CUBO A Mathematical Journal

Vol.11, No¯ 01, (101–121). March 2009

Wrap groups of fiber bundles and their structure

S.V. Ludkovsky

Department of Applied Mathematics,

Moscow State Technical University MIREA, Av. Vernadsky 78,

Moscow 119454, Russia.

email: sludkowski@mail.ru

ABSTRACT

This article is devoted to the investigation of wrap groups of connected fiber bundles.

These groups are constructed with mild conditions on fibers. Their examples are given.

It is shown, that these groups exist and for differentiable fibers have the infinite di-

mensional Lie groups structure, that is, they are continuous or differentiable manifolds

and the composition (f,g) 7→ f−1g is continuous or differentiable depending on a class

of smoothness of groups. Moreover, it is demonstrated that in the cases of real, com-

plex, quaternion and octonion manifolds these groups have structures of real, complex,

quaternion or octonion manifolds respectively. Nevertheless, it is proved that these

groups does not necessarily satisfy the Campbell-Hausdorff formula even locally.

RESUMEN

Este artículo es dedicado a la investigación de grupos Wrap de fibrados conexos. Estos

grupos son construidos con condiciones blandas sobre las fibras, ejemplos son dados. Es

demostrado que estos grupos existen y para fibras diferenciables tienen una estructura

de grupo de Lie infinito dimensional, es decir, son variedades continuas o diferenciables

y la composición (f,g) 7→ f−1g es continua o diferenciable dependiendo de la clase

de suavidad de los grupos. Además es demostrado que en el caso de variedades real,

compleja, cuaternion y octonion esos grupos tienen una estructura de variedad real,

compleja, cuaternion o octonion respectivamente. También es probado que estos grupos

no necesariamente satisfacen la fórmula de Campbell-Hausdorff incluso localmente.



102 S.V. Ludkovsky CUBO
11, 1 (2009)

1 Introduction.

Wrap groups of fiber bundles considered in this paper are constructed with the help of families of

mappings from a fiber bundle with a marked point into another fiber bundle with a marked point

over the fields R, C, H and the octonion algebra O. Conditions on fibers supplied with parallel

transport structures are rather mild here. Therefore, they generalize geometric loop groups of

circles, spheres and fibers with parallel transport structures over them. A loop interpretation is

lost in their generalizations, so they are called here wrap groups. This paper continues previous

works of the author on this theme, where generalized loop groups of manifolds over R, C and H

were investigated, but neither for fibers nor over octonions [15, 23, 21, 22].

Loop groups of circles were first introduced by Lefshetz in 1930-th and then their construc-

tion was reconsidered by Milnor in 1950-th. Lefshetz has used the C0-uniformity on families of

continuous mappings, which led to the necessity of combining his construction with the structure

of a free group with the help of words. Later on Milnor has used the Sobolev’s H1-uniformity,

that permitted to introduce group structure more naturally [27]. Iterations of these constructions

produce iterated loop groups of spheres. Then their constructions were generalized for fibers over

circles and spheres with parallel transport structures over R or C [4].

Wrap groups of quaternion and octonion fibers as well as for wider classes of fibers over R or

C are defined and investigated here for the first time.

Holomorphic functions of quaternion and octonion variables were investigated in [19, 20, 17].

There specific definition of super-differentiability was considered, because the quaternion skew

field has the graded algebra structure. This definition of super-differentiability does not impose

the condition of right or left super-linearity of a super-differential, since it leads to narrow class

of functions. There are some articles on quaternion manifolds, but practically they undermine

a complex manifold with additional quaternion structure of its tangent space (see, for example,

[28, 39] and references therein). Therefore, quaternion manifolds as they are defined below were

not considered earlier by others authors (see also [17]). Applications of quaternions in mathematics

and physics can be found in [6, 9, 10, 14].

In this article wrap groups of different classes of smoothness are considered. Henceforth, we

consider not only orientable manifolds M and N, but also nonorientable manifolds.

In particular, geometric loop groups have important applications in modern physical theories

(see [11, 24] and references therein). Groups of loops are also intensively used in gauge theory.

Wrap groups defined below with the help of families of mappings from a manifold M into another

manifold N with a dimension dim(M) > 1 can be used in the membrane theory which is the

generalization of the string (superstring) theory.

Section 2 is devoted to the definitions of topological and manifold structures of wrap groups.

The existence of these groups is proved and that they are infinite dimensional Lie groups not

satisfying even locally the Campbell-Hausdorff formula (see Theorems 3, 6, 12, Corollaries 5, 8, 9



CUBO
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Wrap groups of fiber bundles and their structure 103

and Examples 10). In the cases of complex, quaternion and octonion manifolds it is proved that

they have structures of complex, quaternion and octonion manifolds respectively.

All main results of this paper are obtained for the first time.

2 Wrap groups of fibers.

To avoid misunderstandings we first give our definitions and notations.

1.1. Note. Denote by Ar the Cayley-Dickson algebra such that A0 = R, A1 = C, A2 = H is

the quaternion skew field, A3 = O is the octonion algebra. Henceforth we consider only 0 ≤ r ≤ 3.

1.2. Definition. A canonical closed subset Q of the Euclidean space X = Rn or of the

standard separable Hilbert space X = l2(R) over R is called a quadrant if it can be given by the

condition Q := {x ∈ X : qj(x) ≥ 0}, where (qj : j ∈ ΛQ) are linearly independent elements of the

topologically adjoint space X∗. Here ΛQ ⊂ N (with card(ΛQ) = k ≤ n when X = R
n) and k is

called the index of Q. If x ∈ Q and exactly j of the qi’s satisfy qi(x) = 0 then x is called a corner

of index j.

If X is an additive group and also left and right module over H or O with the corresponding

associativity or alternativity respectively and distributivity laws then it is called the vector space

over H or O correspondingly.

In particular l2(Ar) consisting of all sequences x = {xn ∈ Ar : n ∈ N} with the finite norm

‖x‖ < ∞ and scalar product (x,y) :=
∑∞
n=1

xny
∗
n

with ‖x‖ := (x,x)1/2 is called the Hilbert space

(of separable type) over Ar, where z
∗ denotes the conjugated Cayley-Dickson number, zz∗ =: |z|2,

z ∈ Ar. Since the unitary space X = A
n

r
or the separable Hilbert space l2(Ar) over Ar while

considered over the field R (real shadow) is isomorphic with XR := R
2
r
n or l2(R), then the above

definition also describes quadrants in An
r

and l2(Ar). In the latter case we also consider generalized

quadrants as canonical closed subsets which can be given by Q := {x ∈ XR : qj(x + aj) ≥ 0,aj ∈

XR,j ∈ ΛQ}, where ΛQ ⊂ N (card(ΛQ) = k ∈ N when dimRXR < ∞).

1.2.2. Definition. A differentiable mapping f : U → U′ is called a diffeomorphism if

(i) f is bijective and there exist continuous mappings f′ and (f−1)′, where U and U′ are

interiors of quadrants Q and Q′ in X.

In the Ar case with 1 ≤ r ≤ 3 we consider bounded generalized quadrants Q and Q
′ in

An
r

or l2(Ar) such that they are domains with piecewise C
∞-boundaries. We impose additional

conditions on the diffeomorphism f in the 1 ≤ r ≤ 3 case:

(ii) ∂̄f = 0 on U,

(iii) f and all its strong (Frechét) differentials (as multi-linear operators) are bounded on

U, where ∂f and ∂̄f are differential (1, 0) and (0, 1) forms respectively, d = ∂ + ∂̄ is an exterior

derivative, for 2 ≤ r ≤ 3 ∂ corresponds to super-differentiation by z and ∂̃ = ∂̄ corresponds to



104 S.V. Ludkovsky CUBO
11, 1 (2009)

super-differentiation by z̃ := z∗, z ∈ U (see [19, 20]).

The Cauchy-Riemann Condition (ii) means that f on U is the Ar-holomorphic mapping.

1.2.3. Definition and notation. An Ar-manifold M with corners is defined in the usual

way: it is a metric separable space modelled on X = An
r

or X = l2(Ar) respectively and is supposed

to be of class C∞, 0 ≤ r ≤ 3. Charts on M are denoted (Ul,ul,Ql), that is, ul : Ul → ul(Ul) ⊂ Ql
is a C∞-diffeomorphism for each l, Ul is open in M, ul ◦ uj

−1 is biholomorphic for 1 ≤ r ≤ 3 from

the domain uj(Ul ∩ Uj) 6= ∅ onto ul(Ul ∩ Uj) (that is, uj ◦ u
−1
l

and ul ◦ u
−1
j

are holomorphic and

bijective) and ul ◦ u
−1
j

satisfy conditions (i − iii) from §1.2.2,
⋃
j
Uj = M.

A point x ∈ M is called a corner of index j if there exists a chart (U,u,Q) of M with x ∈ U

and u(x) is of index indM(x) = j in u(U) ⊂ Q. A set of all corners of index j ≥ 1 is called a

border ∂M of M, x is called an inner point of M if indM(x) = 0, so ∂M =
⋃
j≥1 ∂

jM, where

∂jM := {x ∈ M : indM(x) = j}.

For a real manifold with corners on the connecting mappings ul ◦ u
−1
j

∈ C∞ of real charts

only Condition 1.2.2(i) is imposed.

1.2.4. Terminology. In an Ar-manifold N there exists an Hermitian metric, which in each

analytic system of coordinates is the following
∑
n

j,k=1
hj,kdzjdz̄k, where (hj,k) is a positive definite

Hermitian matrix with coefficients of the class C∞, hj,k = hj,k(z) ∈ Ar, z are local coordinates in

N.

