits a strong HPol∗-expansion (HPol∗-expansion), which is the Cartesianproduct of strong HPol∗-expansions (HPol∗-expansions) of these spaces, then X × Yis a product in the strong coarse shape category (coarse shape category). As aconsequence we show that the Cartesian product of two compact Hausdorff spacesis a product in the coarse shape and strong coarse shape categories. Moreover, weshow that the shape groups and the coarse shape groups commute with the product.

By the fact that every inverse system is isomorphic to a cofinite inverse system [8,Remark 1.1.5], in this paper every inverse system is assumed to be cofinite inversesystem.

2. The strong coarse shape category

In this section we define the strong coarse shape category for topological spaces,denoted by SSh∗(Top) that is coarser than the strong shape category SSh(Top). Theobjects of this category are topological spaces. To define its morphisms similar tostrong shape morphisms, one needs the coarse coherent homotopy category CCHwhich define similar to the coherent homotopy category CH. For the sake of com-pleteness, let us briefly recall the well known notions and main facts concerning thecoherent homotopy category and the strong shape category (see [7]).

Let HTop be the homotopy category of topological spaces, then the objects of thecoherent homotopy category are inverse systems in HTop. To define its morphisms,for a pre-ordered set M , we consider the sets Mm of all increasing (m+1)-tuplesµ = (µ0, . . . , µm), m ≥ 0, where µi ∈M and µ0 ≤ . . . ≤ µm. We call µ is a multiindexof length m. We also define face operators djm : Mm → Mm−1, j = 0, . . . , m, m ≥ 1and degeneracy operators sjm :Mm → Mm+1, j = 0, . . . , m, m ≥ 0 by putting

djm(µ0, . . . , µm) = (µ0, . . . , µj−1, µj+1, . . . , µm),

sjm(µ0, . . . , µm) = (µ0, . . . , µj, µj, µj+1, . . . , µm).

2

Also we consider face operators dmj : ∆m−1 → ∆m, j = 0, . . . , m, m ≥ 1 and degen-eracy operators smj : ∆m+1 → ∆m, j = 0, . . . , m, m ≥ 0 by

dmj (t0, . . . , tm−1) = (t0, . . . , tj−1, 0, tj, . . . , tm−1),

smj (t0, . . . , tm+1) = (t0, . . . , tj−1, tj + tj+1, tj+2, . . . , tm+1).

A coherent mapping f : X = (Xλ, pλλ′ ,Λ) → Y = (Yµ, qµµ′ ,M) is a map whichconsists of an increasing function f : M → Λ and of mappings fµ : Xf(µm) ×∆m →Yµ0

, where µ = (µ0, µ1, . . . , µm) ∈ Mm such that

fµ(x, djt) =

qµ0µ1fd0µ(x, t) j = 0,

fdj µ(x, t) 0 < j < m,

fdmµ(pf(µm−1)f(µm)(x), t) j = m

fµ(x, sjt) = fsj µ(x, t) 0 ≤ j ≤ m.

A coherent homotopy fromX toY is a coarse coherent mapping F = (F, Fµ) : X×I → Y where X×I = (Xλ×I, pλλ′×1,Λ). F connects coherent mapping f = (f, fµ)to f ′ = (f ′, f ′

µ), denoted by f ≃ f ′, provided F ≥ f, f ′ (i.e. F (µ) ≥ f(µ), f ′(µ) for allµ ∈M) and for all x ∈ XF (µm), t ∈ ∆m,

Fµ(x, 0, t) = fµ(pf(µm)F (µm)(x), t)

Fµ(x, 1, t) = f ′

µ(pf ′(µm)F (µm)(x), t).

A morphism in the coherent homotopy category is the homotopy classes of co-herent mapping f which is denoted by [f ].

In order to define the composition h = gf = (h, hν) : X→ Z of coherent mappingsf = (f, fµ) : X → Y and g = (g, gν) : Y → Z = (Zν , rνν′, N), one decomposes ∆m

into m + 1 subpolyhedra Pmi = ∆i ×∆m−i, 0 ≤ i ≤ m where t = (t0, ..., tm) ∈ P

mi ,

provided∑i−1

j=0 tj ≤12≤

∑ij=0 tj.

Also there exist affine homemorphisms cmi : Pmi → ∆m−i×∆i given by mappings

ami : Pmi → ∆m−i and bmi : Pm

i → ∆i by ami (t) = (1− 2(ti+1+ ...+ tm), 2ti+1, ..., 2tm),bmi (t) = (2t0, ..., 2ti−1, 1− 2(t0 + ...+ ti−1)). We now put h = fg : N → Λ and definehν : Xh(νm) ×∆m → Zν0 for ν = (ν0, ..., νm) ∈ Nm and t ∈ Pm

i by putting

hν(x, t) = gν0...νi(fg(νi)...g(νm)(x, ami (t)), b

mi (t)).

The composition of homotopy classes of coherent mappings is defined by compos-ing their representatives, [g][f ] = [gf ].

3

Now, we define the coherent functor C : pro− Top→ CH as follows:Put C(X) = X if f = (f, fµ) : X→ Y is a morphism in pro-Top, put C(f) = [(f, fµ)]where fµ : Xf(µm) ×∆m → Yµ0

is defined by

fµ(x, t) = fµ0pf(µ0)f(µm)(x).

Also, we define the forgetful functor E : CH → pro−HTop. Put E(X) = X andif f = (f, fµ) : X→ Y is a coherent map, then E(f) to be the homotopy class of the(f, fµ0

) : X→ Y.Now, let us to recall some of the main notions concerning the coarse shape cat-

egory and the pro∗-HTop (see [3]). Let X = (Xλ, pλλ′ ,Λ) and Y = (Yµ, qµµ′ ,M) betwo inverse systems in HTop. An S∗-morphism of inverse systems, (f, fn

µ ) : X→ Y,consists of an index function f : M → Λ and of a set of mappings fn

µ : Xf(µ) → Yµ,n ∈ N and µ ∈ M such that for every related pair µ ≤ µ′ in M there exists λ ∈ Λ,λ ≥ f(µ), f(µ′), and there exists an n ∈ N so that, for every n′ ≥ n,

qµµ′fn′

µ′ pf(µ′)λ ≃ fn′

µ pf(µ)λ.

The composition of S∗-morphisms (f, fnµ ) : X → Y and (g, gnν ) : Y → Z =

(Zν , rνν′, N) is a S∗-morphism (h, hnν ) = (g, gnν )(f, fnµ ) : X → Z, where h = fg and

hnν = gnν fng(ν). The identity S∗-morphism on X is a S∗-morphism (1Λ, 1

nXλ

) : X → X,where 1Λ is the identity function and 1nXλ

= 1Xλfor all n ∈ N and λ ∈ Λ.

