( L )-Semigroup Sums †

An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S× {0} and T × {1} by identifying the point (x, 0) in S× {0} with (x, 1) in T × {1} for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group.


Introduction
An H-space is a space X together with a continuous multiplication m : X × X → X and an identity element e ∈ X such that m(e, x) = m(x, e) = x for all x ∈ X. 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 MC( f ) of a quotient morphism f : X → X/N 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 n ≤ 4, 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 n ≤ 5.

(L)-Semigroup Splitting
Let I = [0, 1] denote the unit interval endowed with the operation of multiplication of real numbers.If f : X → Y is a mapping between compact spaces, then the mapping cylinder MC( f ) is the quotient space obtained by taking the disjoint union of X× I and Y and identifying each point (x, 0) ∈ X× I with f (x) ∈ Y.There are natural embeddings i X : X → MC( f ) and i Y : Y → MC( f ), so X and Y may be regarded as disjoint closed subspaces of MC( f ), and it is easy to check that i Y (Y) is a strong deformation retract of MC( f ).In the special case when Y consists of a single point v, the mapping cylinder is called the cone over X, denoted by cone(X).
Let S n denote the unit n-sphere in Euclidean n-space R n .Then, in the following result of Mostert and Shields [1], cone(S n ), n = 0, 1, 3, is homeomorphic to the unit one-ball in the real line R 1 , the unit disk E 2 in the complex plane C, the unit four-ball E 4 in the quaternions H, respectively, and is considered to be a topological monoid with the inherited multiplicative structure.[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 S n , n = 0, 1, 3, and let f : X → X/N = Y be the quotient morphism.Then:

