Connectedness and Punctual Space in Fiber Bundle Space

Connectedness and Punctual Space in Fiber Bundle Space

Connectedness and Punctual Space in Fiber Bundle Space

Mr. H. G. Haloli

ISSN: 2278-0181

Vol. 2 Issue 6, June – 2013

The topology of a fiber bundles intuitively a space E with locally looks like a product space B x F but globally may have a different topological structure. Specifically the similarity between fiber bundle E and the product space B x F is defined using a continuous surjection that satisfies a local trivially condition. Jeffrey M. Lee [7] defined as

Definition 1.1 [7]

Let F, B, and E be Cr manifolds and let rr:E—B be a Cr-map. The quadruple(E, rr, B, F) is called a (locally trivial) Cr-fiber bundle if for each point b B there is an open set U containing b and a Cr diffeomorphism :-1(U)—U x F such that the following diagram commutes

-1

-1

(U) UxF

1 U

In differential geometry, your attention is usually focused on C fiber bundle(that is smooth fiber bundles)

If (E,,B,F) is a smooth fiber bundle, then E is called the total space,

:E—B is called the bundle

projection,

B: is called the base space, F: is called the typical fiber,

For each b B, the Eb-1(b) is called the fiber over b.

Examples of fiber bundles: 1.2:-

1. (Vector bundle): A real n-dimensional vector bundle is a fiber bundle with fiber n and structure group GL (n, ). Similarly an n-dimensional complex vector bundle has fiber Cn and structure group GL (n, ). By introducing a metric in each vector bundle. We may

   reduce the structure group to O(n) and U(n), respectively.

2. (The tangent frame bundle of a Cp manifold M): Let M be a n- dimensional manifold. By a frame at p M we mean on ordered basis [V1,V2,Vn] of the tangent space TpM. Denote by Fp the set of all frames at p. We also consider the space F(M) of all frames at all points of M. The general linear group GL(n, ) acts on each fiber Fb on the right.

   Fb x GL(n, )—Fb, namely for a frame u=[V1,V2,Vn] Fb and for a regular matrix

   g= (gij) GL(n, ), we set, wi= gji Vj, ug=[w1,.,wn].

3. Covering manifold: The covering map : N—M of manifolds becomes a fiber bindle with zero dimensional manifold as fiber.

4. For a smooth manifold B and F. We have the projection 1:E—B is a submersion and each fiber -1(b) is a regular manifold which is diffeomorphic to F.

This discus shows that if both F and B are connected the E is connected.

A global smooth section of a fiber bundle

= (E, , B, F) is a smooth map a:B—E such that oa=IdB i.e. a(b) Eb.

A local smooth section over an open set U is a smooth map a:U—E such that oa = IdU. The set of a smooth section of is denoted by ()

Shigeyuki Morita [12] and [John Lee[6]]

,[Jeffrey Lee[7]] defined the isomorphic property of bundles as.

Definition 1.3

Let i = (Ei, i, Bi, F) (i=1, 2) be two fiber bundles with the same fiber, by a bundle map from 1 to 2 we mean C maps

f-:E1 — E2 . f: B1—B2 such that the diagram

We have (U) = 1o(U), where 1:UxF—U denotes

E E the projection onto the first components.

1 2

We call E The total space

B The Base Space

2

1 F Fiber

projection

B B

1 f 1

is commutative.

In this paper we extended the topological properties of fiber space. Lawrence Conlon [8 ] defined connected property of fiber space. We discussed this property in different ways like simply connected, weakly connected, strongly connected and densely connected. These properties on E, B spaces which are topological manifolds.

In Section 2 we recalled some basic

We call Eb= -1(b) the fiber over b. Instead of (E, , B, f) we may ball :E—B or simply E a fiber bundle or is fiber bundles.

