1. Introduction
An H-space is a space
X together with a continuous multiplication
and an identity element
such that
for all
. If, in addition, the multiplication is associative, then
X is called a topological monoid. A space together with an associative continuous multiplication is called a topological semigroup. A compact
n-manifold
S with connected boundary
B together with a topological monoid structure such that
B is a subsemigroup of
S is called an (L)-semigroup in [
1], p. 117. Such a topological monoid
S can be considered as a mapping cylinder
of a quotient morphism
of a compact connected Lie group
X where
N is a normal sphere subgroup of
X (see [
1,
2,
3]).
In [
2], p. 315, it was shown that every commutative
n-dimensional (L)-semigroup is a retract of a compact connected Lie group, and if
, then every
n-dimensional (L)-semigroup is a retract of a compact connected Lie group. In this note, it is shown that the sum of two commutative (L)-semigroups cannot be a retract of a topological group, nor can the sum of two
n-dimensional (L)-semigroups if
.
2. (L)-Semigroup Splitting
Let = denote the unit interval endowed with the operation of multiplication of real numbers. If is a mapping between compact spaces, then the mapping cylinder is the quotient space obtained by taking the disjoint union of and Y and identifying each point with . There are natural embeddings and , so X and Y may be regarded as disjoint closed subspaces of , and it is easy to check that is a strong deformation retract of . In the special case when Y consists of a single point v, the mapping cylinder is called the cone over X, denoted by .
Let
denote the unit
n-sphere in Euclidean
n-space
. Then, in the following result of Mostert and Shields [
1],
(
,
, is homeomorphic to the unit one-ball in the real line
, the unit disk
in the complex plane
, the unit four-ball
in the quaternions
, respectively, and is considered to be a topological monoid with the inherited multiplicative structure.
Proposition 1 (Mostert and Shields [
1]; also see [
2,
3])
. Let X be a compact connected Lie group with a closed normal subgroup N such that N is isomorphic to , and let be the quotient morphism. Then:- (1)
is a compact manifold with boundary with S being a topological monoid such that is the group of units of S with identity and is the minimal ideal of S with identity .
- (2)
is a locally-trivial fibre bundle over the Lie group as base with fibre , the unit n-ball for .
A compact topological monoid
S of the above type is called an (L)-semigroup in the literature and
S is nonorientable if
and orientable if
,
. (Theorem C in [
1]).
Let S and T be two (L)-semigroups with boundary B, and let be an autohomeomorphism of B. The quotient space obtained by taking the union of and and identifying the point in with in for each is a closed (i.e., compact without boundary) connected n-manifold. Any manifold M obtained in this fashion is said to admit an (L)-semigroup splitting. In the case when h is the identity mapping, we call M the sum of S and T and denote it by . If , then , the double of the manifold S.
A space X is said to be homogeneous if for every , there is an autohomeomorphism h of X such that .
Proposition 2. If M admits an (L)-semigroup splitting, then M admits the structure of a topological monoid iff M is a Lie group.
Proof. If
M is a Lie group, then it is a topological monoid. Thus, suppose
M is a topological monoid. A finite-dimensional homogeneous compact connected monoid admits the structure of a topological group [
4]. If, in addition, it is locally contractible, then it must be a Lie group since a compact connected group is a Lie group iff it is locally contractible [
5]. Since
M is a closed connected
n-manifold, the result follows. □
Proposition 3. Let G be a compact connected Lie group. If M admits an (L)-semigroup splitting, then so does . In particular, if M is an (L)-semigroup sum, then so is .
Proof. Let , and be defined as in the definition of an (L)-semigroup splitting. Then, and are (L)-semigroups with as a boundary, and the correspondence determines an autohomeomorphism of . It follows that admits an (L)-semigroup splitting if M does. In the case when , the identity mapping on B, we obtain . □
Remark 1. It is well known that the fundamental group of an H-
space is Abelian and that a covering space of an H-
space admits an H-
space structure (cf. p. 78 and p. 157 in [6]). According to a famous theorem of J.F.Adams [7], the only spheres that are H-
space are , and it follows that , are the only real projective n-spaces, which admit H-
space structures. We also remark that if a product space is homogeneous, then it admits an H-
space structure iff each factor does (Corollary 2.5 in [8]). Proposition 4. Let B be a compact connected Abelian Lie group and let S, and T be (L)-semigroups with boundary B. Then, the sum does not admit an H-space structure.
Proof. Let
denote the
n-torus, which is the product of
n copies of the circle group
. In the case when
=
, the normal sphere subgroups are
and
. For the two element subgroups
of
, the quotient morphism
/
yields
=
, the classical Möbius band (see Example 2.3(b) in [
2]). When the normal subgroup of
is
, the quotient morphism
yields
, the unit disk in the complex plane
(see Example 2.3(a) in [
2]). Thus, the only two-dimensional (L)-semigroup splittings are 2
=
,
+
=
and 2
=
, the Klein bottle. By Remarks 1,
and
do not admit H-spaces structures, nor does
since its fundamental group
(
is not Abelian (this follows from the fact that the Abelianization of
is
⊕
, the direct sum of the integers and a cyclic group of order two (see [
6], p. 135), but
must contain a copy of
⊕
since the two-torus
is a double covering space of the Klein bottle
).
It follows from Proposition 2.3 that
,
, and
,
are (L)-semigroup sums. Since
and
are the only
-dimensional (L)-semigroups with boundary
(see Corollaries 7.5.4 and 7.5.5 in [
1]), it follows that the (L)-semigroup sum
must be one of the manifolds
,
,
,
,
or
for
. However none of these manifolds admit on H-space structure since a homogeneous product space admits an H-space structure iff each of its factors does (see Corollary 2.5 in [
8]). □
We remark that a retract of a homogeneous H-space admits an H-space structure (cf. Proposition 2.4 in [
8]). Consequently, we have the following corollary.
