Duan's Fixed Point Theorem: Proof and Generalization

Let be an H-space of the homotopy type of a connected, finite CW-complex, any map and the th power map. Duan proved that has a fixed point if . We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a -structure as defined by Hemmi-Morisugi-Ooshima. The conclusion is that and each has a fixed point.


Introduction
Let G be a topological group and f : G → G a map (i.e., a continuous function). Let p k : G → G be the kth power map defined by p k (x) = x k . Recall that a fixed point of f is an element x 0 ∈ G such that f (x 0 ) = x 0 . In 1993 Duan Haibo proved the following interesting fixed point theorem.
Theorem 1.1 [1]. If G is a compact, connected topological group and f : G → G is a map, then for any k ≥ 2, the map p k f : G → G has a fixed point.
This theorem was proved more generally for homotopy-associative H-spaces having the homotopy type of a finite, connected CW-complex (Theorem 2.2). In 1996, Lupton and Oprea [2] gave a new proof of Duan's theorem using rational homotopy theory. In 1997, Hemmi-Morisugi-Ooshima [3] extended Duan's theorem to spaces more general than homotopy-associative H-spaces. In all of the above results, the existence of a fixed point of a map was obtained by showing the Lefschetz number of the map is non-zero.
The purpose of this paper is two-fold. First, we give a new, short proof of Duan's theorem. The proof is elementary in that the only non-trivial result required is the 2 Duan's fixed point theorem: proof and generalization Hopf-Leray-Samelson theorem on the rational cohomology of a homotopy-associative H-space. Secondly, we use rational methods, in particular, a result of Halperin [4], and ideas from [3] to generalize Duan's theorem.

Duan's theorem
We begin by briefly discussing the Lefschetz number and H-spaces. All spaces will be assumed to have the homotopy type of a finite, connected CW-complex (though this assumption can be weakened). The cohomology of a space with coefficients in the additive group of rationals will be written H * (X) = {H n (X)}, so that cohomology will always be taken with rational coefficients. A map f : X → X induces a linear transformation f * n : H n (X) → H n (X). The Lefschetz number is defined by where H i (X) = 0, for i > N, and Tr denotes the trace. Lefschetz's famous fixed point theorem asserts that if L( f ) = 0, then f has a fixed point [5].
Next we state some basic facts about H-spaces. An H-space consists of a space X and a map m : X × X → X (called the multiplication) such that m restricted to each factor is homotopic to the identity map id. For an H-space X, the power map p k : X → X, k ≥ 1, is inductively defined as follows: p 1 = id, and p k is the composition where Δ is the diagonal map. The multiplication m induces a homomorphism m * : is primitive, then it follows immediately from the definitions that The H-space X is said to be homotopy-associative if the maps m(m × id), m(id × m) : an exterior algebra on odd degree generators x 1 ,x 2 ,...,x r which are primitive. With these generalities out of the way, we state an obvious lemma and proceed with Duan's theorem and its proof.

Martin Arkowitz 3
Lemma 2.1. If A is an n × n matrix of rationals and B is a diagonal n × n matrix of rationals, then Tr(AB) = a 11 b 1 + a 22 b 2 + ··· + a nn b n = Tr(BA).
Theorem 2.2 [1]. If X is a homotopy-associative H-space, f : X → X any map and p k : X → X the kth power map, k ≥ 2, then p k f : X → X has a fixed point.
Proof. We show that L(p k f ) = 0. For this we consider the trace of (p k f ) * n = f * n p * n k H n (X) By the theorem of Hopf-Leray-Samelson, where the x i are primitive elements of odd degree |x i | = m i . If n ≥ 1 then a basis of H n (X) consists of elements We examine the matrix of p * n k and f * n with respect to this basis.
Now suppose that there are b (n) 1 basis elements in H n (X) of length one (i.e., those of the form y i ), b (n) 2 basis elements in H n (X) of length two (i.e., those of the form y i1i2 , Next we consider the matrix A of f * n with respect to this basis. Now f * n is obtained by taking the homomorphism on integral n-dimensional cohomology induced by f and tensoring it with the rationals. Thus A is a matrix of integers. Let e (n) 1 be the sum of the first b (n) 1 diagonal entries of A, e (n) 2 the sum of the next b (n) 2 diagonal entries, etc. Then by Lemma 2.1, Tr p k f * n = Tr(AB) = ke (n) 1 + k 2 e (n) 2 + ··· + k r e (n) r .
Let k be an integer such that |k| ≥ 2. Suppose that X is a homotopy-associative Hspace and there is a map μ : Then the previous proof shows that if f : X → X is any map, then μ f and f μ each has a fixed point. We will return to this in Section 4. We note the following immediate consequence of Duan's theorem which appears in [5, Theorem 1, page 49].