A S∗-morphism (f, fnµ ) : X → Y is said to be equivalent to a S∗-morphism

(f ′, f ′nµ ) : X→ Y denoted by (f, fn

µ ) ∼ (f ′, f ′nµ ) provided every µ ∈M admits λ ∈ Λ

and n ∈ N such that λ ≥ f(µ), f ′(µ) and for every n′ ≥ n,

fn′

µ pf(µ)λ ≃ f ′n′

µ pf ′(µ)λ.

The relation ∼ is an equivalence relation. The category pro∗-HTop has as objects allinverse systems X in HTop and as morphisms all equivalence classes f∗ = [(f, fn

µ )].The composition in pro∗-HTop is well defined by putting

g∗f∗ = h∗ = [(h, hnν )].

In particular if (X) and (Y ) are two rudimentary inverse systems of HTop, thenevery set of mappings fn : X → Y , n ∈ N, induces a map f∗ : (X) → (Y ) inpro∗-HTop.

Let p : X → X and p′ : X → X′ be two HPol-expansions of the same spaceX in HTop, and let q : Y → Y and q′ : Y → Y′ be two HPol-expansions of thesame space Y in HTop. Then there exist two natural isomorphisms i : X→ X′ and

4

j : Y → Y′ in pro-HTop. Consequently i∗ = J (i) : X→ X′ and j∗ = J (j) : Y → Y′

are isomorphisms in pro∗-HTop ( where J : pro−HTop→ pro∗−HTop is a functorthat J (X) = X and J ([(f, fµ)]) = [(f, fn

µ )], where fnµ = fµ for all n ∈ N). A

morphism f∗ : X→ Y is said to be equivalent to a morphism f ′∗ : X′ → Y′, denotedby f∗ ∼ f ′∗, provided the following diagram in pro∗-HTop commutes:

Xi∗

−−−→ X′

yf∗ f ′∗

y

Yj∗

−−−→ Y′,

(1)

Now, the coarse shape category Sh∗ is defined as follows: The objects of Sh∗ aretopological spaces. A morphism F ∗ : X → Y is the equivalence class < f∗ > of amapping f∗ : X→ Y in pro∗-HTop.

The composition of F ∗ =< f∗ >: X → Y and G∗ =< g∗ >: Y → Z is definedby the representatives, i.e., G∗F ∗ =< g∗f∗ >: X → Z. The identity coarse shapemorphism on a space X , 1∗X : X → X , is the equivalence class < 1∗X > of the identitymorphism 1∗X in pro∗-HTop.

In the following we introduce the coarse coherent mapping.

Definition 2.1. Let X = (Xλ, pλλ′ ,Λ) and Y = (Yµ, qµµ′ ,M) be inverse systems inHTop. A coarse coherent mapping f∗ = (f, fn

µ ) : X → Y consists of an increasingfunction f :M → Λ, called the index function and of a set of mappings fn

µ : Xf(µm)×∆m → Yµ0

, where n ∈ N and µ = (µ0, µ1, . . . , µm) ∈Mm, such that there exists n ∈ N

so that, for every n′ ≥ n,

fn′

µ (x, djt) =

qµ0µ1fn′

d0µ(x, t) j = 0,

fn′

dj µ(x, t) 0 < j < m,

fn′

dmµ(pf(µm−1)f(µm)(x), t) j = m

(2)

fn′

µ (x, sjt) = fn′

sj µ(x, t) 0 ≤ j ≤ m.

If X and Y are inverse systems in HTop over the same index set Λ and f = 1Λ,then (1Λ, f

nλ) is said to be a level coarse coherent mapping.

Lemma 2.2. With the notation and assumptions of the first of this section, letf∗ = (f, fn

µ ) : X → Y and g∗ = (g, gnν ) : Y → Z be coarse coherent mappingsof inverse systems. Then h∗ = (h, hnν ) : X → Z, where h = fg : N → Λ andhnν : Xh(νm) ×∆m → Zν0, n ∈ N, ν ∈ Nm is given by

hnν (x, t) = gnν0...νi(fng(νi)...g(νm)(x, a

mi (t)), b

mi (t)),

is a coarse coherent mapping.

5

Proof. Taking into account the construction of hnν and coarse coherent properties offnµ and gnν , this lemma is obvious.

The above lemma enables us to define the composition of coarse coherent map-pings of inverse systems. If f∗ = (f, fn

µ ) : X → Y and g∗ = (g, gnν ) : Y → Z aretwo coarse coherent mappings of inverse systems, then g∗f∗ = h∗ = (h, hnν ) : X→ Z,which is define in Lemma 2.2. This composition is associative.

Definition 2.3. A coarse coherent homotopy from X to Y is a coarse coherentmapping F∗ = (F, F n

µ ) : X × I → Y where X × I = (Xλ × I, pλλ′ × 1,Λ). F∗ con-nects coarse coherent mappings f∗ = (f, fn

µ ) and f ′∗ = (f ′, f ′nµ ), denoted by f∗ ≃ f ′∗

provided F ≥ f, f ′ and there exists n ∈ N such that for every n′ ≥ n and for allx ∈ XF (µm), t ∈ ∆m

F n′

µ (x, 0, t) = fn′

µ (pf(µm)F (µm)(x), t)

F n′

µ (x, 1, t) = f ′n′

µ (pf ′(µm)F (µm)(x), t).

A morphism in the coarse coherent homotopy category is the homotopy classesof coarse coherent mapping f∗ which is denoted by [f∗].

Lemma 2.4. Homotopy of coarse coherent mappings is an equivalence relation.

Proof. Reflexivity and symmetry are obvious. To prove transitivity, assume thatF′∗ = (F ′, F ′n

µ ) connects f∗ to f ′∗ and F′′∗ = (F ′′, F ′′nµ ) connects f ′∗ to f ′′∗. Let

F : M → Λ be an increasing function such that F ≥ F ′, F ′′. We define F nµ :

XF (µn) × I ×∆m → Yµ0, µ ∈ Mm, by

F nµ (x, s, t) =

{

F ′nµ (pF ′(µm)F (µm)(x), 2s, t) 0 ≤ s ≤ 1/2,

F ′′nµ (pF ′′(µm)F (µm)(x), 2s− 1, t) 1/2 ≤ s ≤ 1.

Then F∗ = (F, F nµ ) is a coarse coherent homotopy, which connects f∗ to f ′′∗.

The following lemma has a similar argument to the proof of [7, Theorem 2.4].

Lemma 2.5. Let f∗, f ′∗ : X→ Y and g∗, g′∗ : Y → Z be coarse coherent mappings.If f∗ ≃ f ′∗ and g∗ ≃ g′∗, then g∗f∗ ≃ g′∗f ′∗.

By Lemmas 2.4 and 2.5 we can define the composition of homotopy classes ofcoarse coherent mappings by composing their representatives, [g∗][f∗] = [g∗f∗].

6

Let us define functors C∗ : inv − Top → CCH , C∗ : pro − Top → CCH , C∗ :pro∗ − Top→ CCH , called the coarse coherent functors. Put C∗(X) = X, for everyinverse system X in Top. If f is a map, put C∗(f) = [(f, fn

µ )], where

fnµ (x, t) = fn

µ0pf(µ0)f(µm)(x) = fµ0

pf(µ0)f(µm)(x).