As real manifolds we shall consider Riemann manifolds.

In accordance with the definition above for internal points of N it is supposed that they can

belong only to interiors of charts, but for boundary points ∂N it may happen that x ∈ ∂N belongs

to boundaries of several charts. It is convenient to choose an atlas such that ind(x) is the same

for all charts containing this x.

1.3.1. Remark. If M is a metrizable space and K = KM is a closed subset in M of

codimension codimR N ≥ 2 such that M \ K = M1 is a manifold with corners over Ar, then we

call M a pseudo-manifold over Ar, where KM is a critical subset.

Two pseudo-manifolds B and C are called diffeomorphic, if B \ KB is diffeomorphic with

C \ KC as for manifolds with corners (see also [4, 26]).

Take on M a Borel σ-additive measure ν such that ν on M \ K coincides with the Riemann

volume element and ν(K) = 0, since the real shadow of M1 has it.

The uniform space Ht
p
(M1,N) of all continuous piecewise H

t Sobolev mappings from M1

into N is introduced in the standard way [21, 22], which induces Ht
p
(M,N) the uniform space of

continuous piecewise Ht Sobolev mappings on M, since ν(K) = 0, where R ∋ t ≥ [m/2] + 1, m

denotes the dimension of M over R, [k] denotes the integer part of k ∈ R, [k] ≤ k. Then put

H∞
p

(M,N) =
⋂
t>m

Ht
p
(M,N) with the corresponding uniformity.

For manifolds over Ar with 1 ≤ r ≤ 3 take as H
t

p
(M,N) the completion of the family of



CUBO
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Wrap groups of fiber bundles and their structure 105

all continuous piecewise Ar-holomorphic mappings from M into N relative to the H
t

p
uniformity,

where [m/2] + 1 ≤ t ≤ ∞. Henceforth we consider pseudo-manifolds with connecting mappings of

charts continuous in M and Ht
′

p
in M \ KM for 0 ≤ r ≤ 3, where t

′ ≥ t.

1.3.2. Note. Since the octonion algebra O is non-associative, we consider a non-associative

subgroup G of the family Matq(O) of all square q × q matrices with entries in O. More generally

G is a group which has a Ht
p

manifold structure over Ar and group’s operations are H
t

p
mappings.

The G may be non-associative for r = 3, but G is supposed to be alternative, that is, (aa)b = a(ab)

and a(a−1b) = b for each a,b ∈ G.

As a generalization of pseudo-manifolds there is used the following (over R and C see [4, 34]).

Suppose that M is a Hausdorff topological space of covering dimension dim M = m supplied with

a family {h : U → M} of the so called plots h which are continuous maps satisfying conditions

(D1 − D4):

(D1) each plot has as a domain a convex subset U in An
r
, n ∈ N;

(D2) if h : U → M is a plot, V is a convex subset in Al
r

and g : V → U is an Ht
p

mapping,

then h ◦ g is also a plot, where t ≥ [m/2] + 1;

(D3) every constant map from a convex set U in An
r

into M is a plot;

(D4) if U is a convex set in An
r

and {Uj : j ∈ J} is a covering of U by convex sets in A
n

r
, each

Uj is open in U, h : U → M is such that each its restriction h|Uj is a plot, then h is a plot. Then

M is called an Ht
p
-differentiable space.

A mapping f : M → N between two Ht
p
-differentiable spaces is called differentiable if it

continuous and for each plot h : U → M the composition f ◦ h : U → N is a plot of N. A

topological group G is called an Ht
p
-differentiable group if its group operations are Ht

p
-differentiable

mappings.

Let E, N, F be Ht
′

p
-pseudo-manifolds or Ht

′

p
-differentiable spaces over Ar, let also G be an

Ht
′

p
group over Ar, t ≤ t

′ ≤ ∞. A fiber bundle E(N,F,G,π, Ψ) with a fiber space E, a base space

N, a typical fiber F and a structural group G over Ar, a projection π : E → N and an atlas Ψ

is defined in the standard way [4, 26, 35] with the condition, that transition functions are of Ht
′

p

class such that for r = 3 a structure group may be non-associative, but alternative.

Local trivializations φj ◦ π ◦ Ψ
−1
k

: Vk(E) → Vj(N) induce the H
t
′

p
-uniformity in the family

W of all principal Ht
′

p
-fiber bundles E(N,G,π, Ψ), where Vk(E) = Ψk(Uk(E)) ⊂ X

2
(G), Vj(N) =

φj(Uj(N)) ⊂ X(N), where X(G) and X(N) are Ar-vector spaces on which G and N are modelled,

(Uk(E), Ψk) and (Uj(N),φj) are charts of atlases of E and N, Ψk = Ψ
E

k
, φj = φ

N

j
.

If G = F and G acts on itself by left shifts, then a fiber bundle is called the principal fiber

bundle and is denoted by E(N,G,π, Ψ). As a particular case there may be G = A∗
r
, where A∗

r

denotes the multiplicative group Ar \ {0}. If G = F = {e}, then E reduces to N.

2. Definitions. Let M be a connected Ht
p
-pseudo-manifold over Ar, 0 ≤ r ≤ 3 satisfying the

following conditions:



106 S.V. Ludkovsky CUBO
11, 1 (2009)

(i) it is compact;

(ii) M is a union of two closed subsets over Ar A1 and A2, which are pseudo-manifolds and

which are canonical closed subsets in M with A1 ∩ A2 = ∂A1 ∩ ∂A2 =: A3 and a codimension over

R of A3 in M is codimRA3 = 1, also A3 is a pseudo-manifold;

(iii) a finite set of marked points s0,1, ...,s0,k is in ∂A1 ∩ ∂A2, moreover, ∂Aj are arcwise

connected j = 1, 2;

(iv) A1\∂A1 and A2\∂A2 are H
t

p
-diffeomorphic with M\[{s0,1, ...,s0,k}∪(A3\Int(∂A1∩∂A2))]

by mappings Fj(z), where j = 1 or j = 2, ∞ ≥ t ≥ [m/2] + 1, m = dimRM such that H
t ⊂ C0 due

to the Sobolev embedding theorem [25], where the interior Int(∂A1 ∩ ∂A2) is taken in ∂A1 ∪ ∂A2.

Instead of (iv) we consider also the case

(iv′) M, A1 and A2 are such that (Aj \ ∂Aj) ∪ {s0,1, ...,s0,k} are

C0([0, 1],Ht
p
(Aj,Aj))-retractable on X0,q ∩ Aj, where X0,q is a closed arcwise connected subset in

M, j = 1 or j = 2, s0,q ∈ X0,q, X0,q ⊂ KM, q = 1, ...,k, codimR KM ≥ 2.

Let M̂ be a compact connected Ht
p
-pseudo-manifold which is a canonical closed subset in

Al
r

with a boundary ∂M̂ and marked points {ŝ0,q ∈ ∂M̂ : q = 1, ..., 2k} and an H
t

p
-mapping

Ξ : M̂ → M such that

(v) Ξ is surjective and bijective from M̂ \ ∂M̂ onto M \ Ξ(∂M̂) open in M, Ξ(ŝ0,q) =

Ξ(ŝ0,k+q)s0,q for each q = 1, ...,k, also ∂M ⊂ Ξ(∂M̂).

A parallel transport structure on a Ht
′

p
-differentiable principal G-bundle E(N,G,π, Ψ) with

arcwise connected E and G for Ht
p
-pseudo-manifolds M and M̂ as above over the same Ar with

t′ ≥ t + 1 assigns to each Ht
p

mapping γ from M into N and points u1, ...,uk ∈ Ey0 , where y0 is

a marked point in N, y0 = γ(s0,q), q = 1, ...,k, a unique H
t

p
mapping Pγ̂,u : M̂ → E satisfying

conditions (P1 − P5):

(P1) take γ̂ : M̂ → N such that γ̂ = γ ◦ Ξ, then Pγ̂,u(ŝ0,q) = uq for each q = 1, ...,k and

π ◦ Pγ̂,u = γ̂

(P2) Pγ̂,u is the H
t

p
-mapping by γ and u;

(P3) for each x ∈ M̂ and every φ ∈ DifHt
p
(M̂,{ŝ0,1, ..., ŝ0,2k}) there is the equality Pγ̂,u(φ(x)) =

Pγ̂◦φ,u(x), where DifH
t

p
(M̂,{ŝ0,1, ..., ŝ0,2k}) denotes the group of all H

t

p
homeomorphisms of M̂

preserving marked points φ(ŝ0,q) = ŝ0,q for each q = 1, ..., 2k;

(P4) Pγ̂,u is G-equivariant, which means that Pγ̂,uz(x) = Pγ̂,u(x)z for every x ∈ M̂ and each

z ∈ G;

(P5) if U is an open neighborhood of ŝ0,q in M̂ and γ̂0, γ̂1 : U → N are H
t
′

p
-mappings such

that γ̂0(ŝ0,q) = γ̂1(ŝ0,q) = vq and tangent spaces, which are vector manifolds over Ar, for γ0 and

γ1 at vq are the same, then the tangent spaces of Pγ̂0,u and Pγ̂1,u at uq are the same, where

q = 1, ...,k, u = (u1, ...,uk).



CUBO
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Wrap groups of fiber bundles and their structure 107

Two Ht
′

p
-differentiable principal G-bundles E1 and E2 with parallel transport structures

(E1, P1) and (E2, P2) are called isomorphic, if there exists an isomorphism h : E1 → E2 such

that P2,γ̂,u(x) = h(P1,γ̂,h−1(u)(x)) for each H
t

p
-mapping γ : M → N and uq ∈ (E2)y0 , where

q = 1, ...,k, h−1(u) = (h−1(u1), ...,h
−1

(uk)).