Fixed points and eigenvalues
In this section we consider spaces with restricted cohomology and state a result on the Lefschetz number of self maps of such spaces. This result, Theorem 3.1, which may be of some interest in itself, will be used to generalize Duan's theorem in Section 4.
We will always assume for a space X which satisfies (3.2) that 1 < n 1 < n 2 < ··· < n s .
We give examples of such spaces in Examples 3.3(1).
Now let X be a space satisfying (3.2) and f : X → X a map. The vector space of indecomposables I * (H * (X)) can be split into its odd and even degree parts where V = iodd I i (H * (X)) and W = ieven I i (H * (X)). Then I * ( f * ) : I * (H * (X)) → I * (H * (X)) induces linear transformations The following theorem will be proved in Section 5.
Theorem 3.1. Let X be a space satisfying (3.2) and f : X → X a map. Suppose that −1 is not an eigenvalue of f W . Then We make some remarks on the theorem.

Remarks 3.2.
(1) The matrices of the linear transformations f V and f W can be determined from the induced linear transformation f * applied to the algebra generators x 1 ,x 2 ,..., x r , y 1 , y 2 ,..., y s of H * (X). In general, it is difficult to calculate the eigenvalues of a linear transformation since this requires finding the roots of the characteristic polynomial. In applying Theorem 3.1 to show L( f ) = 0, however, it is only necessary to show that −1 and 1 are not roots of the appropriate characteristic polynomials. This is much easier to do.
(2) The theorem holds if r or s = 0. In addition, the conclusion L( f ) = 0 holds without the hypothesis that f V has no eigenvalue equal to 1, provided all n i are odd. This can be seen from the proof.
(3) A result similar to Theorem 3.1 has been proved by Lupton and Oprea [2]. In Remark 5.3 we discuss the relation of their result to our work.
We next give some examples related to Theorem 3.1.

Examples 3.3.
(1) We indicate one way (though not the only way) to construct spaces X satisfying (3.2). Let A be a space such that H * (A) = Λ(x 1 ,...,x r ). For example, A could be the product of any number of the following spaces: homotopy-associative H-spaces and odd dimensional spheres. Let B be a space such that H * (B) = P(y 1 ,..., y s )/ y n1 1 ,..., y ns s . For example, B could be the product of any number of the following spaces: projective spaces and even dimensional spheres. Then X = A × B is a space which satisfies (3.2).
(2) We next show that the hypothesis that −1 is not an eigenvalue of f W is necessary in general. Let X be the complex projective space CP 2n+1 and let f : X → X be a map of degree −1, that is, f * 2 (u) = −u for every u ∈ H 2 (X). Then 1 is not an eigenvalue of f V , −1 is an eigenvalue of f W and L( f ) = 0.
(3) Here we show that the strict inequality 1 < n 1 < n 2 < ··· < n r is necessary in