Therefore if f : X → Y is an isomorphism in pro-Top, then C∗(f) is an isomor-phism in CCH. Also, we define the forgetful functor E∗ : CCH → pro∗ − HTop asfollows: Put E∗(X) = X, for every inverse system X in HTop. If [f∗] = [(f, fn

µ )] ∈CCH(X,Y), put E∗([f∗]) = [(f, fn

µ0)].

Also, we define the functor J ∗ : CH → CCH by J ∗(X) = X, for every inversesystem X in HTop. If [f ] = [(f, fµ)] ∈ CH(X,Y), put J ∗([f ]) = [f∗] = [(f, fn

µ )] ∈CCH(X,Y), where for every n ∈ N, fn

µ = fµ for all µ ∈Mm.Now we are ready to introduce the strong coarse shape category. First, we Recall

that an equivalent definition of a strong HPol-expansion of X . A map p : X → X isa strong HPol-expansion of X provided, for any inverse system Y in HPol and anymorphism [h] : X → Y of CH, there exists a unique morphism [f ] : X → Y of CHsuch that [h] = [f ]C(p) (see [7]). Now, let p : X → X and p′ : X → X′ be twocofinite strong HPol-expansions of the same space X in HTop, and let q : Y → Y

and q′ : Y → Y′ be two cofinite strong HPol-expansions of the same space Y inHTop. We know that the maps [i] : X → X′ and [j] : Y → Y′ are isomorphismsin CH, consequently [i∗] = J ∗([i]) : X → X′ and [j∗] = J ∗([j]) : Y → Y′ areisomorphisms in CCH. A morphism [f∗] : X → Y is said to be equivalent to amorphism [f ′∗] : X′ → Y′, denoted by [f∗] ∼ [f ′∗], provided the following diagram inCCH commutes:

X[i∗]−−−→ X′

y

[f∗] [f ′∗]

y

Y[j∗]−−−→ Y′,

(3)

Now, we define the strong coarse shape category SSh∗ as follows: The objectsof SSh∗ are topological spaces. A morphism F ∗ : X → Y is the equivalence class< [f∗] > of a coarse coherent mapping [f∗] : X→ Y with respect to any choice of apair of strong HPol-expansions p : X → X and q : Y → Y. In other words, a strongcoarse shape morphism F ∗ : X → Y is given by a diagram

X ←−−−p

X

y

[f∗]

Y ←−−−q

Y.

(4)

7

The composition of F ∗ =< [f∗] >: X → Y and G∗ =< [g∗] >: Y → Z is definedby the representatives, i.e., G∗F ∗ =< [g∗f∗] >: X → Z. The identity strong coarseshape morphism on a space X , 1∗X : X → X , is the equivalence class < [1∗X] > of theidentity morphism [1∗X] in CCH.

We say that topological spaces X and Y have the same coarse strong shape type,denoted by SSh∗(X) = SSh∗(Y ), provided there exists an isomorphism F ∗ : X → Yin SSh∗. If there exist strong coarse shape morphisms F ∗ : X → Y and G∗ : Y → Xsuch that G∗F ∗ = 1∗X , then we say that the coarse strong shape of X is dominatedby the coarse strong shape of Y , and we write SSh∗(X) ≤ SSh∗(Y ).

Definition 2.6. Let X be a topological space. A strong HPol∗-expansion of X is amorphism p∗ : X → X, where X is an inverse system in HPol with the followingproperty:For any inverse system Y in HPol and any morphism [h∗] : X → Y of CCH, thereexists a unique morphism [f∗] : X→ Y of CCH such that [h∗] = [f∗]C∗(p∗).

Remark 2.7. For every map f : X → Y in Top and every pair of cofinite strongHPol∗-expansions p∗ : X → X and q∗ : Y → Y, there exists [f∗] : X→ Y in CCH,such that the following diagram in CCH commutes:

X ←−−−−C∗(p∗)

X

y

[f∗] f

y

Y ←−−−−C∗(q∗)

Y.

(5)

Thus every morphism f ∈ HTop(X, Y ) yields an equivalence class < [f∗] > i.e.a strong coarse shape morphism F ∗ : X → Y . If we define S∗(X) = X for everytopological space X and S∗(f) = F ∗ =< [f∗] >, for every map f : X → Y , thenS∗ : HTop→ SSh∗ becomes a functor, called the strong coarse shape functor. Also,we can define the functor S∗ : HTop→ SSh∗ by S∗ = J ∗◦S where S : HTop→ SShis defined in [7] and J ∗ : SSh → SSh∗ is induced by the functor J ∗ defined in thissection.

Therefore if X and Y have the same homotopy type, then they have the samestrong coarse shape type. Also, we have SSh(X) = SSh(Y ) implies that SSh∗(X) =SSh∗(Y ).

Now, we define the functor E∗ : SSh∗ → Sh∗ as follows: Put E∗(X) = X . IfF ∗ =< [f∗] >: X → Y is a morphism in SSh∗, put E∗(F ∗) =< E∗[f∗] >. Thisfunctor implies that if SSh∗(X) = SSh∗(Y ), then Sh∗(X) = Sh∗(Y ) for topological

8

spaces X and Y . Therefore the position of strong coarse shape theory is betweenstrong shape and coarse shape. From this fact and results in [2] we have the followingtheorems.

Theorem 2.8. Let X and Y be topological spaces and let SSh∗(X) ≤ SSh∗(Y ).Then the following assertions hold:(i) If X is connected, then so is Y .(ii) If the shape dimension sd(X) ≤ n, then also sd(Y ) ≤ n.(iii) If X is movable, then so is Y .(iv) If X is n-movable, then so is Y .(v) If X is strongly movable, then so is Y .

Proof. Since SSh∗(X) ≤ SSh∗(Y ) implies that Sh∗(X) ≤ Sh∗(Y ), the results holdby [2, Theorem 3].

Theorem 2.9. Let X and Y be topological spaces and let X be stable. If X and Yhave the same strong coarse shape type, then Y is also stable.

Proof. Since SSh∗(X) = SSh∗(Y ), so Sh∗(X) = Sh∗(Y ). But X is stable, thus Yis stable by [2, Theorem 4].

Theorem 2.10. Let X be a topological space and let P be a polyhedra. ThenSSh∗(X) = SSh∗(P ) if and only if SSh(X) = SSh(P ).

Proof. Let SSh∗(X) = SSh∗(P ), then Sh∗(X) = Sh∗(P ). By [2, Theorem 5],Sh(X) = Sh(P ) and so SSh(X) = SSh(P ) by [7, Theorem 9.19].

The following corollary is a consequence of Theorem 2.10.

Corollary 2.11. Let X or Y be a stable space. Then X and Y have the same strongshape type if and only if they have the same strong coarse shape type.