Definition 2.2 [S. Morita[12]]

Let i = (Ei, i, Bi, F), i = 1, 2 be two fiber bundles with the same fiber. By a bundle map from 1 to 2 we mean C-maps

f-:E1—E2 , f:B1—B2 such that the

diagram

concepts and definitions. E E

In Section 3 we redefined the connected 1 2

property of fiber bundle particularly Base Space (B),

Total space E with different concepts like globally

1

1

connectedness on E, B, (E, , B, f) 2

Lastly we in introduced the cut point and punctured points on the fiber bundles on the Base

Space B. B. Honari[1], D.K. Kamboj and Vinod B f B

1 2

kumar[2] discussed the cut points of Topological

Manifold M. We discussed on fiber bundles. This

is commutative (ie of- = (fo ) and arbitrary fiber –

concept also defined on and its structure group G. 1 2 1

Naoyuki Monden[] introduced punctured surface and mapping which we discussed on space (i.e. = (E, B, F,).

1. Basic Definitions

   A fiber bundle is a topological manifold our general assumption is that the Base B is Topological Manifold.

   Definition 2.1 :-

   Let F be a C, manifold. Suppose there are C manifolds E and B and a C-map :E—B, we call =(E, ,B, F)is differentiable fiber bundle (or a differentiable F bundle) if it satisfies the following conditions

   (Local Trivially) For each point bB, there are open neighborhoods U and a diffeomorphism :-1(U) =- UxF such that for an arbitrary u-1(U).

   (b), b B is a diffeomorphism. If further more f is a diffeomorphism so is f-(vice versa). Also (f–1, f-1) is a bundle map.

   Definition 2.3 [ S.Morita[12]]

   Two fiber bundles i=(Ei,i,B,F) over the same base space B and with the same fiber F are said to be isomorphic is there exists a bundle map

   f-:E1—E2 together with the identity map f:B—B we

   write 1 =- 2. A bundle that is isomorphic to the product bundle BxF is called a trivial bundle.

   Theorem:2.4 (Jhon Lee)(6)

   Let X, Y be topological spaces and let

   f:X—Y be a continuous map. If X is connected, then f(X) is connected.

2. Connectedness in Fiber

   it also satisfy the composition

   In fiber bundle connectedness is the major concept of topological space. Here we define

   g

   [0,1] Ei

   –+ f

   ri

   i Ej

   connectedness of E, B and F, as topological space.

   In this section a general assumption is that connected and path connected are the same concepts.

   Definition 3.1: Connectedness in Base space.

   A topological space B of dimension is said to be connected if there exists continues map g: [0, 1] –+B such that g (1) =b1 ,g (1)=bn for all b1,bn B

   Definition 3.2: Connectedness in total space.

   A topological space E of dimension n is said to be connected if there exists a continuous map -1 :

   fi 0 g(0) =p1 Ej ,fi 0 g(1) =p2 Ej for fixed j

   Definition -3.6-Strongy connected in E:

   Let Ei (i=1.n) be sub space of E, there exists a maps fI :Ei –+ Ej and

   Continuous functions gi: [0,1] –+Ei

   Such that gi (0)=p1

   gi (i)=pn

   for each p1,pn Ei defines the composition fj ogi :[0,1]

   –+ E –+ E

   B –+E, such that -1 (b1)= p1

   , -1 (bn) =p2

   i j

   for all p1, p2

   E, b1,bn B

   The map g and -1 satisfies the composition -1og[0]=p1, -1og[1]=p2

   Definition 3.3: Simply (weakly

   )Connectedness in B.

   Let Bi (i=1..n) be sub bases of fiber bundle , there exists a map between

   fI o gi [0] =p1 Ej fI o gi [1] =pn Ej Definition 3.7 :

   In all the above definition the maps which are defined rom fI :Ei –+ Ej

   are called bundle maps which are commutative in n- dimensitionaly projection

   Some Bi to Bj that is f:Bi–+Bj such that the

   : E –+ B

   g fi

   compositions [0,1] Bi Bj

   Are defined as

   fiog (0) = b1 Bj

   fiog (1) = b2 Bj, for a fixed j.