Let (SME)t,H : (S
M,{s0,q :q=1,...,k}E; N,G, P)t,H be a set of H

t

p
-closures of isomorphism classes

of Ht
p

principal G fiber bundles with parallel transport structure.

3. Theorems. 1. The uniform space (SME)t,H from §2 has the structure of a topological

alternative monoid with a unit and with a cancelation property and the multiplication operation of

Hl
p

class with l = t′ −t (l = ∞ for t′ = ∞). If N and G are separable, then (SME)t,H is separable.

If N and G are complete, then (SME)t,H is complete.

2. If G is associative, then (SME)t,H is associative. If G is commutative, then (S
ME)t,H is

commutative. If G is a Lie group, then (SME)t,H is a Lie monoid.

3. The (SME)t,H is non-discrete, locally connected and infinite dimensional for dimR(N ×

G) > 1.

Proof. If there is a homomorphism θ : G → F of Ht
′

p
-differentiable groups, then there exists

an induced principal F fiber bundle (E×θF)(N,F,πθ, Ψθ) with the total space (E×θF)(E×F)/Y,

where Y is the equivalence relation such that (vg,f)Y(v,θ(g)f) for each v ∈ E, g ∈ G, f ∈ F .

Then the projection πθ : (E ×θ F) → N is defined by πθ([v,f]) = π(v), where [v,f] := {(w,b) :

(w,b)Y(v,f),w ∈ E,b ∈ F} denotes the equivalence class of (v,f).

Therefore, each parallel transport structure P on the principal G fiber bundle E(N,G,π, Ψ)

induces a parallel transport structure Pθ on the induced bundle by the formula Pθ
γ̂,[u,f]

(x) =

[Pγ̂,u(x),f].

Define multiplication with the help of certain embeddings and isomorphisms of spaces of

functions. Mention that for each two compact canonical closed subsets A and B in Al
r
Hilbert spaces

Ht(A, Rm) and Ht(B, Rm) are linearly topologically isomorphic, where l,m ∈ N, hence Ht
p
(A,N)

and Ht
p
(B,N) are isomorphic as uniform spaces. Let Ht

p
(M,{s0,1, ...,s0,k}; W,y0) := {(E,f) : E =

E(N,G,π, Ψ) ∈ W,f = Pγ̂,y0 ∈ H
t

p
: π ◦ f(s0,q) = y0∀q = 1, ...,k; π ◦ f = γ̂,γ ∈ H

t

p
(M,N)} be the

space of all Ht
′

p
principal G fiber bundles E with their parallel transport Ht

p
-mappings f = Pγ̂,y0 ,

where W is as in §1.3.2. Put ω0 = (E0, P0) be its element such that γ0(M) = {y0}, where e ∈ G

denotes the unit element, E0 = N × G, π0(y,g) = y for each y ∈ N, g ∈ G, Pγ̂0,u = P0.

The mapping Ξ : M̂ → M from §2 induces the embedding

Ξ
∗

: Ht
p
(M,{s0,1, ...,s0,k}; W,y0) →֒ H

t

p
(M̂,{ŝ0,1, ..., ŝ0,2k}; W,y0),

where M̂ and Â1 and Â2 are retractable into points.

Let as usually A∨B := ρ(Z) be the wedge sum of pointed spaces (A,{a0,q : q = 1, ...,k}) and

(B,{b0,q : q = 1, ...,k}), where Z := [A × {b0,q : q = 1, ...,k} ∪ {a0,q : q = 1, ...,k} × B] ⊂ A × B,

ρ is a continuous quotient mapping such that ρ(x) = x for each x ∈ Z \ {a0,q × b0,j; q,j = 1, ...,k}

and ρ(a0,q) = ρ(b0,q) for each q = 1, ...,k, where A and B are topological spaces with marked



108 S.V. Ludkovsky CUBO
11, 1 (2009)

points a0,q ∈ A and b0,q ∈ B, q = 1, ...,k. Then the wedge product g ∨ f of two elements

f,g ∈ Ht
p
(M,{s0,1, ...,s0,k}; N,y0) is defined on the domain M ∨ M such that (f ∨ g)(x × b0,q) =

f(x) and (f ∨ g)(a0,q × x) = g(x) for each x ∈ M, where to f,g there correspond f1,g1 ∈

Ht
p
(M̂,{ŝ0,1, ..., ŝ0,2k}; N,y0) such that f1f ◦ Ξ and g1 = g ◦ Ξ.

Let (Ej, Pγ̂j,uj ) ∈ H
t

p
(M,{s0,1, ...,s0,k}; W,y0), j = 1, 2, then take their wedge product

Pγ̂,u1 := Pγ̂1,u1 ∨ Pγ̂2,v on M ∨ M with vq = uqg
−1
2,q
g1,q+k = y0 × g1,q+k for each q = 1, ...,k

due to the alternativity of G, γ = γ1 ∨γ2, where Pγ̂j,uj (ŝj,0,q)y0 ×gj,q ∈ Ey0 for every j and q. For

each γj : M → N there exists γ̃j : M → Ej such that π ◦ γ̃j = γj. Denote by m : G × G → G the

multiplication operation. The wedge product (E1, Pγ̂1,u1 ) ∨ (E2, Pγ̂2,u2 ) is the principal G fiber

bundle (E1 × E2) ×
m G with the parallel transport structure Pγ̂1,u1 ∨ Pγ̂2,v.

The uniform space Ht
p
(J,A3; W,y0) := {(E,f) ∈ H

t

p
(J,W) : π ◦ f(A3) = {y0}} has the H

t

p
-

manifold structure and has an embedding into

Ht
p
(M,{s0,1, ...,s0,k}; W,y0) due to Conditions 2(i − iii), where either J = A1 or J = A2. This

induces the following embedding χ∗ : Ht
p
(M ∨M,{s0,q ×s0,q : q = 1, ...,k}; W,y0) →֒ H

t

p
(M,{s0,q :

q = 1, ...,k}; W,y0).

Analogously considering Ht
p
(M,{X0,q : q = 1, ...,k}; W,y0) = {f ∈ H

t
(M,W) : f(X0,q) =

{y0},q = 1, ...,k} and H
t

p
(J,A3 ∪ {X0,q : q = 1, ...,k}; W,y0) in the case (iv

′
) instead of (iv)

we get the embedding χ∗ : Ht
p
(M ∨ M,{X0,q × X0,q : q = 1, ...,k}; W,y0) →֒ H

t

p
(M,{X0,q : q =

1, ...,k}; W,y0). Therefore, g◦f := χ
∗
(f∨g) is the composition in Ht

p
(M,{s0,q : q = 1, ...,k}; W,y0).

There exists the following equivalence relation Rt,H in H
t

p
(M,{X0,q : q = 1, ...,k}; W,y0):

fRt,Hh if and only if there exist nets ηn ∈ DifH
t

p
(M,{X0,q : q = 1, ...,k}), also fn and hn ∈

Ht
p
(M,{X0,q : q = 1, ...,k}; W,y0) with limn fn = f and limn hn = h such that fn(x) = hn(ηn(x))

for each x ∈ M and n ∈ ω, where ω is a directed set and convergence is considered in Ht
p
(M,{X0,q :

q = 1, ...,k}; W,y0). Henceforward in the case 2(iv) we get s0,q instead of X0,q in the case 2(iv
′
).

Thus there exists the quotient uniform space

Ht
p
(M,{X0,q : q = 1, ...,k}; W,y0)/Rt,H =: (S

ME)t,H. In view of [30, 31] DifH
t

p
(M) is the group

of diffeomorphisms for t ≥ [m/2] + 1. The Lebesgue measure λ in the real shadow of M̂ by the

mapping Ξ induces the measure λΞ on M which is equivalent to ν, since Ξ is the Ht
p
-mapping from

the compact space onto the compact space, λ(∂M̂) = 0 and Ξ : M̂ \ ∂M̂ → M is bijective.

Due to Conditions (P1 − P5) each element f = Pγ̂,u up to a set QM of measure zero,

ν(QM) = 0, is given as f ◦ Ξ
−1 on M \ QM, where π ◦ f = γ̂, γ̂ = γ ◦ Ξ. Denote f ◦ Ξ

−1 also

by f. Thus, for each (E,f) ∈ Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0) the image f(M) is compact and

connected in E.

Therefore, for each partition Z there exists δ > 0 such that for each partition Z∗ with

sup
i
infj dist(Mi,M

∗
j) < δ and (E,f) ∈ H

t
(M,W ; Z), f(s0,q) = uq, there exists (E,f1) ∈

Ht(M,W ; Z∗) with f1(s0,q) = uq for each q = 1, ...,k such that fRt,Hf1, where Mi and M
∗
j

are

canonical closed pseudo-submanifolds in M corresponding to partitions Z and Z∗, Ht(M,W ; Z)

denotes the space of all continuous piecewise Ht-mappings from M into W subordinated to the



CUBO
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Wrap groups of fiber bundles and their structure 109

partition Z such that Z and Z∗ respect Ht
p

structure of M.

Hence there exists a countable subfamily {Zj : j ∈ N} in the family of all partitions Υ such

that Zj ⊂ Zj+1 for each j and limj d̃iamZj = 0. Then

(i) str − ind{Ht(M,{s0,q : q = 1, ...,k}; W,y0; Zj); h
Zi

Zj
; N}/Rt,H = (S

ME)t,H is separable if

N and G are separable, since each space Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0; Zj) is separable.