The following example shows that, in general, the induced function E∗|. : SSh∗(X, Y )→Sh∗(X, Y ) is not injective and the induced function S∗|. : HTop(X, Y )→ SSh∗(X, Y )is not surjective.

Example 2.12. (i) Let X = {∗} and Y be the dyadic solenoid. As in [4, Example17.7.2], one can shows that

card(Sh∗(X, Y )) = 1

whilecard(SSh∗(X, Y )) = c,

9

where c is the cardinal number of real numbers R. Consequently, the induced functionE∗|. : SSh∗(X, Y )→ Sh∗(X, Y ) can not be injective.(ii) Let X = {∗} and Y = {∗}⊔{∗}. By a similar argument to [3, Example 7.4], wecan show that

card(HTop(X, Y )) = 2

whilecard(SSh∗(X, Y )) = c,

where c is the cardinal number of real numbers R. Consequently, the induced functionS∗|. : HTop(X, Y )→ SSh∗(X, Y ) can not be surjective.

3. Products in strong coarse shape and coarse shape

As it is shown by Keesling in 1974 [5] the ordinary shape category doesn’t havethe product, in general. He also proved that if X and Y are compact Hausdorffspaces, then X × Y is a product in the shape category. Also, Mardesic in 2004[6] showed that X × Y is a product in the strong shape category when X and Yare compact Hausdorff spaces. In this section, we intend to study the existence ofproducts in the coarse shape and strong coarse shape categories. We define an HPol∗-expansion for a topological space X and show that if the Cartesian product of twospaces X and Y admits an HPol∗-expansion (strong HPol∗-expansion), which is theCartesian product of HPol∗-expansions (strong HPol∗-expansions) of these spaces,then X×Y is a product in the coarse shape category (strong coarse shape category).In particular, the Cartesian product of two compact Hausdorff spaces is a productin these categories. Finally, we show that the k-th shape groups and the k-th coarseshape groups commute with the product for every k ∈ N.

Definition 3.1. Let X be a topological space. An HPol∗-expansion of X is a mor-phism p∗ : X → X in pro∗-HTop, where X is an inverse system in HPol with thefollowing property:For any inverse system Y in HPol and any morphism h∗ : X → Y in pro∗-HTop,there exists a unique morphism f∗ : X→ Y in pro∗-HTop such that h∗ = f∗p∗.

Remark 3.2. Let p∗ : X → X and q∗ : Y → Y be HPol∗-expansions of X and Yrespectively, for every morphism f : X → Y in HTop, there is a unique morphismf∗ : X→ Y in pro∗-HTop such that the following diagram commutes in pro∗-HTop.

X ←−−−p∗

X

yf∗ f

y

Y ←−−−q∗

Y.

(6)

10

If we take other HPol∗-expansions p′∗ : X → X′ and q′∗ : Y → Y′, we obtainanother morphism f ′∗ : X′ → Y′ in pro∗-HTop such that f ′∗p′∗ = q′∗f and wehave f∗ ∼ f ′∗. Hence every morphism f ∈ HTop(X, Y ) yields an equivalence class< [f∗] > i.e. a coarse shape morphism F ∗ : X → Y , we denoted by S∗(f). If we putS∗(X) = X for every topological space X, we obtain a functor S∗ : HTop → Sh∗,called the coarse shape functor.

Now, we intend to prove that X × Y is a product in the coarse shape categorywith some conditions. First consider some notations.

Let p∗ : X → X = (Xλ, pλλ′,Λ) and q∗ : Y → Y = (Yµ, qµµ′ ,M) be mappings ofpro∗-HTop. Then X×Y = (Xλ×Yµ, pλλ′× qµµ′ ,Λ×M) is an inverse system and setof the mappings pnλ×q

nµ : X×Y → Xλ×Yµ form a mapping p∗×q∗ : X×Y → X×Y

of pro∗-HTop.To define the canonical projection π∗

X fix an index µ ∈M . Let µπ∗

X : X×Y → X

be the mapping which consists of fµ : Λ → Λ × M by fµ(λ) = (λ, µ) and set ofmappings of projections µπn

Xλ: Xλ×Yµ → Xλ, λ ∈ Λ. If µ′ is another index fromM ,

then µπ∗

X and µ′

π∗

X are congruent mappings. Hence they induce the same morphismof pro∗-HTop which is denoted by π∗

X.

Lemma 3.3. Let X and Y be inverse systems of spaces and (Z) be a rudimentarysystem. Let f∗ : (Z)→ X and g∗ : (Z)→ Y be morphisms of pro∗-HTop, then thereexists a unique morphism h∗ : (Z)→ X×Y such that π∗

Xh∗ = f∗ and π∗

Yh∗ = g∗.

Proof. Let f∗ and g∗ be given by mappings fnλ and gnµ . To prove existence of h∗,

consider the mapping h∗ = f∗×g∗ : (Z)→ X×Y of pro∗-HTop, given by mappingshnµλ = fn

λ × gnµ : Z → Xλ × Yµ, (λ, µ) ∈ Λ ×M . Note that, for (λ, µ) ∈ Λ ×M , one

has µπnXλ

(hnµλ) = fnλ and hence π∗

Xh∗ = f∗. Analogously, π∗

Yh∗ = g∗.

To prove uniqueness, assume that we have a morphism h∗ : (Z) → X ×Y suchthat π∗

Xh∗ = f∗ and π∗

Yh∗ = g∗. Let h∗ is given by mappings hnν : Z → Xλ×Yµ, where

ν = (λ, µ). Note that hnν must be of the form hnν = h′nν ×h′′nν , where h′nν : Z → Xλ and

h′′nν : Z → Yµ. Since µπnXλhnν = h′nν , we see that the equality πXh

∗ = f∗ implies thath′nν ∼ fn

λ . Analogously, h′′nν ∼ gnµ. However, this implies that hnν = h′nν ×h

′′nν ∼ fn

λ×gnµ

and hence h∗ = f∗ × g∗.

Theorem 3.4. If X and Y admit HPol∗-expansions p∗ : X → X and q∗ : Y → Y

respectively such that p∗×q∗ : X × Y → X×Y is an HPol∗-expansion, then X × Yis a product in the coarse shape category.

Proof. Let πX : X × Y → X and πY : X × Y → Y denote the canonical projections.We are going to show that X × Y with maps S∗(πX) and S∗(πY ) is a product in

11

Sh∗. Let Z be a topological space and let F ∗ : Z → X and G∗ : Z → Y be coarseshape morphisms. We must prove that there exists a unique coarse shape morphismH∗ : Z → X × Y such that S∗(πX)H

∗ = F ∗ and S∗(πY )H∗ = G∗.