Theta spaces
In this section we will use Theorem 3.1 to extend Duan's theorem to spaces X which satisfy (3.2). In order to do this it is necessary to describe a map X → X which plays the role of the power map of H-spaces. This has been done by Hemmi-Morisugi-Ooshima [3]. We begin this section by summarizing their work (with some small changes in terminology).
For the remainder of the paper we will use X to denote a space which satisfies (3.2) of Section 3 and will use Y to denote an arbitrary space (of the homotopy type of a finite, connected CW-complex). (4.1) Let θ : {m 1 ,m 2 ,...,m t } → Z be an integer-valued function. Then a θ-structure on Y is a map μ θ : Y → Y such that There is a long list of θ-spaces in [3] and we mention some of them below. All θ functions in the following list have the form θ(m i ) = k e(mi) , for some integer k and function e.
(i) H-spaces and co-H-spaces have constant θ-structure given by the power map.
(ii) Semi-simple Lie groups G and their classifying spaces BG have θ-structure given by the unstable Adams operations ψ k on BG and Ωψ k on ΩBG = G, for certain k. (iii) Complex and Quaternionic Grassman manifolds G p,q with some restrictions on p and q have θ-structure. (iv) The Stiefel manifolds U(2n + 2)/U(2n) have constant θ-structure k if and only if k ≡ 0,1,5(mod 8). In addition, the existence of θ-structure on a large class of spaces is obtained from the following corollary of Theorem 1 in [3]: If X is a simply-connected space which satisfies (3.2) of Section 3, then there exists infinitely many θ-structures on X.
In Theorem 2 of [3] the authors consider self maps f : Y → Y of a θ-space, for certain θ, and show the existence of fixed points of f μ θ and μ θ f . The restrictions on θ are that θ(m i ) = k e(mi) , where e(m i ) = (b − a)m i + 2a − b, for b ≥ a ≥ 1 and |k| ≥ 2. We prove a similar theorem below (by different methods) which restricts the spaces to those satisfying (3.2) but allows a much larger class of functions θ. Proof. We apply Theorem 3.1 to μ θ f : X → X. We decompose I * (H * (X)) = V ⊕ W into odd and even parts and first consider and so f W (w) = (−1/θ(m i ))w. Thus −1/θ(m i ) is an eigenvalue of f W which is a rational number. But f W is induced by a map f : X → X and so, as noted in the proof of Theorem 2.2, with respect to some basis of W, f W is represented by an integral matrix (see also [2, §3] (1). If μ θ is a θ-structure on A and μ θ is a θ-structure on B, then μ θ × μ θ is a θ-structure on A × B. More specifically, suppose A is a homotopy-associative H-space with μ θ the kth power map and B is a product of even dimensional spheres and projective spaces with μ θ a θ-structure which is constant at l (for example, the product of maps of degree l). If k and l are both = 0, ±1, then Theorem 4.2 applies to the θ-structure μ θ × μ θ on X.

Proof of Theorem 3.1
We state a special case of a theorem of Halperin which will be needed to prove Theorem 3.1. This requires the use of rational homotopy theory, in particular, Sullivan minimal models (see [9] and [10]). For a space X which satisfies (3.2), one can construct the minimal model ᏹ of X. This has the following properties: ᏹ is a free-commutative, graded, differential algebra with generators x 1 ,...,x r , y 1 ,..., y s (which are in one-one correspondence with the generators of H * (X) and have the same degree) and generators z 1 ,...,z s with |z i | = |y i |n i − 1. Then ᏹ = Λ x 1 ,...,x r ,z 1 ,...,z s ⊗ P y 1 ,..., y s , with |x i | and |z i | odd and |y i | even. Note that a vector space basis for ᏹ consists of all x η1 1 ··· x ηr r y λ1 1 ··· y λs s z τ1 1 ··· z τs s , where 0 ≤ η i , τ i ≤ 1 and 0 ≤ λ i . The differential d on ᏹ is defined by: dx i = 0, dy i = 0 and dz i = y ni i . Clearly H * (ᏹ,d) = H * (X). We split the vector space I * (ᏹ) of indecomposables of ᏹ into the direct sum of an odd degree part O and an even degree part E. We identify O = x 1 ,...,x r ,z 1 ,...,z s and E = y 1 ,..., y s . A map f : X → X induces a homomorphism φ : ᏹ → ᏹ. This determines I * (φ) : I * (ᏹ) → I * (ᏹ) and thence φ O : O → O and φ E : E → E. We now state a special case of Halperin's theorem for spaces which satisfy (3.2).