The space str−ind{Ht(M,{s0,q : q = 1, ...,k}; W,y0; Zj); h
Zi

Zj
; N} is complete due to Theorem

12.1.4 [29], when N and G are complete. Each class of Rt,H-equivalent elements is closed in it. Then

to each Cauchy net in (SME)t,H there corresponds a Cauchy net in str−ind{H
t
(M ×[0, 1],{s0,q ×

e× 0; W,y0; Zj ×Yj); h
Zi×Yi
Zj ×Yj ; N} due to theorems about extensions of functions [25, 33, 38], where

Yj are partitions of [0, 1] with limj d̃iam(Yj) = 0, Zj × Yj are the corresponding partitions of

M × [0, 1]. Hence (SME)t,H is complete, if N and G are complete.

If f,g ∈ Ht(M,X) and f(M) 6= g(M), then

(ii) infψ∈DifHtp(M,{s0,q :q=1,...,k}) ‖f ◦ ψ − g‖Ht(M,X) > 0. Thus equivalence classes < f >t,H

and < g >t,H are different. The pseudo-manifold M̂ is arcwise connected. Take η : [0, 1] → M̂ an

Ht
p
-mapping with η(0) = ŝ0,q and η(1) = ŝ0,k+q, where 1 ≤ q ≤ k. Choose in M̂ H

t

p
-coordinates

one of which is a parameter along η. Therefore, for each gq,gk+q ∈ G there exists Pγ̂,u with

Pγ̂,u(s0,q) = y0 × gq and Pγ̂,u(s0,k+q) = y0 × gk+q for each q = 1, ...,k. Since E and G are arcwise

connected, then N is arcwise connected and (SME)t,H is locally connected for dimRN > 1. Thus,

the uniform space (SME)t,H is non-discrete.

The tangent bundle THt
p
(M,E) is isomorphic with Ht

p
(M,TE), where TE is the Ht

′−1
p

fiber

bundle, t′ ≥ t + 1. There is an infinite family of fα ∈ H
t

p
(M,TE) with pairwise distinct images in

TE for different α such that fα(M) is not contained in
⋃
β<α

fβ(M), α ∈ Λ, where Λ is an infinite

ordinal. Therefore, T(SME)t,H is an infinite dimensional fiber bundle due to (ii) and inevitably

(SME)t,H is infinite dimensional.

Evidently, if f ∨ g = h ∨ g or g ∨ f = g ∨ h for {f,g,h} ⊂ Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0),

then f = h. Thus χ∗(f ∨ g) = χ∗(h ∨ g) or χ∗(g ∨ f) = χ∗(g ∨ h) is equivalent to f = h

due to the definition of f ∨ g and the definition of equal functions, since χ∗ is the embedding.

Using the equivalence relation Rt,H gives < f >t,H ◦ < g >t,H=< h >t,H ◦ < g >t,H or

< g >t,H ◦ < f >t,H=< g >t,H ◦ < h >t,H is equivalent to < h >t,H=< f >t,H. Therefore,

(SME)t,H has the cancelation property.

Since G is alternative, then a2,q[a
−1
2,q

(a2,q+k(a
−1
2,q
a1,q+k))]a2,q+k(a

−1
2,q
a1,q+k), hence P1 ∨ (P2 ∨

P2) = (P1 ∨P2)∨P2; also a2,q[a
−1
2,q

(a1,q+k(a
−1
1,q
a1,q+k))]a1,q+k(a

−1
1,q
a1,q+k), consequently, P1 ∨(P1 ∨

P2) = (P1 ∨ P1) ∨ P2 and inevitably for equivalence classes (aa)b = a(ab) and b(aa) = (ba)a for

each a,b ∈ (SME)t,H. Thus (S
ME)t,H is alternative.

If G is associative, then the parallel transport structure gives (f ∨ g) ∨ h = f ∨ (g ∨ h) on

M ∨ M ∨ M for each {f,g,h} ⊂ Ht
p
(M,{s0,q : q = 1, ...,k; W,y0). Applying the embedding χ

∗ and

the equivalence relation Rt,H we get, that (S
ME)t,H is associative < f >ξ ◦(< g >ξ ◦ < h >ξ) =



110 S.V. Ludkovsky CUBO
11, 1 (2009)

(< f >ξ ◦ < g >ξ)◦ < h >ξ.

In view of Conditions 2(i − iv) there exists an Ht
p
-diffeomoprhism of (A1 \ A3) ∨ (A2 \ A3)

with (A2 \A3) ∨ (A1 \A3) as pseudo-manifolds (see §1.3.1). For the measure ν on M naturally the

equality ν(A3) = 0 is satisfied. If M
′ - is the submanifold may be with corners or pseudo-manifold,

accomplishing the partition Z = Zf of the manifold M, then the codimension M
′ in M is equal

to one and ν(M′) = 0. For the point s0,q in (M \ A3) ∪ {s0,q} there exists an open neighborhood

U having the Ht
p
-retraction F : [0, 1] × U → {s0,q}. Hence it is possible to take a sequence of

diffeomorphisms ψn ∈ DifH
t

p
(M,{s0,q : q = 1, ...,k}) such that limn→∞ diam(ψn(U)) = 0.

Let w0 be a mapping w0 : M → W such that w0(M) = {y0 ×e}. Consider w0 ∨(E,f) for some

(E,f) ∈ Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0). If (E,f) ∈ H

t

p
(M,{s0,q : q = 1, ..,k}; W,y0) with the

natural positive t ∈ N, then f is bounded relative to the uniformity of the uniform space Ht
p
(M; E).

If Un is a sequence of bounded open or canonical closed subsets in M such that limn diam(Un) = 0,

then limn→∞ ν(Vn) = 0 for the sequence of ν-measurable subsets Vn such that Vn ⊂ Un. Therefore,

for each bounded sequence {gn : gn ∈ H
t

p
(M; E); n ∈ N} there exists the limit limn→∞ gn|Un = 0

relative to the Ht
p

uniformity, where Un is subordinated to the partition of M into H
t submanifolds.

Then if {gn : gn ∈ H
t

p
(M,{s0,q : q = 1, ...,k}; E,y0); n ∈ N} is a bounded sequence such that gn

converges to g ∈ Ht
p
(M,{s0,q : q = 1, ...,k}; N,y0) on M \ Wk for each k relative to the H

t

p
-

uniformity, the given open Wk in M, where k,n ∈ N and limn→∞ ν(Wn △ Un) = 0, then gn
converges to g in the uniform space Ht

p
(M,{s0,q : q = 1, ...,k}; E,y0).

Mention that for each marked point s0,q in M there exists a neighborhood U of s0,q in M

such that for each γ1 ∈ H
t

p
(M,{s0,q : q = 1, ...,k}; N,y0) there exists γ2 ∈ H

t

p
such that they are

Rt,H equivalent and γ2|U = y0. Therefore, if C is an arcwise connected compact subset in M of

codimension codimRC ≥ 1 such that s0,q ∈ C, then the standard proceeding shows that for each

γ1 ∈ H
t

p
there exists γ2 ∈ H

t

p
such that γ1Rt,Hγ2 and γ2|C = y0. Since C is compact, then each

its open covering has a finite subcovering and hence

(Y0) there exists an open neighborhood U of C in M such that for each γ1 there exists γ2 such

that γ1Rt,Hγ2 and γ2|U = y0.

There exists a sequence ηn ∈ DifH
t

p
(M,{s0,q : q = 1, ...,k}) such that limn→∞ diam(ηn(A2 \

∂A2)) = 0 and wn,fn ∈ H
t

p
(M,{s0,q : q = 1, ...,k}; E,y0) with

(iii) limn→∞ fn = f, limn→∞ wn = w0 and limn→∞ χ
∗
(fn ∨wn)(η

−1
n

) = f due to π◦f(s0,q) =

s0,q in the formula of differentiation of compositions of functions (over H and O see it in [19, 20, 17]).

In more details, the sequence ηn as a limit of ηn(A2) produces a pseudo-submanifold B in M of

codimension not less than one such that B can be presented with the help of the wedge product of

spheres and compact quadrants up to Ht
p
-diffeomorphism with marked points {s0,q : q = 1, ...,k},

but as well B may be a finite discrete set also. Then by induction the procedure can be continued

lowering the dimension of B. Particularly there may be circles and curves in the case of the

unit dimension. Two quadrants up to an Ht
p

quotient mapping gluing boundaries produce a

sphere. Thus the consideration reduces to the case of the wedge product of spheres. The case



CUBO
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Wrap groups of fiber bundles and their structure 111

of spheres reduces to the iterated construction with circles, since the reduced product S1 ∧ Sn

is Ht
p

homeomorphic with Sn+1 (see Lemma 2.27 [37] and [4]). For the particular case of the

n-dimensional sphere Mn = S
n take M̂n = D

n, where Dn is the unit ball (disk) in Rn or in a n

dimensional over R subspace in Al
r
, D1 = [0, 1] for n = 1. But S

n \ s0 has the retraction into the

point in Sn, where s0 ∈ S
n, n ∈ N.

Therefore, w0 ∨ (E,f) and (E,f) belong to the equivalence class < (E,f) >t,H: {g ∈

Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0) : (E,f)Rt,Hg} due to (iii) and (Y0). Thus, < w0 >t,H ◦ <

g >t,H=< g >t,H.