We will first prove uniqueness of H∗. Assume that H∗ : Z → X × Y has thedesired properties. Consider the Hpol∗-expansion p∗ × q∗ : X × Y → X × Y andlet h∗ : (Z)→ X×Y be the morphism of pro∗-HTop associated with H∗. Similarly,consider the Hpol∗-expansions p∗ : X → X and q∗ : Y → Y and let f∗ : (Z)→ X andg∗ : (Z) → Y be morphisms of pro∗-HTop associated with F ∗ and G∗, respectively.Note that the canonical projection µπn

Xλ: Xλ × Yµ → Xλ satisfies the equality

µπnXλ

(pnλ × qnµ) = pnλπnX , for all n ∈ N. Therefore π∗

X(p∗ × q∗) = p∗πX i.e. the

following diagram is commutative:

X×Y ←−−−−p∗×q∗

X × Y

y

π∗

XπX

y

X ←−−−p∗

X.

(7)

It follows by Remark 3.2 that S∗(πX) =< π∗

X > i.e. the morphism π∗

X is associatedwith the shape morphism S∗(πX) and so the morphism π∗

Xh∗ is associated with the

shape morphism S∗(πX)H∗ = F ∗. However, we already know that f∗ is associated

with F ∗. Consequently, π∗

Xh∗ = f∗.

An analogous argument shows that π∗

Yh∗ = g∗. We now apply Lemma 3.3 and

conclude that h∗ is unique. However, this implies that also H∗ is unique, because itis associated with h∗.

To prove existence of H∗, choose f∗ and g∗ as above and put h∗ = f∗ × g∗.Then define H∗ : Z → X × Y as the coarse shape morphism associated with h∗.Arguing as before, π∗

Xh∗ is associated with S∗(πX)H

∗. By Lemma 3.3, π∗

Xh∗ = f∗.

Consequently, S∗(πX)H∗ is associated with f∗. Since also F ∗ is associated with f∗,

we conclude that S∗(πX)H∗ = F ∗. Analogously, S∗(πY )H

∗ = G∗.

Now, we want to see that if p∗ and q∗ are HPol∗-expansions, then in whichconditions p∗ × q∗ : X × Y → X×Y is an HPol∗-expansion. For this end we provesome general results on expansions of products.

Theorem 3.5. Let p∗ : X → X be an HPol∗-expansion, if Y is a compact Hausdorff,then p∗ × 1 : X × Y → X× Y is an HPol∗-expansion.

Proof. Let P = (Pν , rνν′, N) be an inverse system in HPol and let h∗ : X × Y → P

given by (h, hnν ) be a morphism of pro∗-HTop. We consider the map h∗ : X → PY

12

given by (h, hnν ), where hnν : X → P Y

ν is given by hnν (x)(y) = hnν (x, y) for every(x, y) ∈ X × Y . Since Pν is a polyhedra for every ν ∈ N and Y is a compactHausdorff, P Y

ν is a polyhedra by [8, Theorem 1]. Thus PY is an inverse system inHPol. Since p∗ : X → X is an HPol∗-expansion, there exists a map f∗ : X→ PY ofpro∗-HTop such that f∗p∗ = h∗. Let f∗ be given by (f, fn

ν ), then f∗ : X × Y → P

induced by (f, fnν ) which f : N → Λ and fn

ν : Xf(ν) × Y → Pν is a morphism ofpro∗-HTop such that f∗(p∗ × 1) = h∗. Therefore p∗ × 1 : X × Y → X × Y is anHPol∗-expansion.

Note that the morphism p∗ : X → X is given by a set of mappings pnλ : X → Xλ,for λ ∈ Λ and n ∈ N such that there exists k ∈ N with pλλ′pk

′

λ′ = pk′

λ for every k′ ≥ k.The following theorem is an equivalent definition of an HPol∗-expansion.

Theorem 3.6. Let X be a topological space. A morphism p∗ : X → X is an HPol∗-expansion of X if and only if for every polyhedron P it has the following properties:(C1) For every set of mappings hn : X → P , there is λ ∈ Λ, m ∈ N and a set ofmappings fk : Xλ → P such that fm′

pm′

λ ≃ hm′

for every m′ ≥ m.(C2) Whenever for a λ ∈ Λ and for two sets of mappings fn

0 , fn1 : Xλ → P there

exists k ∈ N which the mappings fk′

0 pk′

λ and fk′

1 pk′

λ are homotopic for every k′ ≥ k,then there exists λ′ ≥ λ and m ∈ N such that the mappings fm′

0 pλλ′ and fm′

1 pλλ′ arehomotopic for every m′ ≥ m.

Proof. Let p∗ : X → X be an HPol∗-expansion of X and P be a polyhedron. Wemust verify conditions (C1) and (C2). Let hn : X → P , n ∈ N, be a set of mappings.This maps induce a map h∗ : (X) → (P ) of pro∗-HTop. Since p∗ is an HPol∗-expansion, there exists a unique morphism f∗ : X → (P ) of pro∗-HTop such thath∗ = f∗p∗. So there exists λ ∈ Λ and m ∈ N such that fm′

pm′

λ ≃ hm′

for everym′ ≥ m. To verify the property (C2), let fn

0 , fn1 : Xλ → (P ) be two sets of mappings

and let there is k ∈ N which the mappings fk′

0 pk′

λ and fk′

1 pk′

λ are homotopic for everyk′ ≥ k. This maps induce two maps f∗0 , f

∗

1 : X → (P ) of pro∗-HTop such thatf∗0p

∗ = f∗1p∗. By uniqueness it follows that f∗0 = f∗1 i.e. there exist λ′ ≥ λ and m ∈ N

such that fm′

0 pm′

λλ′ ≃ fm′

1 pm′

λλ′ for every m′ ≥ m.Conversely, let Y = (Yµ, qµµ′ ,M) be an inverse system in HPol and h∗ : X → Y

be a morphism of pro∗-HTop determined by hnµ : X → Yµ. By the property (C1)

there is λ ∈ Λ, mµ ∈ N and a set of mappings fkµ : Xλ → Yµ such that fm′

µ pm′

λ ≃ hm′

µ

for every m′ ≥ mµ. We define f :M → Λ by f(µ) = λ and claim that the morphismsfkµ : Xf(µ) → Yµ determine a morphism f∗ : X→ Y.

To proof the claim, for µ′ ≥ µ we have, fm′

µ′ pm′

f(µ′) ≃ hm′

µ′ for every m′ ≥ mµ′ . Put

m0 = max{mµ, mµ′}. Then for every m′ ≥ m0 we have

13

qµµ′fm′

µ′ pm′

f(µ′) ≃ qµµ′hm′

µ′ ≃ hm′

µ ≃ fm′

µ pm′

f(µ).

Let λ ≥ f(µ), f(µ′), then there is m1 ∈ N such that for every m′ ≥ m1, pf(µ′)λpm′

λ ≃pm

′

f(µ′) and there is m2 ∈ N such that for every m′ ≥ m2, pf(µ)λpm′

λ ≃ pm′

f(µ). Now, if

put m = max{m0, m1, m2}, then for every m′ ≥ m we have

qµµ′fm′

µ′ pf(µ′)λpm′

λ ≃ fm′

µ pf(µ)λpm′

λ .