   Definition 3.4: Strongy connected in B

   Let Bi(i=1n)be sub bases of fiber bundle .there exists a map fj

   Bi–+Bj for all i and j such that the composition map fj o gi (0) = bi Bj

   fj o gi (1)=bi Bj, for all i and j.

   Definition 3.5: Simply (weakly) Connectedness in E.

   Let Ei (i=1.n) be sub spaces of E such that there exists a maps fI ;Ei –+ Ej such that

   fI ;Ei –+ Ej is defined as

   fi(pi)=pj, for all pi Ei, pj Ej.

   1. i i

      j:Ej–+ Bj

      These maps and projection are diffeomorphisms and n-projection of fiber bundle .

      This shows that,

      fi-1 , f i-1 are also bundle maps .

      These concepts of bundle map forms a densely connected space in .

      Definition 3.8: Densely connected (fiber bundle)

      Let Ei ,Bi,i are the subtotal spaces , subbase space and projection maps respectively and a continuous map, g:[0,1] –+Ei or Bi ,satisfies

      g(0)=ei Ei or bi Bi for all i.

      These bundle map and projections forms densely connection between them then such enrich structure is called densely connected .

      Theorem 3.9

      Let :E –+ B be a projection map , if E is connected then B is connected .

      Proof :

      Let E and B are topological spaces. The map :E –+B be a continues map .

      A total space E is connected, then there exists a continuous map g

      Defined g:[0,1] –+E such that g(0)=P1=-1 (b1) and

      g (1)=P2= -1(b2)for all P1, P2 E andb1,b2 B.

      Theorem 3.11 :

      Let E and B be a topological space.

      :E–+ B is a bundle projection if E is simply connected then B is simply connected.

      Proof :

      Let Ei(i=1,. n) be sub space of E such that there exists a map fi : Ei –+ Ej defined as fi (Pi)=Pj Pi,Pj Ei ,Ej respectively

      By definition of simply connected in E

      The composition , fI o g[0]=P2 Ej for fixed j

      fI o g[1]=Pn Ej

      We know is projection map which is always continues map also bijective map

      g

      [0,1] Ei–+ fi

      _–+Ej E

      Therefore the image of connected space is connected under continuous map.(see theorem 2.4)

      [For basic theorem see [Jhon Lee[6] [p-67] theorem

      :4,3.]

      fi (g)(0)=P2 = fi (P1)=P2,

      fi g(1)= fi (Pn-1)=Pn

      These composition maps are continuous in Ej s.

      Ei Ej

      T[i

      T[j

      f

      f

      Bi Bj

      i

      As E is connected and B be a topological space :E

      –+ B which is continuous. Therefore B is connected. Corollary 3.10 :

      Let E and B be a topological spaces and

      :E –+B is bijective map which is continuous if B is connected then E is connected.

      Proof :

      Let E1and B are topological spaces. :E

      –+B is bijective map and continuous then -1:B–+E is also continuous map .(see theorem2.3).

      Then by (Jhon Lee[6] [p-67] theorem4.3 & theorem 3.9) E is connected.

      Now Ej ,EjE , consider the projections map generally

      i :Ei –+ Bi B are projections from Ei to Bi Then the composition

      j. ofi og(0)=bj , j ofi og (1)=bj for some i

      Therefore these composition maps are continuous j o fi og ,this shows the path between some points in B

      .i.e.j o fi

      i

      This commutative diagram shows that

      j ofi (Pi)=bj , Pi Ei bi Bj similarly j o fi og(0)=bj1 ————–(i)

      for some n fixed j

      j o fiog(1)=bjn —————(ii) These two equation gives the composition map j o fi o g (0)& j o fi o g (1) given a path

      Pj Ej , for fixed i

      OR

      fi o i1 (bi)=Pj , bi Bi ,

      i-1 o fi-1 (bj)=Pi , Pi

      between some point bj1 and bjn respectively

      B is simply connected space

      Theorem 3.12

      Let E be total space and B base space and the projection :E— B is bijective map.

