Some Theorems On The Sheaf Of Higher Homotopy Groups

Some Theorems On The Sheaf Of Higher Homotopy Groups

Erdal GÜNER1

Summary: In this paper, constructing the sheaf of higher homotopy groups on a connected and locally path connected topological space, its some characterizations are examined. Let the pairs

(

,

and

(

,

be given.If the mapping

is a sheaf isomorphism, we show that there exists an isomorphism between the pairs and .

H

_n

X H

₂

)

X H

_{1 n} 1 n2

)

f H

∗

_:

_n

_→

H

_n 1 2

(

X H

₁

,

_n

X H

_n 2

H

_n

(

X H

₁

,

_n

)

1

(

X H

2

,

)

f H

n

H

n ∗

_:

_→

1 2 1

)

)

1

(

2

,

)

Keywords: Higher Homotopy Group, Sheaf of Abelian Groups, Regular Covering Space, Sheaf Isomorphism, Covariant Functor.

Yüksek Homotopi Gruplarının Demetleri Üzerine Bazı

Teoremler

Özet: Bu çalışmada, irtibatlı lokal eğrisel irtibatlı bir topolojik uzay üzerinde yüksek homotopi gruplarının demeti oluşturularak bazı karakterizasyonları incelenmiştir.

ve iki çift olsun. Eğer bir demet izomorfizmi ise ve çiftleri arasında bir izomorfizim olduğu gösterilmiştir.

n2

(

X H

₂

,

(

X H

₁

,

_n

)

_n

2

Anahtar Kelimeler: Yüksek Homotopi Grubu, Abelian Grupların Demeti, Regüler Örtü Uzayı, Demet İzomorfizmi, Kovaryant Funktor.

Introduction

Let X be a connected and locally path connected space. Then X is a path connected and has only are path component, that is X . For an arbitrary fixed point , we will consider X as a pointed topological space (X,c) unless otherwise stated. Let x be any point of X and

c X

∈

π

n

( , )

X x

be higher homotopy group of X with respect to x and

H

_n

V

x X

=

∈

π

n

( , )

X x

σ

. Clearly, is a set over X and the mapping defined by for any is an onto projection.

H

x n n

H

⊂

Ψ

:H

_n

→

X

Ψ

(

σ

x

)

=

x

x

∈

(

H

n

)

We introduce on

H

_n a natural topology as follows: Let

x

₀ an arbitrary fixed point of X,

W W x

=

( )

₀ be a path connected open neighborhood of

x

₀ and

σ

x0

=

[ ]

α

x0 be a homotopy

class of

(H )

_{n x0}. Since X is path connected, there exists a path

γ

with initial point and with terminal point x, for every . Therefore, the path

x

₀

x W

∈

γ

determines an isomorphism

γ

*:(

H

_{n x}

)

₀

[ ]

(

→

H

_{n x}

)

defined by

γ

*

( )

[ ]

α

_x0

=

[ ]

β

_xfor any

α

x0

s x

( )

= γ

*

α

[ ]

n x

H

₀

(

)

∈

1

(H

H

∈

[ ]

( )

x0

[ ]

γ =

:

→

( )

x W

∈

c W

∈

( )

[ ]

α

x0

x W

∈

]

γ

α

x0

Ψ ο

s

=

1

_W

Γ

( ,

W H

x X

∈

(

_n

, )

Ψ

W

( ):

[ ]

H

_n

4

W B s

W

=

∈

,

∈

Γ

( ,

)

H

_n

Ψ

)

Ψ

H

H

_n

H

_n

Γ

_n

)

H

_n

(

H

_{n x}

)

=

π

n

( , )

X x

H

x

∈

X

( ,

W H

H

n n

+

:H

⊕

n x

=

[

H

_n

5

]

(

)

π

_n

( ,

X x

)

H

_n 1

Ι

ψ

−1

_{( )}

n

>

1

s x

₁

( )

₀

=

s

₂

(

H

_n

x

₀

)

x

∈

H

_n

W W

₁

,

₂

X

H

_n

W

₂

∈

Ι

,

s

₁

=

1

∈Γ

,

x

₀

W

₁

10

]

∈Ι

s

i

ψ

s W s W

_i

( ): (

_i

)

→

W

∈Ι

ψ

(

H

_n

,

ψ

)

]

H

_n

H

_n

Γ

( ,

X H

_n

)

H

n x

≅

Γ

( ,

X H

_n

)

≅

T

.