By property (C2), there exist λ′ ≥ λ and k0 ∈ N such that the mappings qµµ′fk′

µ′ pf(µ′)λ′

and fk′

µ pf(µ)λ′ are homotopic for every k′ ≥ k0.

Theorem 3.7. Let 1 × q∗ : X × Y → X × Y and p∗ × 1 : X × Yµ → X × Yµbe HPol∗-expansions for every µ ∈ M . Then p∗ × q∗ : X × Y → X × Y is anHPol∗-expansion.

Proof. We must verify conditions (C1) and (C2) for p∗ × q∗. Let P be a polyhedraand let hn : X × Y → P , n ∈ N, be a set of mappings. By property (C1) for1 × q∗ : X × Y → X × Y, there exist µ0 ∈ M , m0 ∈ N and a set of mappings

gk : X × Yµ0→ P , k ∈ N, such that gm

′

0(1 × qm′

0

µ ) ≃ hm′

0 for every m′

0 ≥ m0. Byproperty (C1) for p∗ × 1 : X × Yµ0

→ X × Yµ0, there exist λ0 ∈ Λ, m1 ∈ N and a

set of mappings fk : Xλ0× Yµ0

→ P , k ∈ N, such that fm′

1(pm′

1

λ0× 1) ≃ gm

′

1 for every

m′

1 ≥ m1. Putm = max{m0, m1}. It follows that for every m′ ≥ m, fm′

(pm′

λ0×qm

′

µ0) =

fm′

(pm′

λ0× 1)(1 × qm

′

µ0) ≃ gm

′

(1 × qm′

µ0) ≃ hm

′

. Therefore fm′

(pm′

λ0× qm

′

µ0) ≃ hm

′

. Thisestablishes property (C1).

To establish the property (C2), consider λ ∈ Λ, µ ∈ M and let fk0 , f

k1 : Xλ×Yµ →

P be two sets of morphisms such that there exists k1 ∈ N that fk0 (p

kλ × qkµ) ≃

fk1 (p

kλ × q

kµ) for every k ≥ k1. Consider the mapping gni = fn

i (pnλ × 1) : X × Yµ → P ,

i = 0, 1. Note that gni (1 × qnµ) = fni (p

nλ × qnµ) : X × Y → P and thus gk0(1 × qkµ)

is homotopic to gk1(1 × qkµ) for every k ≥ k1. Since 1 × q∗ : X × Y → X × Y isan HPol∗-expansion, property (C2) for 1 × q∗ yields µ′ ≥ µ and m1 ∈ N such thatgm

′

0 (1× qm′

µµ′) ≃ gm′

1 (1× qm′

µµ′) for every m′ ≥ m1 i.e. fm′

0 (pm′

λ × qm′

µµ′) ≃ fm′

1 (pm′

λ × qm′

µµ′).Note that pnλ×q

nµµ′ = (1×qnµµ′)(pnλ×1). Then putting kni = fn

i (1×qnµµ′) : Xλ×Yµ → P ,

i = 0, 1, we have km′

0 (pm′

λ × 1) ≃ km′

1 (pm′

λ × 1) for every m′ ≥ m1. Now, property(C2) for p∗ × 1 : X × Yµ′ → X × Yµ′ yields an index λ′ ≥ λ and m ∈ N such thatkm

′

0 (pm′

λλ′×1) ≃ km′

1 (pm′

λλ′×1), for every m′ ≥ m. But kni (pnλλ′×1) = fn

i (pnλλ′×qnµµ′) and

we obtain fm′

0 (pm′

λλ′ × qm′

µµ′) and fm′

1 (pm′

λλ′ × qm′

µµ′) are homotopic, for every m′ ≥ m.

The following result is a consequence of Theorems 3.4, 3.5 and 3.7.

14

Corollary 3.8. If X and Y are compact Hausdorff spaces. Then X×Y is a productin the coarse shape category Sh∗.

In the rest of the paper we intend to obtain the same results as above resultsin the strong coarse shape category. We will show that X × Y is a product in thestrong coarse shape category with some conditions.

Lemma 3.9. Let X and Y be inverse systems of spaces and (Z) be a rudimentarysystem. Let [f∗] : (Z) → X and [g∗] : (Z) → Y be morphisms of CCH, then thereexists a unique morphism [h∗] : (Z) → X × Y such that C∗(π∗

X)[h∗] = [f∗] and

C∗(π∗

Y)[h∗] = [g∗].

Proof. First note that there is no loss of generality if we assume that the indexingsets Λ and M have initial elements λ∗ and µ∗, respectively [6, Corollary 2].

Let f∗ : (Z) → X and g∗ : (Z) → Y be coarse coherent mappings given byfnλ: Z ×∆m → Xλ0

, λ ∈ Λm and gnµ : Z ×∆m → Yµ0, µ ∈ Mm, respectively. They

determine a mapping h∗ : (Z) → X × Y given by the mappings hnν = fnλ× gnµ :

Z × ∆m → Xλ0× Yµ0

, where ν = λ × µ = ((λ0, µ0), ..., (λm, µm)) ∈ (Λ ×M)m forλ = (λ0, ..., λm) ∈ Λm and µ = (µ0, ..., µm) ∈Mm. Now, we must verify that h∗ is alsoa coarse coherent mapping. For this, since f∗ and g∗ are coarse coherent mappings,there exist n1, n2 ∈ N such that Equation (2) holds for fn′

λand gn

′′

µ for every n′ ≥ n1

and n′′ ≥ n2. By putting n = max{n1, n2} and using the proof of [6, Corollary 2],one can see that hnν satisfies in Equation (2) and so h∗ = f∗ × g∗ : (Z) → X ×Y isa coarse coherent map.

To verify that C∗(π∗

X)[h∗] = [f∗], recall that a representative of π∗

X is the mappingµ∗πn

X, which consists of the index function λ 7→ (λ, µ∗) and of the first projectionsµ∗πn

Xλ: Xλ × Yµ∗

→ Xλ, λ ∈ Λ. Therefore the composition C∗(π∗

X)[h∗] has as

representative the coarse coherent mapping k∗ : (Z) → X given by the mappingsknλ: Z × ∆m → Xλ0

, where knλ= πXλ0

hnλ×µ∗

= πXλ0(fn

λ× gnµ∗

) = fnλ

and µ∗ =

(µ∗, ..., µ∗). Consequently k∗ = f∗ and hence C∗(π∗

X)[h∗] = [k∗] = [f∗]. The equality

C∗(π∗

Y)[h∗] = [g∗] is verified analogously.

Now, we show that [h∗] is unique. Assume that f∗ : (Z) → X, g∗ : (Z) → Y

and h∗ : (Z) → X × Y are coarse coherent maps such that C∗(π∗

X)[h∗] = [f∗] and

C∗(π∗

Y)[h∗] = [g∗]. We must show that [h∗] = [f∗ × g∗]. It suffices to exhibit two

coarse coherent mappings k′∗ : (Z) → X and k′′∗ : (Z) → Y such that [k′∗] = [f∗],[k′′∗] = [g∗] and the coarse coherent mapping k∗ = k′∗ × k′′∗ : (Z)→ X×Y has theproperty that [k∗] = [h∗].