      If B is simply connected than E is also simply connected.

      Proof:

      Let Bi are sub bases space of B and Ei are sub space of E.

      As B is simply connected by definition there exists a maps fi :Bi — Bj

      And the continuous map g:[0,1] —Bi ,the composition is defined

      fi og (0)=bj1 Bj ,

      i.e. fi [bi]=bj1 , for a fixed j. fiog(1)=bjn ,

      i.e. fi[bin]= bjn

      consider the projection

      i :Ei — Bi is bijective implies i-1 :Bi — Ei is continuous

      The following commutative map shows that

      case i

      Ei , bj — Bj for fixed j Since f i-1 , fi are bundle maps

      case ii

      -1j fi (bi)=Pj Pj Ej , bj Bi for fixed i,

      fi-1 o -1j (bj)=pi for fixed j

      These cases show that case i is considered with composition map of g,

      g:[0,1]—Bi

      and composition

      fi o -1i o g (0) =Pj1 fi o -1i o g (1)=Pjn

      This shows that there exists a simple path between Pj1 to Pjn in Ej E

      similarly case i

      -1i o f-1i (g(0))=Pi1 -1i o fi (g(1))=Pin

      This shows that there exists path in Ei E.

      Similarly for case ii we will get the subspace Ei and Ej are simply connected.

      Therefore E is simply connected space.

      Theorem 3.12

      If :E–+B be a projection map and E is strongly connected then B is so.

      Proof

      Let Ei (i=1.n)are subspaces of E. Bi are the spaces of B and the projection

      :E–+B a bundle projection which is bijective and continuous map.

      As E is strongly connected there exists map g:[0,1]

      –+E and bundle maps fi from Ei to Ej that is

      fi:Ei–+Ej defined as fi[Pi]=Pj

      Therefore this shows that all Bi s are connected with continuous maps fi

      Therefore B is strongly connected.

      Corollary 3.13

      If :E–+B be a bijective map & B is strongly connected then E is so.

3. Cut Points and Punctured Points in Fiber Bundles.

A Topological spaces E and B are connected which contains at least two points. A point which becomes separate or disconnected is a cut point space

Ei Ej

T[i

T[j

f

f

Bi

i

The composition map fI og:[0,1]–+Ej is a bundle map which is also continuous map

The commutative

By definition of a strongly connectedness in E. j ofj[Pi]=bj Bj

i ofi[Pi]=bj Bj

Therefore the composition map

o f : E –+B which is continuous for any point P E

Bj

with different dimensions of topological space like 1 dimension or more, which are defined in connectedness paper.

The concept of cut point and punctured point in fiber space are same concept as in topological space but they depend only on base space. We defined cut point on base space as

Definition 4.1 Cut point in B

A point x B is said to be a cut point if B

{x} has a separation or B becomes disconnected. That is by removal of a point x from the B becomes

1. i i j

   i i disconnected as subspaces.

   The image of connected space under continuous map fI o j is connected & a continuous map

   g:[0,1]–+Ei

   j o fi o g [0]=bj 1 Bj j o fi o g [1]=bj n Bj OR

   fi oi o g(0) =bi 1 Bj fi oi o g(1) =bi n Bj

   Therefore each element of base space connected by a path

   Lemma 4.2

   Let B be a topologically connected space, x B be any point of B, then by removal of x from B, B becomes disconnected but disconnected set satisfies the following properties.

   1. If B1 and B2 are two separated sets then {xn} B1 B2 = B and B1 B2

      = x

   2. The separated sets B1 and B2 are connected.

Theorem 4.3

Let B be connected topological space and x ct x B with condition B1, B2 B and

B1 B2 {x} = B, B1 B2 = x then there exists a path between this to subspace B1 and B2 passing through x.

Let B be connected topological space x be a cut point of B which forms a separation of

B =B1/B2.

Separated space (sets) are connected.

Let xis B1 , yis B2 are points in separated sets for some i.