The pseudo-manifold M ∨ M \ {s0,q × s0,j : q,j = 1, ...,k} has the H
t

p
-diffeomorphism ψ (see

definition in §1.3.1) such that ψ(x,y) = (y,x) for each (x,y) ∈ (M × M \ {s0,q × s0,j : q,j =

1, ...,k}). Suppose now, that G is commutative. Then (f ∨ g) ◦ ψ|(M×M\{s0,q×s0,j :q,j=1,...,k}) =

g ∨ f|(M×M\{s0,q×s0,j :q,j=1,...,k}). On the other hand, < f ∨ w0 >t,H=< f >t,H=< f >t,H ◦ <

w0 >t,H=< w0 >t,H ◦ < f >t,H, hence, < f ∨ g >t,H=< f >t,H ◦ < g >t,H=< f ∨ w0 >t,H

◦ < w0 ∨ g >t,H=< (f ∨ w0) ∨ (w0 ∨ g) >t,H=< (w0 ∨ g) ∨ (f ∨ w0) >t,H due to the existence of

the unit element < w0 >t,H and due to the properties of ψ. Indeed, take a sequence ψn as above.

Therefore, the parallel transport structure gives (g ∨ f)(ψ(x,y)) = (g ◦ f)(y,x) for each x,y ∈ M,

consequently, (f ◦ g)Rt,H(g ◦ f) for each f,g ∈ H
t

p
(M,{s0,q : q = 1, ...,k}; W,y0). The using of the

embedding χ∗ gives that (SME)t,H is commutative, when G is commutative.

The mapping (f,g) 7→ f ∨ g from Ht
p
(M,{s0,q : q = 1, ...,k}; W,y0)

2 into Ht
p
(M ∨ M \

{s0,q × s0,j : q,j = 1, ...,k}; W,y0) is of class H
t

p
. Since the mapping χ∗ is of class Ht

p
, then

(f,g) 7→ χ∗(f∨g) is the Ht
p
-mapping. The quotient mapping from Ht

p
(M,{s0,q : q = 1, ...,k}; W,y0)

into (SME)t,H is continuous and induces the quotient uniformity, T
b
(SME)t,H has embedding into

(SMTbE)t,H for each 1 ≤ b ≤ t
′ − t, when t′ > t is finite, for every 1 ≤ b < ∞ if t′ = ∞, since E

is the Ht
′

p
fiber bundle, TbE is the fiber bundle with the base space N. Hence the multiplication

(< f >t,H,< g >t,H>) 7→< f >t,H ◦ < g >t,H=< f ∨ g >t,H is continuous in (S
ME)t,H and is of

class Hl
p

with l = t′ − t for finite t′ and l = ∞ for t′ = ∞.

4. Definition. The (SME)t,H from Theorem 3.1 we call the wrap monoid.

5. Corollary. Let φ : M1 → M2 be a surjective H
t

p
-mapping of Ht

p
-pseudo-manifolds over

the same Ar such that φ(s1,0,q) = s2,0,a(q) for each q = 1, ...,k1, where {sj,0,q : q = 1, ...,kj} are

marked points in Mj, j = 1, 2, 1 ≤ a ≤ k2, l1 ≤ k2, l1 : card φ({s1,0,q : q = 1, ...,k1}). Then there

exists an induced homomorphism of monoids φ∗ : (SM2E)t,H → (S
M1E)t,H. If l1 = k2, then φ

∗ is

the embedding.

Proof. Take Ξ1 : M̂1 → M1 with marked points {ŝ1,0,q : q = 1, ..., 2k1} as in §2, then

take M̂2 the same M̂1 with additional 2(k2 − l1) marked points {ŝ2,0,q : q = 1, ..., 2k3} such that

ŝ1,0,q = ŝ2,0,q for each q = 1, ..,k1, k3 = k1 + k2 − l1, then φ ◦ Ξ1 := Ξ2 : M̂2 → M2 is the

desired mapping inducing the parallel transport structure from that of M1. Therefore, each γ̂2 :

M̂2 → N induces γ̂1 : M̂1 → N and to Pγ̂2,u2 there corresponds Pγ̂1,u1 with additional conditions

in extra marked points, where u1 ⊂ u2. The equivalence class < (E2, Pγ̂2,u2 ) >t,H∈ (S
M2E)t,H



112 S.V. Ludkovsky CUBO
11, 1 (2009)

gives the corresponding elements < (E1, Pγ̂1,u1 ) >t,H∈ (S
M1E)t,H, since DifH

t

p
(M̂1,{ŝ0,q : q =

1, ..., 2k2}) ⊂ DifH
t

p
(M̂1,{ŝ0,q : q = 1, ..., 2k3}). Then φ

∗
(< (E2, Pγ̂2,u2 ) ∨ (E1, Pη̂2,v2 ) >t,H

) = φ∗(< (E2, Pγ̂2,u2 ) >t,H)φ
∗
(< (E1, Pη̂2,v2 ) >t,H), since f2 ◦ φ(x) for each x ∈ Ξ1(M̂1 \ ∂M̂1)

coincides with f1(x), where fj corresponds to Pγj,y0×e (see also the beginning of §3).

If l1 = k2, then M̂1 = M̂2 and the group of diffeomorphisms DifH
t

p
(M̂1,{ŝ0,q : q = 1, ..., 2k1})

is the same for two cases, hence φ∗ is bijective and inevitably φ∗ is the embedding.

6. Theorems. 1. There exists an alternative topological group (WME)t,H containing the

monoid (SME)t,H and the group operation of H
l

p
class with l = t′ − t (l = ∞ for t′ = ∞). If N

and G are separable, then (WME)t,H is separable. If N and G are complete, then (W
ME)t,H is

complete.

2. If G is associative, then (WME)t,H is associative. If G is commutative, then (W
ME)t,H

is commutative. If G is a Lie group, then (WME)t,H is a Lie group.

3. The (WME)t,H is non-discrete, locally connected and infinite dimensional for dimR(N ×

G) > 1. Moreover, if there exist two different sets of marked points s0,q,j in A3, q = 1, ...,k,

j = 1, 2, then two groups (WME)t,H,j, defined for {s0,q,j : q = 1, ...,k} as marked points, are

isomorphic.

4. The (WME)t,H has a structure of an H
t

p
-differentiable manifold over Ar.

Proof. If γ ∈ Ht
p
(M,{s0,q : q = 1, ...,k}; N,y0), then for u ∈ Ey0 there exists a unique

hq ∈ G such that Pγ̂,u(ŝ0,q+k) = uqhq, where hq = g
−1
q
gq+k, y0 × gq = Pγ̂,u(ŝ0,q), gq ∈ G. Due

to the equivariance of the parallel transport structure h depends on γ only and we denote it by

h(E,P)(γ) = h(γ) = h, h = (h1, ...,hk). The element h(γ) is called the holonomy of P along γ and

h(E,P)(γ) depends only on the isomorphism class of (E, P) due to the use of DifHt
p
(M̂; {ŝ0,q : q =

1, ..., 2k}) and boundary conditions on γ̂ at ŝ0,q for q = 1, ..., 2k.

Therefore, h(E1,P1)(E2,P2)(γ) = h(E1,P1)(γ)h(E2,P2)(γ) ∈ Gk, where Gk denotes the direct

product of k copies of the group G. Hence for each such γ there exists the homomorphism h(γ) :

(SME)t,H → G
k, which induces the homomorphism h : (SME)t,H → C

0
(Ht

p
(M,{s0,q : q =

1, ...,k}; N,y0),G
k
), where C0(A,Gk) is the space of continuous maps from a topological space A

into Gk and the group structure (hb)(γ) = h(γ)b(γ) (see also [4] for Sn).

Thus, it is sufficient to construct (WMN)t,H from (S
MN)t,H. For the commutative monoid

(SMN)t,H with the unit and the cancelation property there exists a commutative group (W
MN)t,H.

Algebraically it is the quotient group F/B, where F is the free commutative group generated by

(SMN)t,H, while B is the minimal closed subgroup in F generated by all elements of the form

[f + g] − [f] − [g], f and g ∈ (SMN)t,H, [f] denotes the element in F corresponding to f (see also

about such abstract Grothendieck construction in [13, 36]).

By the construction each point in (SMN)t,H is the closed subset, hence (S
MN)t,H is the

topological T1-space. In view of Theorem 2.3.11 [7] the product of T1-spaces is the T1-space. On

the other hand, for the topological group G from the separation axiom T1 it follows, that G is the



CUBO
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Wrap groups of fiber bundles and their structure 113

Tychonoff space [7, 32]. The natural mapping η : (SMN)t,H → (W
MN)t,H is injective. We supply

F with the topology inherited from the topology of the Tychonoff product (SMN)Z
t,H

, where each

element z in F has the form z =
∑
f
nf,z[f], nf,z ∈ Z for each f ∈ (S

MN)t,H,
∑
f
|nf,z| < ∞. By

the construction F and F/B are T1-spaces, consequently, F/B is the Tychonoff space. In particular,

[nf] − n[f] ∈ B, hence (WMN)t,H is the complete topological group, if N and G are complete,

while η is the topological embedding, since η(f + g) = η(f) + η(g) for each f,g ∈ (SMN)t,H,

η(e) = e, since (z + B) ∈ η(SMN)t,H, when nf,z ≥ 0 for each f, and inevitably in the general case

z = z+ − z−, where (z+ + B) and (z− + B) ∈ η(SMN)t,H.

Using plots and Ht
′

p
transition mappings of charts of N and E(N,G,π, Ψ) and equivalence

classes relative to DifHt
p
(M,{s0,q : q = 1, ...,k}) we get, that (W

ME)t,H has the structure of the

Ht
p
-differentiable manifold, since t′ ≥ t.

The rest of the proof and the statements of Theorems 6(1-4) follows from this and Theorems

3(1-3) and [21, 22].

7. Definition. The (WME)t,H = (W
M,{s0,q :q=1,...,k}E; N,G, P)t,H from Theorem 6.1 we

call the wrap group.