We define k′∗ : (Z)→ X by k′nλ: Z ×∆m → Xλ0

, λ = (λ0, ..., λm), which definedby k′n

λ= πXλ0

hnλ×µ∗

, where µ∗ = (µ∗, ..., µ∗) and k′′∗ : (Z) → Y consists of k′′nλ

:

15

Z ×∆m → Yµ0, µ = (µ0, ..., µm), defined by k′′nµ = πYµ0

hnλ∗×µ

, where λ∗ = (λ∗, ..., λ∗).

By similar argument of [6, Corollary 2], one can show that the equalities [k′∗] = [f∗],[k′′∗] = [g∗] and [k∗] = [h∗] are hold.

Theorem 3.10. If X and Y admit strong HPol∗-expansions p∗ : X → X andq∗ : Y → Y respectively such that p∗ × q∗ : X × Y → X × Y is a strong HPol∗-expansion, then X × Y is a product in the strong coarse shape category.

Proof. Let πX : X × Y → X and πY : X × Y → Y denote the canonical projections.We want to show that X × Y with maps S∗(πX) and S

∗(πY ) is a product in SSh∗.Let Z be a topological space and let F ∗ : Z → X and G∗ : Z → Y be strong coarseshape morphisms. We must prove that there exists a unique strong coarse shapemorphism H∗ : Z → X × Y such that S∗(πX)H

∗ = F ∗ and S∗(πY )H∗ = G∗.

We will first prove uniqueness of H∗. Assume that H∗ : Z → X × Y has thedesired properties. Consider the strong Hpol∗-expansion p∗ × q∗ : X × Y → X×Y

and let [h∗] : (Z)→ X×Y be the morphism of CCH associated with H∗. Similarly,consider the strong Hpol∗-expansions p∗ : X → X and q∗ : Y → Y and let [f∗] :(Z) → X and [g∗] : (Z) → Y be morphisms of CCH associated with F ∗ and G∗,respectively. By applying the functor C∗ on Diagram (7), we have C∗(π∗

X)C∗(p∗ ×

q∗) = C∗(p∗)πX . It follows by Remark 2.7 that S∗(πX) =< C∗(π∗

X) > and sothe morphism C∗(π∗

X)[h∗] is associated with the shape morphism S∗(πX)H

∗ = F ∗.However, we know that [f∗] is associated with F . Consequently, C∗(π∗

X)[h∗] = [f∗].

An analogous argument shows that C∗(π∗

Y)[h∗] = [g∗]. We now apply Lemma 3.9

and conclude that [h∗] is unique.To prove existence of H∗, choose f∗ and g∗ as above and put h∗ = f∗ × g∗.

Then define H∗ : Z → X × Y as the strong coarse shape morphism associated with[h∗]. Arguing as before, C∗(π∗

X)[h∗] is associated with S∗(πX)H

∗. By Lemma 3.9,C∗(π∗

X)[h∗] = [f∗]. Consequently S∗(πX)H

∗ is associated with [f∗]. Also, since F ∗ isassociated with [f∗], we conclude that S∗(πX)H

∗ = F ∗. Analogously, S∗(πY )H∗ =

G∗.

Now, we explain two conditions (SC1) and (SC2) for a morphism p∗ : X → X

and polyhedron P which are equivalent to the definition of strong HPol∗-expansionfor a topological space X (Definition 2.6).(SC1)=(C1),(SC2) For λ ∈ Λ, let fn

0 , fn1 : Xλ → P be two sets of mappings and let there exists

k ∈ N such that F k′ : X × I → P be a homotopy which connects fk′

0 pk′

λ to fk′

1 pk′

λ forevery k′ ≥ k. Then there exist λ′ ≥ λ and m ∈ N such that for every m′ ≥ m thereexists a homotopy Hm′

: Xλ′ × I → P which connects fm′

0 pλλ′ to fm′

1 pλλ′ , and there

16

is a homotopy Mm′

: X × I × I → P which connects Hm′

(Pm′

λ′ × I) to Fm′

and isstationary on X × ∂I.

Theorem 3.11. Let p∗ : X → X satisfies in conditions (SC1) and (SC2) and let Ybe a compact Hausdorff space. Then p∗ × 1 : X × Y → X × Y is also satisfies inconditions (SC1) and (SC2).

Proof. By converting map f : X × Y → P to f : X → PY and note that if Pis a polyhedron and Y is a compact Hausdorff space, then PY is a polyhedron [8,Theorem 1], the result holds (see [7, Theorem 7.5]).

Theorem 3.12. Let X be a topological space. A morphism p∗ : X → X is a strongHPol∗-expansion of X if and only if for every polyhedron P , it has the properties(SC1) and (SC2).

Proof. To proof the theorem, we need the m-dimensional version, m ≥ 1, of property(SC2). For a mapping p∗ : X → X and polyhedron P , this is the following statement.(SC2)m If λ ∈ Λ and fn : Xλ × ∂∆

m → P , F n : X × ∆m → P are mappings thatthere exists k ∈ N such that for every k′ ≥ k,

F k′|X×∂∆m = fk′(pk′

λ × 1),

then there exist λ′ ≥ λ and m ∈ N such that for every m′ ≥ m there exists ahomotopy Hm′

: Xλ′ ×∆m → P such that

Hm′

|Xλ×∂∆m = fm′

(pλλ′ × 1),

Hm′

(pm′

λ′ × 1) ≃ Fm′

(rel(X × ∂∆m)).

Note that property (SC2)1 coincides with (SC2).Now, let p∗ satisfies in conditions (SC1) and (SC2). By the same argument of

[7, Lemma 8.3] and using Theorem 3.11, one can show that p∗ has property (SC2)m,for every m ≥ 1. Then by the same argument of [7, Lemma 8.5] and [7, Lemma 8.6],p∗ : X → X has the following property:For every inverse system Y in HPol and every morphism [f∗] : X → Y of CCH,there exists a unique morphism [h∗] : X → Y of CCH such that [f∗] = [h∗]C∗(p∗).Therefore the map p∗ is a strong HPol∗-expansion of X . The converse is similar tothe proof of [7, Theorem 8.2].

Now, we will to see that if p∗ and q∗ are strong HPol∗-expansions, then in whichconditions p∗ × q∗ : X × Y → X×Y is a strong HPol∗-expansion.

17

Corollary 3.13. Let p∗ : X → X be a strong HPol∗-expansion, if Y is a compactHausdorff, then p∗ × 1 : X × Y → X× Y is a strong HPol∗-expansion.

Proof. This is an immediate consequence of Theorems 3.11 and 3.12. Also, by similarargument of the proof of Theorem 3.5 and by applying the universal property ofstrong HPol∗-expansion p∗ : X → X the result holds.

