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Monoids of self-maps of topological spherical space forms

A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.


Introduction
Let X be a pointed space, and let M(X) denote the pointed homotopy set [X, X]. Then M(X) is a monoid under the composition of maps. The monoid M(X) is obviously fundamental for understanding the space X. Invertible elements of M(X) form a group, which is the group of self-homotopy equivalences of X, denoted by E(X). The groups of selfhomotopy equivalences have been intensely studied so that there are a lot of results on them. There is a comprehensive survey on them [8]. However, despite its importance, not much is known about the monoids of self-maps M(X), and in particular, there are only two cases that we know an explicit description of M(X): the case X is a sphere or a complex projective spaces. Notice that M(X) has not been determined even in the case X is a real projective space or a lens space.
A topological spherical space form is, by definition, the quotient space of a sphere by a free action of a finite group. Then real projective spaces and lens spaces are typicial examples of such. We refer to [4] for details about topological spherical space forms. The purpose of this paper is to determine the monoids of self-maps of topological spherical space forms.
We recall basic facts about free actions of finite groups on spheres. Let G be a finite group acting freely on S n . Then it is well known that If n is even, then G must be a cyclic group of order 2, and so S n /G is homotopy equivalent to RP n . Suppose n is odd. Then every orientation reversing self-map of S n has a fixed point by the Lefschetz fixed-point theorem, and so the action of G on S n is orientation-preserving. Hence S n /G is an oriented compact connected manifold.
First, we state the main theorem in odd dimension. Let G be a finite group acting freely on S 2n+1 . We introduce a new monoid out of a finite group G. Let α ∈ End(G). By (1.1), the induced map α * : H 2n+2 (BG; Z) → H 2n+2 (BG; Z) is identified with an element of Z/|G|, which gives rise to a monoid homomorphism where (Z/m) × denotes the monoid of integers mod m under multiplication. Let M α be a subset d(α) + |G|Z of Z. Then we can define a new monoid by such that the product of x ∈ M α and y ∈ M β is xy ∈ M αβ . Clearly, the identity element of Now we are ready to state the main theorem in odd dimension.
Here is an important remark. It is well known that lens spaces of the same dimension with the same π 1 can have different homotopy types. See [6,Theorem VI]. Then different free actions of the same finite group G on the same sphere S 2n+1 can produce topological spherical sphere forms of different homotopy types. However, Theorem 1.1 implies that the monoid of self-maps does not distinguish actions.
When G is abelian, the map d in (1.2) is explicitly given in terms of the order of G and the integer n, where G acts freely on S 2n+1 . Then a more precise description of M(S 2n+1 /G) is available, which will be shown in Section 3. For example, one gets: Next, we state the main theorem in even dimension. As mentioned above, each of topological spherical forms of dimension 2n is of the homotopy type of RP 2n . Then the even dimensional case is covered by the following.
Remark 1.4. In [5] McGibbon claimed that M(RP 2n ) has cardinality two, but this is false as pointed out by Fred Cohen. This was fixed in [3] by Iriye, Matsushita and the first author, but the monoid structure was not considered.
Finally, we present two corollaries of Theorem 1.1. The monoid of self-maps is not abelian in general. See [1] for instance. But by Theorem 1.
Thus one gets: Let E(G, n) denote the group of invertible elements of M (G, n). Then by Theorem 1.1, Thus we obtain the following, which reproves the result of Smallen [9] and Plotnick [7], where their results seem to exclude the case |G| ≤ 2.
Corollary 1.6. Let G be a finite group acting freely on S 2n+1 . Then Acknowledgement: The first author is supported by JSPS KAKENHI No. 17K05248.