H

_n

1

]

n

⊂

[ ]

x

= β

)

n c

. Let us now define a mapping

s W

such that for every . If , then we define

s c

, by taking

[ ]

. It is seen that, the mapping s depends on both the homotopy classes

H

_n

*

_c

[ ]

_c

=

γ

α

=

α

and

[

. Suppose that the homotopy class

[

is chosen as arbitrary fixed, for each . So, the mapping s depends on only the homotopy class

[ ]

]

γ

. s is well defined and . Let us denote the totality of the mapping s defined over W by _n

)

.

Let B be a basis of path connected open neighborhoods for each . Then,

{

}

T

_n

s

H

_n

is a topology base on . In this topology, the mapping

Ψ

and s are continuous and is a local homomorphism. Thus,

(

,

is a sheaf over X. (or only ) is called “The Sheaf of Higher Homotopy Groups over X” [6,7] . For any open set W

⊂

X , an element s of

( ,

W H

is called a section of the sheaf over W. The group

is called the stalk of the sheaf _n for any . The set

Γ

_n is an abelian group with pointwise addition operation. Thus, the operation is continuous for every stalk of . Moreover, the group

→

H

H

)

is abelian for . Hence, is a sheaf of abelian groups over X

)

n

The sheaf satisfies the following properties:

1.

Any two stalks of are isomorphic with each other.

2.

Let be any open sets,

s

W

and

s

W

.If for any point then over the whole

W

W

.

X

⊂

(

s

₂

(

H

_n

)

2

∈Γ

2

,

[

1 2

3.

Let W

⊂

X be an open set. Every section over W can be extended to a global section over X.

4.

Let be any point and W=W(x) be an open set. Then

W

=

V

(

W

and i

is a topological mapping for every

i

. Hence, W=W(x) is evenly covered by . Thus, is an abelian covering space of X

[

9

.

)

5.

A topological stalk preserving mapping of onto itself is called a sheaf isomorphism or a cover transformation, and the set of all cover transformation of is denoted by T. Clearly, T is a group and isomorphic to the group .Hence,

(

)

Thus, T is transitive and is a regular covering space of X

[

.

Characterization

be the corresponding sheaves, respectively. Let us denote these as the pairs and .

(

X H

₁

,

_n 1

)

)

n2

(

X H

₂

,

_n 2

f

* ( (

H

( ,

f f

∗

(

X H

₂

,

_n 2

( , ):

_{f f}

*

_X

(

X H

₁

,

_n 1

f

*

f H

*:

_n 1

→

:

X H

₁

(

X H

₂

,

(

H

_n

)

_x 1 1

⊂

*

x

₂

∈

X

₂

X

₂

s W

We begin by giving the following definitions.

Definition. 2.1. Let

f H

*:

_n1

→

H

be a mapping. If

f*

is continuous, a homomorphism on each stalk of and maps every stalk of into stalk of , then it is called a sheaf homomorphism.

H

_n1

H

_n1

H

_n2

Let be a continuous mapping and be a sheaf homomorphism. If for each , then f* is called a stalk preserving homomorphism with respect to f

[

.

f X

:

₁

→

X

n1

) )

x1

⊂

(

2 n2 n2

)

)

2 n2 n

f H

*:

_n

H

1

→

1

H

_n

)

_{f x( )} 2 1

x

1

∈

X

]

2

Definition 2.2. Let be a sheaf homomorphism. If f* is homeomorphism then f* is called a sheaf isomorphism

[

.

f H

*:

_n

H

1

→

]

8

(

X H

,

Definition 2.3. Let the pairs _{1 n} and be given. If

1

(

X H

2

,

n2

1.

The mapping

f X

:

₁

→

X

is continuous,

2.

The mapping

f H

*:

_n

H

is continuous,

1

→

3.

The mapping

f H

*:

_n

H

is stalk preserving with respect to f.

1

→

2

4.

The mapping

f H

* _{n x}

H

_{n x}

H

_{n f x} ( )

(

₁

) :(

_{1 1}

)

₁

→

(

₂

)

₁

(

X H

,

)

n1

→

2 n2

is a homomorphism for every

then is called a homomorphism between the pairs and .

x

₁

∈

X

₁

)

):(

X H

₁

,

)

)

Definition 2.4. Let the pairs

(

)

, and the homomorphism

be given. If the mappings f and

X H

₁

,

_n1

(

X H

₂

,

_n2

)

)

n2

(

₁

,

H

_n1

) (

→

X H

₂

,

f

* _{are homemorphisms, then (f,} ) is called an isomorphism between the pairs

(

X H

₁

,

_n

)

and

(

₂

,

[

.