8. Corollary. There exists the group homomorphism h : (WME)t,H → C
0
(Ht

p
(M,{s0,q : q =

1, ...,k}; N,y0),G
k
).

Proof follows from §6 and putting hf
−1

(γ)(hf (γ))−1.

9. Corollary. If M1 and M2 and φ satisfy conditions of Corollary 5, then there exists a

homomorphism φ∗ : (WM2E)t,H → (W
M1E)t,H. If l1 = k2, then φ

∗ is the embedding.

10. Remarks and examples. Consider examples of M which satisfy sufficient conditions

for the existence of wrap groups (WME)t,H. Take M, for example, D
n

R
, Sn

R
\ V with s0 ∈ ∂V ,

Dn
R
\ Int(Dn

b
) with s0 ∈ ∂D

n

b
and 0 < b < R < ∞, where Sn

R
denotes the sphere of the dimension

n > 1 over R and radius R, V is Ht
p
-diffeomorphic with the interior Int(Dn

R
) of the n-dimensional

ball Dn
R

:= {x ∈ Rn :
∑
n

k=1
x2
k
≤ R} or in n dimensional over R subspace in Al

r
and is the

proper subset in Sn
R

:= {x ∈ Rn+1 :
∑
n+1

k=1
x2
k

= R}. Instead of sphere it is possible to take an Ht
p

pseudo-manifold Qn homeomorphic with a sphere or a disk, particularly, Milnor’s sphere. Indeed,

divide M by the equator {x1 = 0} into two parts A1 and A2 and take A3 = {x ∈ M : x1 = 0} ∪P ,

where s0 ∈ ∂A1 ∩ ∂A2, while P = ∅, P = ∂V , P = ∂D
n

b
correspondingly. Then take also V and

Dn
b

such that their equators would be generated by the equator {x1 = 0} in S
n

R
or Dn

R
respectively

or more generally Qn.

Take then M = Qn\
⋃
l

k=1
Vk, where Vk are H

t

p
-diffeomorphic to interiors of bounded quadrants

in Rn or in n dimensional subspace in Aa
r
, where l > 1, l ∈ N, ∂Vk ∩ ∂Vj = {s0} and Vk ∩ Vj = ∅

for each k 6= j, diam(Vk) ≤ b < R/3. In more details it is possible make a specification such that

if l is even, then [l/2] − 1 among Vk are displayed above the equator and the same amount below

it, two of Vk have equators, generated by equators {x1 = 0} in Q
n. If l odd, then [(l− 1)/2] among

Vk are displayed above and the same amount below it, one of Vk has equator generated by that of

{x1 = 0} in Q
n, s0 ∈

⋂
k
∂Vk ∩ {x ∈ M : x1 = 0}.



114 S.V. Ludkovsky CUBO
11, 1 (2009)

Divide M by the equator {x1 = 0} into two parts A1 and A2 and let A3 = {x ∈ M : x1 = 0}∪P ,

where P =
⋃
l

k=1
∂Vk. Then either A1 \ A3 and A2 \ A3 are H

t

p
diffeomorphic as pseudo-manifolds

or manifolds with corners and Ht
p

diffeomorphic with M \ [{s0} ∪ (A3 \ Int(∂A1 ∩ ∂A2))] =: D or

2(iv′) is satisfied, since the latter topological space D is obtained from Qn by cutting a non-void

connected closed subset, n > 1, consequently, D is retractable into a point.

In a case of a usual manifold M the point s0 ∈ ∂M (for ∂M 6= ∅) may be a critical point,

but in the case of a manifold with corners this s0 is the corner point from ∂M, since for x ∈ ∂M

there is not less than one chart (U,u,Q) such that u(x) ∈ ∂Q, M \ ∂M =
⋃
k
u
−1
k

(Int(Qk)),

∂M ⊂
⋃
k
u
−1
k

(∂Qk). Further, if M satisfies Conditions 2(i−v) or (i−iii, iv
′,v), then M ×Dm

R
=

P also satisfies them for m ≥ 1, since Dm
R

is retractable into the point, taking as two parts

Aj(K) = Aj(M) × D
m

R
of P , where j = 1, 2, Aj(M) are pseudo-submanifolds of M. Then

A1(P) ∩ A2(P) = (A1(M) ∩ A2(M)) × D
m

R
and it is possible to take A3(P) = A3(M) × D

m

R
,

s0(P) ∈ s0(M) × {x ∈ D
m

R
: x1 = 0}. In particular, for M = S

1 and m = 1 this gives the filled

torus.

This construction can be naturally generalized for non-orientable manifolds, for example, the

Möbius band L, also for M := L \ (
⋃
β

j=1
Vj) with the diameter bj of Vj less than the width

of L, where each Vj is H
t

p
diffeomorphic with an interior of a bounded quadrant in R2, s0,q ∈

∂L ∩ (
⋂
a1+...+aq

j=a1+...+aq−1+1
∂Vj), a0 := 0, a1 + ... + ak = β, q = 1, ...,k, since ∂L is diffeomorphic

with S1, also S1 \ {s0,q} is retractable into a point, consequently, A1 and A2 are retractable into

a point. For L take M̂ = I2, then take a connected curve η̂ consisting of the left side {0} × [0, 1]

joined by a straight line segment joining points {0, 1} and {1, 0} and then joined by the right side

{1} × [0, 1]. This gives the proper cutting of M̂ which induces the proper cutting of L and of M

with A3 ⊃ η ∪ ∂L up to an H
t

p
diffeomorphism, where η := Ξ(η̂), hence the Möbius band L and

M satisfy Conditions 2(i − iii, iv′,v).

Take a quotient mapping φ : I2 → S1 such that φ({s0,1,s0,2}) = s0 ∈ S
1, s0,1 = (0, 0),

s0,2 = (0, 1) ∈ I
2, where I = [0, 1], hence there exists the embedding φ∗ : (WS

1
,s0E)t,H →֒

(WI
2
,{s0,1,s0,2}E)t,H.

The Klein bottle K has M̂ = I2 with twisting equivalence relation on ∂I2 so it satisfies

sufficient conditions. Moreover, K is the quotient φ : Z → K of the cylinder Z with twisted

equivalence relation of its ends S1 using reflection relative to a horizontal diameter. Thus A3 ⊃

φ(S1). Therefore, there exists the embedding φ∗ : (WK,{s0}E)t,H → (W
Z,{s0,1,s0,2}E)t,H, where

s0,1,s0,2 ∈ ∂Z, φ({s0,1,s0,2}) = s0.

Take a pseudo-manifold Qn Ht
p
-diffeomorphic with Sn for n ≥ 2, cut from it β non-intersecting

open domains V1, ...,Vβ H
t

p
-diffeomorphic with interiors of bounded quadrants in Rn, s0,q ∈⋂

a1+...+aq

j=a1+...+aq−1+1
∂Vj, a0 := 0, a1 + ... + ak = β, q = 1, ...,k. Then glue for V1, ...,Vl, 1 ≤ l ≤ β,

by boundaries of slits Ht
p
-diffeomorphic with Sm−1 the reduced product L ∨ Sn−2, since ∂L = S1,

S1 ∧ Sn−2 is Ht
p
-diffeomorphic with Sn−1 [37]. We get the non-orientable Ht

p
-pseudo-manifold M,

satisfying sufficient conditions.



CUBO
11, 1 (2009)

Wrap groups of fiber bundles and their structure 115

Since the projective space RPn is obtained from the sphere by identifying diametrically op-

posite points. Then take M Ht
p
-diffeomorphic with RPn for n > 1 also M with cut V1, ...,Vβ

Ht
p
-diffeomorphic with open subsets in RPn, s0,q ∈ (

⋂
a1+...+aq

j=a1+...+aq−1+1
∂Vj) ∩ {x ∈ M : x1 = 0},

Vj ∩ Vl = ∅ for each j 6= l, a0 := 0, a1 + ... + ak = β, q = 1, ...,k. Then Conditions 2(i − v) or

(i − iii, iv′,v) are also satisfied for RPn and M.

In view of Proposition 2.14 [37] about H-groups [X,x0; K,k0] there is not any expectation or

need on rigorous conditions on a class of acceptable M for constructions of wrap groups (WME)t,H.

If M1 is an analytic real manifold, then taking its graded product with generators {i0, ..., i2r−1}

of the Cayley-Dickson algebra gives the Ar manifold (see [19, 17, 18]). Particularly this gives l2
r

dimensional torus in Al
r

for the l dimensional real torus T2 = (S
1
)
l as M1.

Consider T2. It can be slit along a closed curve (loop) C H
∞
p

-diffeomorphic with S1 and

marked points s0,q ∈ C ⊂ T2 such that C rotates on the surface of T2 = S
1
R
× S1

b
on angle π

around S1
b

while C rotates on 2π around S1
R
, such that C rotates on 4π around S1

R
that return to

the initial point on C, where 0 < b < R < ∞, q = 1, ...,k, k ∈ N. Therefore, the slit along C of

T2 is the non-orientable band which inevitably is the Möbius band with twice larger number of

marked points {sL
0,j

: j = 1, ..., 2k} ⊂ ∂L.