Corollary 1. Let B be a compact connected Abelian Lie group, and let S and T be (L)-semigroups with boundary B. Then, the sum is not a retract of a topological group.
Proposition 5. If M is a manifold that admits an (L)-semigroup splitting and is either two-dimensional or orientable and three-dimensional, then the following statements are equivalent:
- (1)
M is a retract of a topological group.
- (2)
M admits an H-space structure.
- (3)
M is a Lie group.
Proof. In the two-dimensional case, the collection of (L)-semigroup sums coincides with the collection of spaces that admit (L)-semigroup splittings since the connected sum of two surfaces is independent of the homeomorphism h used to form the connected sum. Thus, the only surfaces that admit (L)-semigroup splittings are , , and , and the result follows for surfaces.
The remark following the proof of Proposition 4 shows that (1) implies (2), and since the topological group is a retract of itself, (3) implies (1). Thus, it suffices to show that
. As was noted in the proof of Proposition 4, the only orientable three-dimensional (L)-semigroup is the solid torus
. It follows that
M must be a
-lens space
where the degenerate cases
and
are included (see p. 234 in [
9]). It follows from a theorem of William Browder (p. 140 in [
10]) that only
and
admit H-space structures. Since each of these spaces is a Lie group, the result follows. □
Lemma 1. Let X be a closed n-manifold, which is the total space of a locally-trivial fibre bundle over a compact Lie group G. Then, X does not admit an H-space structure.
Proof. Suppose
X does admit an H-space structure, and consider the fibre bundle
. This sequence extends to a fibration sequence
(cf. [
11], p. 409). Since
X is a (compact metric) ANR-space (see [
12]), it has the homotopy type of a finite complex ([
13], Corollary 44.2), and it follows from a theorem of W.Browder ([
14]) that
, where
denotes the second homotopy group of
X. Exactness yields a surjection from
onto
. An element of
mapping to a generator of
is represented by a map
whose composition with the map
is homotopic to the identity mapping 1
on
. Consequently, there is a homotopy retraction
(i.e.,
is homotopic to
). Since a loop space admits an H-space structure, we may assume that
is an H-space with identity
e, and we may assume that
(since
is a homogeneous space when viewed as a loop group).
Define a mapping
by
for
, where
denotes the product of
x and
y in the H-space
. The maps
given by
and
are homotopic to the identity mapping
, and therefore,
e is a homotopy identity of
. For CW complexes the existence of a homotopy identity can be used as the definition of an H-space (see [
11], p. 291). Consequently,
admits an H-space structure, and this contradiction completes the proof of the lemma. □
Proposition 6. Let be an (L)-semigroup as defined in Proposition 1 where X is a compact connected Lie group and is a quotient morphism with N being a closed normal subgroup of X, which is isomorphic to . Then, the double does not admit an H-space structure.
Proof. By Proposition 1 is a locally-trivial bundle over the compact connected Lie group Y, and it follows that its double is a locally-trivial bundle over Y. Consequently, by Lemma 1, does not admit an H-space structure. □
Corollary 2. Let denote the quotient morphism where is the unitary group, is its centre, and is the projective unitary group. Then, if and , the double does not admit an (H)-space structure.
Proof. The elements of are the complex unitary matrices, and its centre is isomorphic to since its elements are diagonal matrices equal to multiplied by the identity matrix. It follows from Proposition 6 that does not admit an H-space structure. □
Theorem 1. No (L)-semigroup sum of dimension admits an H-space structure.
Proof. Proposition 4 shows that the result is true for all n-dimensional (L)-semigroup sums of the form where both S and L have a compact connected Abelian Lie group boundary B. Thus, we need only consider admissible n-dimensional non-Abelian boundaries B with . Hence, B must be one of , , and (we note that does not qualify as an admissible boundary for an (L)-semigroup since it does not contain normal subgroups of the form , ).
In [
2], it is shown that the (L)-semigroups with boundary
are
and the four-dimensional Möbius manifold
(which is homeomorphic to
with the interior of a four-dimensional Euclidean ball removed). It follows (see [
2]) that the (L)-semigroups with boundaries
,
,
are
,
,
,
,
,
,
,
, and the corresponding (L)-semigroup sums are
,
,
,
,
,
,
,
,
,
,
,
. Since a retract of a homogeneous H-space admits an H-space structure (cf. [
8], Prop. 2.4), it follows that no product containing a copy of
,
,
,
of
as a factor can admit an H-space structure. This leaves only
for consideration. However, its fundamental group
is the free product of
with itself, which is non-Abelian, so
does not admit an H-space structure. Finally, the only five-dimensional (L)-semigroup sum with boundary
is the manifold
in Corollary 2, which does not admit an H-space structure. □
Corollary 3. No (L)-semigroup sum of dimension is a retract of a topological group.
Proof. It was noted above that every retract of a homogeneous H-space admits an H-space structure. Since a topological group is an H-space, the result follows from Theorem 1. □
In [
15], a space homeomorphic to a retract of a topological group is called a GR-space (often referred to as a retral space in the literature). Clearly AR-spaces and topological groups themselves are GR-spaces, and in [
2], it was shown that
and
are GR-spaces. Since GR-spaces are preserved by topological products, it follows that products of
,
,
,
, and topological groups are GR-spaces. This will include all the (L)-semigroups mentioned in this note excluding (L)-semigroups with boundary
,
. This suggests two questions.
- (a)
Is every (L)-semigroup a retract of a topological group?
- (b)
Does every (L)-semigroup sum fail to admit an H-space structure?