Consider a function g:[0, 1]–+B defined as g(0) = x, g(1) = y.

The xis and yis are any other point in B, particularly in two compartments in B like B1 and B2.

As B is connected which implies B1 and B2 are also connected, interconnected by the point xct which is called a cut point of B.

B1 and B2 are interconnected then some points of B1 and some points of B2 are connected to a path y.

y:[0, 1]–+B, y(0) = x B, y(1) = y

Let the family of paths yi(1) = yi for all xi B1 yi B2. But B1 and B2 are connected by a single point all paths yIs are passing through the point which is common in B1 and B2 which cut point of B set xct.

Therefore B is connected with cut point. There exists a path passing through cut point of B.

Corollary 4.4

If B is a path connected then B is connected.

Corollary 4.5

If B is connected with cut point then B is path connected.

Definition 4.6 Cutput in E

Let E be a total space, B be a base space for fiber space. The projection :E–+B is continuous and bijective, a point x B is cut point of B then -1(x) E is called cut point of E.

That is the pre image of cut point of base space is cut point of total space E.

Definition 4.7 Punctured point in E and B

A point P(x) is said to a punctured point of B if removal of P(x) from B, B becomes hole space, but B is connected.

That is P(x) be a punctured point of B. Then

{B P(x)} is connected.

Theorem 4.8

Let B be connected a topological manifold (space). A point P(x) be a punctured point of B then

{B-P(x)} is connected.

Proof

Let B be a topological base space which is connected. There exists a function f:[0, 1]–+B satisfies f(0) = x, f(1) = y for all x, y B.

Let xp be a point other than x and y of B is a punctured point of B. All the neighborhood point of xp are connected with B. There exists a path between them.

Any two points except xp of B are connected by a path.

By a removal of a point xp from B the remaining points of B are connected by a path.

Therefore B is a path connected space and is connected.

Theorem 4.9

Let E topologically connected total space. A point P(x) B a punctured point of B then -1(Px) is punctured in E.

Proof

Let E be Topological connected total space. :E–+B a projection map which is continuous and bijection. P(xp) B be a punctured point of B then projection map :E–+B and -1:B–+E also a continuous map. Satisfies -1(x) = F1 for all F1 E. for all x B

But P(x) is punctured point of B. -1(P(x)) is punctured point of E. That is F1 is a punctured of E.

Theorem 4.10

If :E—B is a projection map and continuous map and B is a path connected space then E is also.

Theorem 4.11

Let E be topological connected total space. A point P(x) B a punctured point of B and B is connected then there exists a punctured point – 1(P(x)) E and E is connected as arc wise. (path connected).

Proof:

Let E be topological connected total space. A point P(x) B a punctured of B. B is connected, there exists a continuous function f:[0, 1]—B such that

f(0) = x, f(1) = y for all x,

Corollary 4.12

Let E and B be a connected topological spaces.

A point -1(x) E is punctured in E and :E—B projection map then x B which need not be punctured point of B.

Conclusion:

Connectedness of base space, total space gives connectedness of fiber bundle which is base of further work, and application in Biology.

y B

:E—B is continuous and bijective. This

implies -1:B—E such that -1(x) = F1, -1(y) = F2 for (F1) = x, (F2) = y, for all F1, F2 E

As B is connected there exists a path y:[0, 1]—B satisfies y(0) = x, y(1) = y.

If P(x) B be a punctured point of B.

There does not exists a path to punctured point. But its neighborhood points are connected with B which forms a boundary of punctured space (p(x)).

We have boundary of any space which is connected. There exists a path between any two points in neighborhood of P(x).

Therefore B is connected, B-P(x) is also connected by a path.

Therefore the continuous map -1:(P(x))—Fp is defined as P(x) = (Fp)

But neighborhood of Fp are connected with E space as B is connected E is also connected.

By continuous map :E—B

Therefore E is connected which implies every point between ant two points except punctured point Fp are connected by a path

Therefore E is path connected.

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