Proposition 1 (Mostert and Shields
(1) S = MC( f ) is a compact manifold with boundary i X (X) with S being a topological monoid such that H(S) = i X (X) is the group of units of S with identity i X (1 X ) and M(S) = i Y (Y) is the minimal ideal of S with identity i Y (1 Y ).
(2) S = MC( f ) is a locally-trivial fibre bundle over the Lie group Y = X/N as base with fibre F = cone(N), the unit n-ball for n = 1, 2, 4.
A compact topological monoid S of the above type is called an (L)-semigroup in the literature and S is nonorientable if N = S 0 and orientable if N = S n , n = 1, 3. C in [1]).
Let S and T be two (L)-semigroups with boundary B, and let h : B → B be an autohomeomorphism of B. The quotient space obtained by taking the union of S × {0} and T × {1} and identifying the point (x, 0) in S × {0} with (h(x), 1) in T × {1} for each x ∈ B 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 S + T. If S = T, then S + S = 2S, the double of the manifold S.
A space X is said to be homogeneous if for every a, b, ∈ X, there is an autohomeomorphism h of X such that h(a) = b.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 M × G.In particular, if M is an (L)-semigroup sum, then so is M × G.
Proof.Let M, S, T, and h : B → B be defined as in the definition of an (L)-semigroup splitting.Then, S × G and T × G are (L)-semigroups with B × G as a boundary, and the correspondence (x, g) → (h(x), g) determines an autohomeomorphism of B × G.It follows that M × G admits an (L)-semigroup splitting if M does.In the case when h = 1 B , 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 S n , n = 0, 1, 3, 7, and it follows that RP n , n = 0, 1, 3, 7, 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 S + T does not admit an H-space structure.
Proof.Let T n denote the n-torus, which is the product of n copies of the circle group S 1 .In the case when B = T 1 = S 1 , the normal sphere subgroups are S 0 and S 1 .For the two element subgroups S 0 of S 1 , the quotient morphism f :S 1 → S 1 / S 0 yields MC( f ) = M 2 , the classical Möbius band (see Example 2.3(b) in [2]).When the normal subgroup of S 1 is S 1 , the quotient morphism f :S 1 → S 1 / S 1 = {1} yields MC( f ) = E 2 , the unit disk in the complex plane C (see Example 2.3(a) in [2]).Thus, the only two-dimensional (L)-semigroup splittings are 2E 2 = S 2 , E 2 + M 2 = RP 2 and 2M 2 = K 2 , the Klein bottle.By Remarks 1, S 2 and RP 2 do not admit H-spaces structures, nor does K 2 since its fundamental group Π 1 (K 2 ) is not Abelian (this follows from the fact that the Abelianization of Π 1 (K 2 ) is Z ⊕ Z 2 , the direct sum of the integers and a cyclic group of order two (see [6], p. 135), but Π 1 (K 2 ) must contain a copy of Π 1 (T 2 ) =Z ⊕ Z since the two-torus T 2 is a double covering space of the Klein bottle K 2 ).
It follows from Proposition 2.3 that S 2 × T n , RP 3 × T n , and L)-semigroup sums.Since E 2 × T n and M 2 × T n are the only (n + 2)-dimensional (L)-semigroups with boundary B =T n+1 (see Corollaries 7.5.4 and 7.5.5 in [1]), it follows that the (L)-semigroup sum S + T must be one of the manifolds S 2 , RP 2 , K 2 , S 2 × T n , RP 2 × T n or K 2 × T n for n = 1, 2, • • • .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 S + T 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.
(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 S 2 , RP 2 , and K 2 , 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 (2) ⇒ (3).As was noted in the proof of Proposition 4, the only orientable three-dimensional (L)-semigroup is the solid torus E 2 × S 1 .It follows that M must be a (p, q)-lens space L(p, q) where the degenerate cases L(0, 1) = S 2 × S 1 and L(1, q) = S 3 are included (see p. 234 in [9]).It follows from a theorem of William Browder (p.140 in [10]) that only L(1, q) = S 3 and L(2, 1) = RP 3 = SO(3) 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 S 2 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 S 2 → X → G.This sequence extends to a fibration sequence • • • ΩG → S 2 → X → G (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 Π 2 (X) = 0, where Π 2 (X) denotes the second homotopy group of X. Exactness yields a surjection from Π 2 (ΩG) onto Π 2 (S 2 ).An element of Π 2 (ΩG) mapping to a generator of Π 2 (S 2 ) is represented by a map S 2 → ΩG whose composition with the map Ω(G) → S 2 is homotopic to the identity mapping 1 S 2 on S 2 .Consequently, there is a homotopy retraction r : ΩG → S 2 (i.e., r| S 2 is homotopic to 1 S 2 ).Since a loop space admits an H-space structure, we may assume that ΩG is an H-space with identity e, and we may assume that e ∈ S 2 (since ΩG is a homogeneous space when viewed as a loop group).
Define a mapping m : S 2 × S 2 → S 2 by m(x, y) = r(xy) for x, y ∈ S 2 , where xy denotes the product of x and y in the H-space ΩG.The maps S 2 → S 2 given by x → m(x, e) and x → m(e, x) are homotopic to the identity mapping 1 S 2 , and therefore, e is a homotopy identity of S 2 .For CW complexes the existence of a homotopy identity can be used as the definition of an H-space (see [11], p. 291).Consequently, S 2 admits an H-space structure, and this contradiction completes the proof of the lemma.Proposition 6.Let S = MC( f ) be an (L)-semigroup as defined in Proposition 1 where X is a compact connected Lie group and f : X → X/N = Y is a quotient morphism with N being a closed normal subgroup of X, which is isomorphic to S 1 .Then, the double 2S does not admit an H-space structure.
Proof.By Proposition 1 S = MC( f ) is a locally-trivial E 2 bundle over the compact connected Lie group Y, and it follows that its double is a locally-trivial S 2 bundle over Y. Consequently, by Lemma 1, 2S does not admit an H-space structure.
Corollary 2. Let f : U(n) → U(n)/ZU(n) = PU(n) denote the quotient morphism where U(n) is the unitary group, Z(U(n)) is its centre, and PU(n) is the projective unitary group.Then, if n > 1 and S = MC( f ), the double 2S does not admit an (H)-space structure.
Proof.The elements of U(n) are the complex n × n unitary matrices, and its centre Z(U(n)) is isomorphic to S 1 since its elements are diagonal matrices equal to e iθ multiplied by the identity matrix.It follows from Proposition 6 that 2S does not admit an H-space structure.Theorem 1.No (L)-semigroup sum of dimension n ≤ 5 admits an H-space structure.
Proof.Proposition 4 shows that the result is true for all n-dimensional (L)-semigroup sums of the form S + L 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 n = 3, 4. Hence, B must be one of S 3 , S 1 × S 3 , S 1 × SO(3) and U(2) (we note that SO(3) does not qualify as an admissible boundary for an (L)-semigroup since it does not contain normal subgroups of the form S n , n = 0, 1, 3).
In [2], it is shown that the (L)-semigroups with boundary S 3 are E 4 and the four-dimensional Möbius manifold M 4 (which is homeomorphic to RP 4 with the interior of a four-dimensional Euclidean ball removed).It follows (see [2]) that the (L)-semigroups with boundaries S 3 , S 3), and the corresponding (L)-semigroup sums are S 4 , RP 4 , 2M 4 . 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 S 2 , S 4 , RP 2 , RP 4 of K 2 as a factor can admit an H-space structure.This leaves only 2M 4 for consideration.However, its fundamental group Π 1 (2M 4 ) is the free product of Π 1 (RP 4 ) = Z 2 with itself, which is non-Abelian, so 2M 4 does not admit an H-space structure.Finally, the only five-dimensional (L)-semigroup sum with boundary U(2) is the manifold 2U(2) in Corollary 2, which does not admit an H-space structure.Corollary 3. No (L)-semigroup sum of dimension n ≤ 5 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 M 2 and M 4 are GR-spaces.Since GR-spaces are preserved by topological products, it follows that products of E 2 , E 4 , M 2 , M 4 , and topological groups are GR-spaces.This will include all the (L)-semigroups mentioned in this note excluding (L)-semigroups with boundary U(n), n ≥ 2. 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?