Mapping degree
In this section, we characterize self-maps of odd dimensional topological spherical space forms in terms of mapping degrees and the induced maps on the fundamental groups.
Recall that we can define the mapping degree of a map f : X → Y , denoted by deg(f ), in the following cases: (2) X and Y are oriented compact connected manifolds of dimension n.
In particular, we can define the mapping degrees of self-maps of topological spherical space forms of odd dimension.
Let G be a group, and let X, Y be G-spaces. We denote the set of G-equivariant homotopy classes of G-equivariant maps from X to Y by [X, Y ] G . The following can be easily deduced from the equivariant Hopf degree theorem [10,Theorem 8.4.1].
The following lemma is proved by Olum [6,Theorem IIIc] in the case that |G| is odd, where we impose nothing on |G|.
Moreover, X is simply connected since the map F 2n+1 → BG is an isomorphism in π 1 . Then X ≃ S 2n+1 . Now there is a homotopy commutative diagram where rows are homotopy fibrations. Then S 2n+1 /G ≃ F 2n+1 , and so the map g : S 2n+1 /G → F is identified with the inclusion of the (2n + 1)-skeleton.
Let α and k be as in the statement. Then there is a homotopy commutative diagram Then α * = k on H 2n+1 (F ; Z) ∼ = Z and π 1 ( α) = α. Thus the restriction of α to S 2n+1 /G, which is identified with the (2n + 1)-skeleton of F , is the desired map.
Proof of Theorem 1.1. For α ∈ End(G), let By Lemmas 3.1 and 3.2, M(S 2n+1 /G) = α∈End(G) N α as a set. For f ∈ N α and g ∈ N β , one has By Lemmas 2.2, 3.1 and 3.2, the map is well-defined and bijective. Then one gets a bijection M(S 2n+1 /G) → M (G, n). Moreover, this map is a monoid homomorphism by (3.1). Thus the proof is complete.
We describe M α in M (G, n) when G is cyclic. Let C m denote a cyclic group of order m, and consider a free action of C m on S 2n+1 . Since First, we set notation that we are going to use in this section. Let p : S n → RP n denote the universal covering, and let q : RP n → S n be the pinch map onto the top cell. Let j : RP n−1 → RP n denote the inclusion. We collect well known facts about real projective spaces that we are going to use. See [2] for the proof. (1) The mod 2 cohomology of RP n is given by H * (RP n ; Z/2) = Z/2[w]/(w n+1 ), |w| = 1.
(3) The composite First, we determine M(RP 2n ) as a set in a way different from [3]. Since π 1 (RP 2n ) ∼ = Z/2 and End(Z/2) ∼ = (Z/2) × , we have the following decomposition Let f ∈ M 0 . Then f lifts to S 2n , implying that f factors as the composite for some integer k ∈ Z. Thus M 0 = p * • q * (π 2n (S 2n )). There is an exact sequence of pointed sets induced from the covering C 2 → S 2n p − → RP 2n . Since [RP 2n , C 2 ] = * , one sees that p * : [RP 2n , S 2n ] → M(RP 2n ) is injective by considering the action of [RP 2n , C 2 ] on [RP 2n , S 2n ]. On the other hand, there is a diagram in which the column and the row are exact sequences of pointed sets induced from the cofibration in Lemma 4.1 (4). Since [ΣRP 2n−2 , S 2n ] = * , the map Σq * : π 2n (S 2n ) → [ΣRP 2n−1 , S 2n ] is surjective. For k ∈ Z, let a 2k ∈ M(RP 2n ) be the composite (4.1). Then one gets the following by Lemma 4.1 (3).  Proof. First, we reproduce the construction of the map b 2l+1 in [3]. Let S 1 = {z ∈ C | |z| = 1}. Consider the antipodal action of C 2 on S 1 and the canonical free action of C 2 on S 0 . Then the diagonal C 2 -action on S 1 * S 0 * · · · * S 0 2n−1 is identified with the antipodal action on is a C 2 -map of degree 2l + 1. Then we get b 2l+1 ∈ M 1 .
Next, we show the uniqueness of b 2l+1 . Let b ′ 2l+1 ∈ M 1 be a map which lifts to a map S 2n → S 2n of degree 2l + 1. Clearly, this lift is a C 2 -map. Then the lifts of b 2l+1 and b ′ 2l+1 are C 2 -maps S 2n → S 2n of the same degree 2l + 1, and so by Lemma 2.1, these lifts are C 2 -equivariantly homotopic. Thus b 2l+1 and b ′ 2l+1 are homotopic, completing the proof. Now we are ready to determine M 1 .
Next, we determine the monoid structure of M(RP 2n ).
Lemma 4.6. In M(RP 2n ), Proof. By definition, a 2k a 2k ′ is the composite Then the first equality follows from Lemma 4.1 (3).
There is a homotopy commutative diagram: The composite of the bottom maps is b 2l+1 a 2k and the composite around the upper perimeter is a 2k(2l+1) . Then one gets b 2l+1 a 2k = a 2k(2l+1) .