Therefore, for M = T2 as M̂ take a quadrant in R
2 with 2k pairwise opposite marked

points ŝ0,q and ŝ0,q+k on the boundary of M̂, q = 1, ...,k, k ∈ N. Suitable gluing of boundary

points in ∂M̂ gives the mapping Ξ : M̂ → T2, Ξ(ŝ0,q) = Ξ(ŝ0,q+k) = s0,q, q = 1, ...,k. Proper

cutting of M̂ into Âj, j = 1, 2, or of L induces that of T2. Thus we get a pseudo-submanifold

A3(T2) =: A3 ⊃ C, while A1 and A2 are retractable into a marked point s0,q ∈ C for each q,

hence T2 satisfies Conditions 2(i − iii, iv
′,v). In view of Corollary 9 there exists the embedding

φ∗ : (WT2,{s0,q :q=1,...,k}E)t,H → (W
L,{sL0,q :q=1,...,2k}E)t,H, where φ : L → T2 is the quotient

mapping with φ({sL
0,q
,sL

0,q+k
}){s0,q}, q = 1, ...,k.

For the n-dimensional torus Tn in A
a

r
with n > 2 take a n−1-dimensional surface B such that

each its projection into T2 is H
t

p
-diffeomorphic with C for a loop C as above. Therefore, the slit

along B up to a Ht
p
-diffeomorphism gives M0 := L × I

n−2 for even n or M0 := S
1 × In−1 for odd

n, where I = [0, 1]. Since Im is retractable into a point, where m ≥ 1. Thus we lightly get for Tn

a pseudo-submanifold A3 ⊃ B and two A1 and A2 retractable into points and satisfying sufficient

Conditions 2(i − iii, iv′,v), where M̂ = In up to a Ht
p
-diffeomorphism, s0,q ∈ B ⊂ A3 := A3(Tn),

{sM0
0,q
,s
M0

0,q+k
} ⊂ ∂M0, q = 1, ...,k, k ∈ N. Proper cutting of M̂ into Âj, j = 1, 2, induces that of

Tn. Thus there exists an H
t

p
quotient mapping φ : M0 → Tn with φ({s

M0
0,q
,s
M0

0,q+k
}) = {s0,q} and

the embedding φ∗ : (WTn,{s0,q :q=1,...,k}E)t,H →֒ (W
M0,{s

M0
0,q :q=1,...,2k}E)t,H due to Corollary 9.

More generally cut from Tn open subsets Vj which are H
t

p
diffeomorphic with interiors of

bounded quadrants in Rn embedded into Al
r
, j = 1, ...,β, such that s0,q ∈ B∩(

⋂
a1+...+aq

j=a1+...+aq−1+1
∂Vj),

Vj∩Vi = ∅ for each j 6= i, Vj∩B = ∅ for each j, where B is defined up to an H
t

p
diffeomorhism, a0 :=

0, a1 + ... + ak = β, q = 1, ...,k, that gives the manifold M2. Then from M0 cut analogously corre-

sponding Vj,b, such that s0,q ∈ B ∩ (
⋂
a1+...+aq

j=a1+...+aq−1+1
∂Vj,1), s0,q+k ∈ B ∩ (

⋂
a1+...+aq

j=a1+...+aq−1+1
∂Vj,2),



116 S.V. Ludkovsky CUBO
11, 1 (2009)

Vj,b1 ∩ Vi,b2 = ∅ for each j 6= i or b1 6= b2, a0 := 0, a1 + ... + ak = β, q = 1, ...,k, j = 1, ...,β,

b = 1, 2, that produces the manifold M1. We choose Vj,b such that for the restriction φ : M1 → M2

of the mapping φ there is the equality φ(Vj,1 ∪ Vj,2) = Vj for each j, φ({s
M1
0,q
,s
M1

0,q+k
}) = {s0,q}.

This gives the embedding φ∗ : (WM2,{s0,q :q=1,...,k}E)t,H →֒ (W
M1,{s

M1
0,q :q=1,...,2k}E)t,H.

Another example is M3 obtained from the previous M2 with 2k marked points and 2β cut

out domains Vj, when s0,q is identified with s0,q+k and each ∂Vj is glued with ∂Vj+β for each

j ∈ λq ⊂ {d : a1 + ... + aq−1 + 1 ≤ d ≤ a1 + ... + aq}, q = 1, ...,k, k ∈ N, by an equvalence

relation υ. Such M3 is obtained from the torus Tn,m with m holes instead of one hole in the

standard torus Tn,1 = Tn cutting from it Vj with j ∈ {1, ..., 2β} \ (
⋃
q=1,...,k

λq), where m =

m1 +...+mk, mq := card(λq). For Tn and M2 the surface B is H
t

p
diffeomorphic with (∂L)×In−2

for even n or S1 × In−1 for odd n. Take A3 ⊃ B ∪ (
⋃
j∈λq υ(∂Vj)), it is arcwise connected

and contains all marked points. Therefore, M3 satisfies conditions of §2 and there exists the

embedding υ∗ : (WM3,{s
M3
0,q :q=1,...,k}E)t,H →֒ (W

M2,{s
M2
0,q :q=1,...,2k}E)t,H. This also induces the

embedding (WTn,m,{s
Tn,m
0,q :q=1,...,k}E)t,H →֒ (W

Tn,{sTn0,q :q=1,...,2k−1}E)t,H such that each element

g ∈ (WTn,m,{s
Tn,m
0,q :q=1,...,k}E)t,H can be presented as a product g = (..(g1g2)...gm) of m elements

gj ∈ (W
Tn,{sTn0,q :q=1,...,2k−1}E)t,H, gj =< fj >t,H, supp(π ◦ fj) ⊂ Bj, B1 ∪ ... ∪ Bm = Tn,

Bi ∩ Bj = ∂Bi ∩ ∂Bj for each i 6= j, each Bj is a canonical closed subset in Tn, s0,1 ∈ B1,

s0,2q,s2q+1 ∈ Bd for m1 + ... + m0 + 1 ≤ d ≤ m1 + ... + mq, q = 1, ...,k − 1, where m0 := 0.

Evidently, in the general case for different manifolds M and N wrap groups may be non

isomorphic. For example, as M1 take a sphere S
n of the dimension n > 1, as M2 take M1 \ K,

where K is up to an Ht
p
-diffeomorphism the union of non intersecting interiors Bj of quadrants

of diameters d1, ...,ds much less, than 1, K = B1 ∪ ... ∪ Bl, l ∈ N. Let N be a δ-enlargement for

M2 in R
n+1 relative to the metric of the latter Euclidean space, where 0 < δ < min(d1, ...,dl)/2.

Then the groups (WM1N)t,H and (W
M2N)t,H are not isomorphic. This lightly follows from the

consideration of the element b :=< f >t,H∈ (W
M2N)t,H, where f : M2 → N is the identity

embedding induced by the structure of the δ-enlargement.

Recall, that for orientable closed manifolds A and B of the same dimension m the degree

of the continuous mapping f : A → B is defined as an integer number deg(f) ∈ Z such that

f∗[A] = deg(f)[B], where [A] ∈ Hm(A) or [B] ∈ Hm(B) denotes a generator, defined by the

orientation of A or B respectively [5]. Consider mappings fj : S
n → N such that Vj ⊃ ∂Bj ∩ N,

where Vj is a domain in R
n+1 bounded by the hyper-surface fj(Bj), fj is w0 on each Bi with

i 6= j, while the degree of the mapping fj from S
n onto fj(S

n
) is equal to one. If there would be

an isomorphism θ : (WM2N)t,H → (W
M1N)t,H, then θ(b) would have a non trivial decomposition

into the sum of non canceling non zero additives, which is induced by mappings fj : S
n → N.

Nevertheless, an element b in (WM2N)t,H has not such decomposition.

If two groups G1 and G2 are not isomorphic, then certainly (W
ME; N,G1, P)t,H and

(WME; N,G2, P)t,H are not isomorphic.

The construction of wrap groups can be spread on locally compact non compact M satisfying



CUBO
11, 1 (2009)

Wrap groups of fiber bundles and their structure 117

conditions 2(ii − iv) or (ii, iii, iv′) changing (v) such that M̂ is locally compact non-compact Ht
p
-

domain in Al
r
, its boundary ∂M̂ may happen to be void. For this it is sufficient to restrict the

family of functions to that of with compact supports f : M → W relative to w0 : M → W ,

that is suppw0 (f) := clM{x ∈ M : f(x) 6= y0 × e} is compact, clMA denotes the closure of

a subset A in M. Then classes of equivalent elements are given with the help of closures of

orbits of the group of all Ht
p

diffeomorphisms g with compact supports preserving marked points

DifHt
p,c

(M,{s0,q : q = 1, ...,k}) that is suppid(g) := clM{x ∈ M : g(x) 6= x} are compact, where

id(x) = x for each x ∈ M. Then wrap groups (WME)t,H for manifolds M such as hyperboloid of

one sheet, one sheet of two-sheeted hyperboloid, elliptic hyperboloid, hyperbolic paraboloid and

so on in larger dimensional manifolds over Ar. For non compact locally compact manifolds it is

possible also consider an infinite countable discrete set of marked points or of isolated singularities.

These examples can be naturally generalized for certain knotted manifolds arising from the given

above.

Milnor and Lefshetz have used for M = S1 and G = {e} the diffeomorphism group preserving

an orientation and a marked point of S1. So their loop group L(S1,N) may be non-commutative.

The iterated loop group L(S1,L(Sn−1,N)) is isomorphic with L(Sn,N), where the latter group

is supplied with the uniformity from the iterated loop group, so n times iterated loop group of S1

gives loop group of Sn [4]. For dimRM > 1 orientation preservation loss its significance. Here

above it was used the diffeomorphism group without any demands on orientation preservation of

M such that two copies of M in the wedge product already are not distinguished in equivalence

classes and for commutative G it gives a commutative wrap group.