1

X H

n2

)

3

]

Teorem 2.1. Let the pairs and be given. If the mapping

is given as a sheaf homomorphism, then there exists a unique continuous mapping such that the pair (f,

(

X H

₁

,

_n 1

)

)

n 2

(

X H

₂

,

_n 2

H

f X

2

X

1

→

f

*_{) is a homomorphism between the pairs}

₍

_,

and . n1

)

n2

)

2

H

2

⊂

Proof. To prove this teorem, we must first find a mapping . However, for each

there exists a stalk , since

f X

:

₁

→

X

) )

(

H

_x1

H

_n2 n1

(

)

( (

)

*

H

_{n2 x2}

⊂

H

_n2

∋

f

_{n1 x}

f

is stalk preserving. Therefore, to any point there uniquely corresponds a point . If we denote this correspondence by ,then we obtain a mapping .

x

₁

∈

X

₁

x

( )

₁

=

₂

X

f x

f X

:

₁

→

₂

Let us now show that the mapping f is continuous. Let be an open set. We may be prove that the set is an open set in . Since W is an open set in , there exists the arcwise connected open sets

W

in ,

W

⊂

f X

(

₁

)

f

−1

_{( )}

W

_X

1 i

X

2

i

∈Ι

, such that

W

i

=

∈

W

i

Υ

Ι . Thus

s W

i i i 2

_{( )}

₌

2

₍

₎

∈

Υ

Ι

is an open set in

H

_n2, for a section

s

2

W H

_n . However,

2

(

)

(

)

f

s W

f

s W

i i i *−

_{( )}

₌

*−

₍

₎

∈ 1 ₂ 1 ₂

Υ

Ι

is an open set in

H

_n , since

1

f

* _{is continuous. Thus, there exists the arcwise connected} open sets

V

_iin

X

₁,

i

∈Ι

, such that

(

)

f

s W

s V

i i i *−

_{( )}

₌

_{( )}

∈ 1 _{2 1}

Υ

Ι

where ‘ s are section over

s

1_i

V

_i for each

i

∈Ι

. Hence

(

)

(

)

ψ

1 2 1

f

s W

V

i i *−

_{( )}

=

∈

Υ

Ι

is an open set in

X

₁. Let us now show that

f

W

V

i i − ∈

=

1

_{( )}

_Υ

Ι

.

)

=

₂

σ

x 1. Let . Then, there exists only one point . Hence

and there is an element ,

, for an

x

₁

_∈

f

−1

₍

W

s W

2

_{( )}

=

∈

i

)

i

∈

x

₂

∈

X

₂

∋

f x

( )

₁

x

(

)

σ

x

f

s W

f

σ

x 1 ₂ −

∋

*

_{( )}

₍

s x

2 _x 2 2

( )

σ

σ

x1

s V

i 1

∈

(

1

∈

1

=

2 *

₎

Ι

, since

f

(

s W

)

s V

. Hence i i i *−

_{( )}

_{( )}

=

∈ 1 _{2 1}

Υ

Ι

ψ σ

1

(

x1

)

=

x

1

∈

V

i . Therefore

f

W

i − ∈

⊂

1

_{( )}

_Υ

Ι

V

i

.

2. Let

x

i i 1

∈

∈

V

Υ

Ι . Then

x

1

∈

V

i and

s x

i

H

, for an

i

1 1 1 1

( ) (

∈

n

)

x

∈Ι

. Therefore and

ψ

(

)

f s x

* _i1

_{( )}

1

∈

s W

2

( )

2

(

f s

*

(

1i

( )

x

1

)

)

=

x

2

∈

W.

From the definition of f,

f x

( )

₁

=

x

₂. Thus

)

. Also, 2

)

)

)

)

)

n2 * * 1 * 1 * 1 * 3 2 3

x

₁

_∈

f

−1

₍

W

Υ

i∈

V

i

f

W

−

⊂

Ι 1

_{( )}

_.

Thus the mapping is continuous. On the other hand it can be shown that the pair is a homomorphism between the pairs and

(

,

, and f is unique,

since .

f X

:

₁

→

X

( , )

_{f f}

*

₍

_{X H}

_,

n 1 1

X H

2 n2

f

ο

ψ

1

=

ψ

2

ο

f

*

We can now state the following theorem.

Theorem 2.2. Let the pairs , and be given. If the

mappings and are sheaf homomorphisms, then there exists a homomorphism between the pairs and such that , .