Theorem 3.14. Let 1 × q∗ : X × Y → X ×Y and p∗ × 1 : X × Yµ → X × Yµ bestrong HPol∗-expansions, for every µ ∈ M . Then p∗ × q∗ : X × Y → X × Y is astrong HPol∗-expansion.

Proof. We must verify conditions (SC1) and (SC2) for p∗ × q∗. The verification ofcondition (SC1)=(C1) performed in the proof of Theorem 3.7. It thus remains toverify (SC2). To establish this property, let λ ∈ Λ, µ ∈M and let fn

0 , fn1 : Xλ×Yµ →

P be two sets of morphisms and let there exists k ∈ N that F k′ : X×Y ×I → P be ahomotopy which connects fk′

0 (pk′

λ ×qk′

µ ) to fk′

1 (pk′

λ ×qk′

µ ) for every k′ ≥ k. Consider the

mapping gni = fni (p

nλ × 1) : X × Yµ → P , i = 0, 1. Since gni (1× q

nµ) = fn

i (pnλ × q

nµ) we

see that the homotopy F k′ also connects gk′

0 (1× qk′

µ ) to gk′

1 (1× qk′

µ ). By the property(SC2) for 1×q∗ : X×Y → X×Y there exist µ′ ≥ µ and m1 ∈ N such that for everym′ ≥ m1 there exists a homotopy Hm′

: X×Yµ′×I → P which connects gm′

0 (1×qµµ′)to gm

′

1 (1× qµµ′). Moreover, there exists a homotopy Mm′

: X×Y × I× I → P whichconnects Hm′

(1× qm′

µ′ × 1) to Fm′

which is stationary on X ×Y × ∂I. Now, considerthe mappings kni = fn

i (1 × qµµ′) : Xλ × Yµ′ → P , i = 0, 1. Note that kni (pnλ × 1) =

gni (1× qµµ′), i = 0, 1. Therefore, Hm′

also connects km′

0 (pm′

λ × 1) to km′

1 (pm′

λ × 1) forevery m′ ≥ m1. Consequently, property (SC2) for p∗ × 1 : X × Yµ′ → X× Yµ′ yieldsan index λ′ ≥ λ and m2 ∈ N such that for every m′ ≥ m2 there exists a homotopyKm′

: Xλ′×Yµ′×I → P which connects km′

0 (pm′

λλ′×1) to km′

1 (pm′

λλ′×1) and a homotopyNm′

: X×Y ×I×I → P which connects Km′

(pm′

λ′ ×1×1) to Hm′

which is stationaryon X × Yµ′ × ∂I. Since kni (pλλ′ × 1) = fn

i (pλλ′ × qµµ′), we see that Km′

connectsfm′

0 (pλλ′×qµµ′) to fm′

1 (pλλ′×qµµ′). Clearly Nm′

(1×qm′

µ′ ×1×1) : X×Y ×I×I → P

connects Km′

(pm′

λ′ × 1 × 1)(1 × qm′

µ′ × 1) to Hm′

(1 × qm′

µ′ × 1) and is stationary onX × Y × ∂I. Now, put m = max{m1, m2}. Consequently, the juxtaposition ofhomotopies Nm′

(1×qm′

µ′ ×1×1)∗Mm′

is a homotopy on X×Y ×I which is stationary

on X × Y × ∂I and connects Km′

(pm′

λ′ × qm′

µ′ × 1) to Fm′

for every m′ ≥ m.

The following result is a consequence of Theorems 3.10, 3.14 and Corollary 3.13.

Corollary 3.15. If X and Y are compact Hausdorff spaces, then X×Y is a productin the strong coarse shape category SSh∗.

18

Theorem 3.16. Let X × Y be a product of topological spaces X and Y in Sh∗, thenπ∗

k(X × Y )∼= π∗

k(X)× π∗

k(Y ), for every k ∈ N.

Proof. Let S∗(πX) : X × Y → X and S∗(πY ) : X × Y → Y be the induced coarseshape morphisms of canonical projections and assume that φX : π∗

k(X×Y )→ π∗

k(X)and φY : π∗

k(X × Y ) → π∗

k(Y ) be the induced homomorphisms by S∗(πX) andS∗(πY ), respectively. Then there is a homomorphism φ = (φX , φY ) : π

∗

k(X × Y ) →π∗

k(X) × π∗

k(Y ). We define homomorphism ψ : π∗

k(X) × π∗

k(Y ) → π∗

k(X × Y ) byψ([F ∗], [G∗]) = [(F ∗, G∗)] where (F ∗, G∗) : I → X × Y is a unique coarse shapemorphism that S∗(πX)(F

∗, G∗) = F ∗ and S∗(πY )(F∗, G∗) = G∗ since X × Y is a

product in Sh∗. Indeed, if F ∗ =< f∗ = (f, fnλ ) > and G∗ =< g∗ = (g, gnµ) >, then

(F ∗, G∗) =< (f∗, g∗) >, where (f∗, g∗) is given by (f, g)nλµ = fnλ × g

nµ : I → Xλ × Yµ

(see Theorem 3.4). Note that the homomorphism ψ is well define. Infact, if H∗

1 =<h∗1 >: F ∗ ≃ F ′∗ and H∗

2 =< h∗2 >: G∗ ≃ G′∗, then the coarse shape morphism

H∗ : I × I → X × Y which is associated by h∗ = h∗

1 × h∗

2 connects the coarse shapemorphism (F ∗, G∗) to (F ′∗, G′∗). It is routine to check that φ and ψ are inverse.

Remark 3.17. (i) By a similar argument to the above theorem, if X×Y is a productof topological spaces X and Y in Sh, then πk(X × Y ) ∼= πk(X) × πk(Y ), for everyk ∈ N.(ii) If X and Y are compact Hausdorff spaces, then X × Y is a product in Sh∗ andSh by Corollary 3.8 and [6, Theorem 10], respectively. Therefore in this case πk andπ∗

k commute with product, for every k ∈ N.

References

[1] F. W. Bauer, A shape theory with singular homology, Pacific. J. Math., 64(1976) 25-65.

[2] N. Koceic Bilan, On some coarse shape invariants, Topology Appl., 157(2010) 2679-2685.

[3] N. Koceic Bilan, N. Ugleic, The coarse shape, Glas. Mat., 42:62 (2007)145-187.

[4] R. Geoghegan, Topological methods in group theory, GTM-243, Springer-Verlag, New York, 2008.

[5] J.E. Keesling, Products in the shape category and some applications, Sym-pos. Math. Istituto Nazionaledi Alta Matematica, 1973, vol. 16, AcademicPress, New York, 1974, pp. 133-142.

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[6] S. Mardesic, Strong expansions of products and products in strong shape,Topology Appl., 140 (2004) 81-110.

[7] S. Mardesic, Strong Shape and Homology, Springer-Verlag, Berlin, Heidel-berg, New York, 2000.

[8] S. Mardesic, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.

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