Mention for comparison homotopy groups. The group πq(X) for a topological space X with

a marked point x0 in view of Proposition 17.1 (b) [2] is commutative for q > 1. For q = 1 the

fundamental group π1(X) may be non-commutative, but it is always commutative in the particular

case, when X = G is an arcwise connected topological group (see §49(G) in [32]).

11. Proposition. Let L(S1,N) be an H1
p

loop group in the classical sense. Then the iterated

loop group L(S1,L(S1,N)) is commutative.

Proof. Consider two elements a,b ∈ L(S1,L(S1,N)) and two mappings f ∈ a, g ∈ b,

(f(x))(y) = f(x,y) ∈ N, where x,y ∈ I = [0, 1] ⊂ R, e2πx ∈ S1. An inverse element d−1 of

d ∈ L(S1,N) is defined as the equivalence class d−1 =< h− >, where h ∈ d, h−(x) : h(1 − x).

Then

(1) f(x, 1 −y) = (f(x))(1 −y) ∈ a−1 and g(x, 1 −y) = (g(x))(1 −y) ∈ b−1 for L(S1,L(S1,N))

and symmetrically

(2) (f(y))(1 − x) = f(1 − x,y) ∈ a−1 and (g(y))(1 − x) = g(1 − x,y) ∈ b−1. On the other

hand, f ∨ g corresponds to ab, and g ∨ f corresponds to ba, where the reduced product S1 ∧ S1

is Ht
p
-diffeomorphic with S2 in the sense of pseudo-manifolds up to critical subsets of codimension

not less than two.

Consider (S1 ∨S1) ∧ (S1 ∨S1) and (f ∨w0) ∨ (w0 ∨g) and (g ∨w0) ∨ (w0 ∨f) and the iterated



118 S.V. Ludkovsky CUBO
11, 1 (2009)

equivalence relation R1,H. This situation corresponds to M̂ = I
2 divided into four quadrats by

segments {1/2} × [0, 1] and [0, 1] × {1/2} with the corresponding domains for f, g and w0 in the

considered wedge products, where < f ∨ w0 >=< w0 ∨ f >=< f > is the same class of equvalent

elements.

Since G = {e}, (ab)−1 = b−1a−1, then g(1−x,y)∨f(1−x,y) is in the same class of equivalent

elements as g(x, 1 − y) ∨ f(x, 1 − y). But due to inclusions (1, 2) < g(1 − x,y) ∨ f(1 − x,y) >=<

f(x,y)∨g(x,y) >−1 and < f(x,y)∨g(x,y) >=< g(x, 1−y)∨f(x, 1−y) >−1 and < h(x,y) >−1=<

h(x, 1 − y) >=< h(1 − x,y) > for h ∈ ab, consequently, < h(x,y) >=< h(1 − x, 1 − y) > and

< (f ∨ g)(x, 1 − y) >< f(x, 1 − y) ∨ g(x, 1 − y) >∈ (ab)−1, since (x,y) 7→ (1 − x, 1 − y) interchange

two spheres in the wedge product S2 ∨ S2. Hence a−1b−1 = b−1a−1 and inevitably ab = ba.

12. Theorem. Let M and N be connected both either C∞ Riemann or Ar holomorphic

manifolds with corners, where M is compact and dimM ≥ 1 and dimN > 1. Then (WMN)t,H
has no any nontrivial continuous local one parameter subgroup gb for b ∈ (−ǫ,ǫ) with ǫ > 0.

Proof. Suppose the contrary, that {gb : b ∈ (−ǫ,ǫ)} with ǫ > 0 is a local nontrivial one

parameter subgroup, that is, gb 6= e for b 6= 0. Then to gδ for a marked 0 < δ < ǫ there corresponds

f = fδ ∈ H
∞
p

such that < f >t,H= g
δ, where f ∈ Ht

p
. If f(U) = {y0 × e} for a sufficiently small

connected open neighborhood U of s0,q in M, then there exists a sequence f ◦ψn in the equivalence

class < f >t,H with a family of diffeomorphisms ψn ∈ DifH
t

p
(M; {s0,q : q = 1, ...,k}) such that

limn→∞ diamψn(U) = 0 and
⋂∞
n=1

ψn(U) = {s0,q}. If h(x) 6= y0, then in view of the continuity

of h there exists an open neighborhood P of x in M such that y0 /∈ h(P). Consider the covariant

differentiation ∇ on the manifold M (see [12]). The set Sh of points, where ∇
kh is discontinuous

is a submanifold of codimension not less than one, hence of measure zero relative to the Riemann

volume element in M. For others points x in M, x ∈ M \ Sh, all ∇
kh are continuous.

Take then open V = V (f) in M such that V ⊃ U and ∇k
ν
f|∂V 6= 0 for some k ∈ N, where

∇νf(x) := limz→x,z∈M\V ∇νf(z), ν is a normal (perpendicular) to ∂V in M at a point x in the

boundary ∂V of V in M. Practically take a minimal k = k(x) with such property. Since M

is compact and ∂V := cl(V ) ∩ cl(M \ V ) is closed in M, then ∂V is compact. The function

x 7→ k(x) ∈ N is continuous, since f and ∇lf for each l are continuous. But N is discrete,

hence each ∂qV := {x ∈ ∂V : k(x) = q} is open in V . Therefore, ∂V is a finite union of ∂qV ,

1 ≤ q ≤ qm, where qm : maxx∈∂V k(x) < ∞ for f = fδ, since ∂V is compact. Thus, there exists

a subset λ ⊂ {1, ...,qm} such that ∂V =
⋃
q∈λ ∂qV and ∂qV 6= ∅ for each q ∈ λ. If ∇

lf(x) = 0

for l = 1, ...,k(x) − 1 and ∇k(x)f(x) 6= 0, then ∇k(x)f(ψ(y)) = ∇k(x)(ψ(y)).(∇ψ(y))⊗k(x) 6= 0 for

y ∈ M such that ψ(y) = x, since ∇ψ(y) 6= 0, where ψ ∈ DifH∞
p

(M; {s0,q : q = 1, ...,k}).

We can take ǫ > 0 such that {gb : b ∈ (−ǫ,ǫ)} ⊂ U, where U = −U is a connected symmetric

open neighborhood of e in (WMN)t,H. Since g
b1 + gb2 = gb1+b2 for each b1,b2,b1 + b2 ∈ (−ǫ,ǫ),

then limt→0 g
b

= e for the local one parameter subgroup and in particular limm→∞ g
1/m

= e,

where m ∈ N. Take δ = δm = 1/m and f = fm ∈ H
∞
p

such that < fm >t,H= g
1/m. On the other

hand, jg1/m = gj/m for each j < mǫ, j ∈ N, hence fj/m(M) = f1/m(M) for each j < mǫ, since

f ∨ h(M ∨ M) = f(M) ∨ h(M) and using embedding η of (SMN)t,H into (W
MN)t,H.



CUBO
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Wrap groups of fiber bundles and their structure 119

The function |∇
k(x)
ν fδ(x)| for x ∈ ∂V is continuous by δ due to the Sobolev embedding

theorem [25], 0 < δ < ǫ, consequently, infx∈∂V |∇
k(x)
ν fδ(x)| > 0, since ∂V is compact. We

can choose a family fδ such that z
(l)

(δ,x) := ∇lfδ(x) is continuous for each 0 ≤ l ≤ k0 by

(δ,x) ∈ (−ǫ,ǫ) × M, since {gb : b ∈ (−ǫ,ǫ)} is the continuous by b one parameter subgroup, where

k0 := qm(δ0). Therefore, for this family there exists a neighborhood [−ǫ + c,ǫ − c] such that

δ0 ∈ [−ǫ+c,ǫ−c] ⊂ (−ǫ,ǫ) with 0 < c < ǫ/3 such that qm(δ) ≤ k0 for each δ ∈ [−ǫ+c,ǫ−c] with a

suitable choice of V (fδ), since N is discrete. On the other hand, supx∈∂V (fδ),0<δ≤ǫ−c |∇
k(x)
ν fδ(x)| ≤

sup
x∈M,0<δ≤ǫ−c |∇

k(x)
ν fδ(x)| =: B < ∞, since M and [−ǫ + c,ǫ − c] are compact.

Therefore, for this family there exists a neighborhood [−ǫ+c,ǫ−c] such that δ0 ∈ [−ǫ+c,ǫ−c] ⊂

(−ǫ,ǫ) with 0 < c < ǫ/3 such that qm(δ) ≤ k0 for each δ ∈ [−ǫ + c,ǫ − c] with a suitable choice of

V (fδ), since N is discrete.

Then lim
δ→0,δ>0|∇

k(x)
ν fδ(x)| =: b > 0 for x ∈ ∂V with a suitable choice of V = V (fδ), since

M is connected, dimM ≥ 1 and infm∈N diamfj/m(M) > 0 for a marked δ0 = j/m0 < ǫ with

j,m > m0 ∈ N mutually prime, (j,m) = 1, (j,m0) = 1. To < fl/m >t,H there corresponds

< f1/m >t,H ∨...∨ < f1/m >t,H=:< f1/m >
∨l
t,H

which is the l-fold wedge product. Thus there

exists C = const > 0 for M such that |∇
k(x)
ν fl/m(x)| ≥ Cl infy∈∂V (f1/m) |∇

k(y)
ν f1/m(y)| ≥ Clb,

where C > 0 is fixed for a chosen atlas At(M) with given transition mappings φi ◦ φ
−1
j

of charts.

Consider δ0 ≤ l/m < ǫ − c and m and l tending to the infinity. Then this gives B ≥ Clb for

each l ∈ N, that is the contradictory inequality, hence (WMN)t,H does not contain any non trivial

local one parameter subgroup.

Received: January 2008. Revised: August 2008.

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