(

X H

₁

,

_n 1

H

n2

→

(

X H

₁

,

_n

)

1

(

X H

₂

,

_n 2 n3

(

X H

₃

,

_n

)

3

(

X H

₃

,

_n 3

f

= ο

f

f H

_{1 n}

H

1 *

_:

_→

_{f H}

2 *

_:

f

2 1

f

f

f

*

₌

* 2

ο

1

Proof. Since the mappings are continuous, the mapping is also continuous. By Theorem 2.1, there is a continuous mapping f from into . Clearly preserves the stalk with respect to f and is a homomorphism on each stalk. Hence the pair

( ,

is a homomorphism between the pairs

(

,

and

(

,

. Now, let us show that . Since, for any stalk , there is a stalk

(

)

and for any stalk

(

)

, there is a stalk ,

f f

₁*

_,

₂ 1

H

_n 1

H

2

f

₂*

_ο

f

f

₂*

_ο

* n2

∋

1

((

*

f

₂

X

₁

X

₃

)

n3 n2 x2

H

H

f

(

f

H

_n 2

f

₂*

_ο

f

(

H

_n

)

_x 1 1

H

_{n x} 2 2

⊂

)

* *

f f

₂

ο

f

₁

f

= ο

f

) )

H

_{1 x1 n}

⊂

⊂

)

(

H

_n

)

_x 3 3

)

X H

_n1

X H

₃

(

_n

)

_x 3 3

f

2 1

(

H

₂

)

_x

⊂

n

(

H

H

_{n 2}

)

n3 x2

⊂

∋

⊂

(

_f

*

_f

*

)((

_H

) )

_{f f H}

*

( (

*

) )

_f

*

((

_H

) )

(

_H

)

n x n x n x n x 2

ο

1 1 1

=

2 1 1 1

⊂

2 2 2

⊂

3 3,

f

₁

H

_{n x}

H

_{n x} 1 1 2 2 *

₍₍

_{) )}

_⊂

₍

₎

_,

_f

_H

_H

_. n x n x 2 2 2 3 3 *

₍₍

_{) )}

_⊂

₍

₎

So,

(

f

₂

ο

f

₁

)( )

x

₁

=

f f x

₂

(

₁

( )

₁

)

=

x

₃. Therefore .

)

)

n2

)

2

f

₂

ο =

f

₁

f

Now, we can give the following theorem.

Theorem 2.3. Let the pairs

(

,

and

(

,

be given. If the mapping is a sheaf isomorphism, then there exists an isomorphism between the pairs and

(

,

.

X H

_{1 n} 1

X H

2 n2

f H

*

_:

_n

H

1

→

(

X H

₁

,

_n

)

1

X H

2 n2

Proof. It follows from the theorem 2.1 that, there exists a continuous mapping . Let us now show that f is a bijection. In fact, for any two elements , if

, then there is a stalk

(

)

. However, this is impossible, since

f X

:

₁

→

X

₂

x y

₁

,

₁

∈

X

₁

⊂

H

,

x

₂

∈

X

₂

∋

f x

( )

₁

=

f y

( )

₁

=

x

f

*

₍₍

H

_n

_{) )}

_x

f

*

₍₍

H

1 1

H

_{n x} 2 2 n2

H

n1

) ) (

y1 n2

)

x2

=

=

f

* _{is one-to-one.}

Therefore . On the other hand, for each stalk

(

, there exists a stalk since

x

₁

=

y

(

_H

)

_f

*

((

n1 x1

∋

1

) ) (

)

H

_{n x}

H

_{n x} 1 1

=

2 2

)

_x 2 2

H

_n

f

*

∈

∋

X

is onto. It follows from this reason that, for each , there exists an element Hence f is a bijection. By Theorem 2.1, there is a continuous mapping , since is continuous. It is similarly shown that g is a bijection. On the other hand, it can be shown that Therefore f is a homeomorphism.

x

₂

∈

X

₂

x

₁

X

₁

f x

( )

₁

x

2

→

1

f

*−1

g

=

2

=

.

n2 2 n3

g X

:

f

−1

Clearly,

f

*−1 preserves the stalk with respect to f. Thus the pair

( , )

f f

* is an isomorphism. Now, let C be the category of the sheaves of higher homotopy groups and sheaf homomorphisms and D be the category of the connected and locally path connected topological spaces and continuous mappings. Then, we can define a mapping

F C

:

→

D

as follows:

For any sheaf

H

_n and every morphism

f H

*

_:

_n

H

, let F( ) = X and

1

→

H

n

F f

( )

*

₌

f X

:

_→

X

1 . Then,

1.

If

f

*

₌

₁

_Hn , then .

1

F

(

1

Hn1

)

=

1

x1

2.

If

f H

_{1 n}

H

_n and are two sheaf homomorphisms, then

1 2 *

_:

_→

_{f H}

_H

n 2 2 *

_:

_→

F f

(

₂*

ο

f

₁*

)

=

F f

( )

₂*

ο

F f

( )

₁* .

Thus, we can state the following theorem.

Theorem 2.4. There is a covariant functor from the category of the sheaves of higher

homotopy groups and sheaf homomorphisms to the category of the connected and locally path connected topological spaces and continuous mappings.

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A

₁

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A

₁

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