DUAN’S FIXED POINT THEOREM: PROOF AND
GENERALIZATION
MARTIN ARKOWITZ
Received 25 July 2004; Revised 6 January 2005; Accepted 21 July 2005
LetXbe an H-space of the homotopy type of a connected, finite CW-complex,f :X→X any map and pk:X→Xthekth power map. Duan proved thatpkf :X→Xhas a fixed point ifk≥2. We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spacesXwhose 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 aθ-structureμθ:X→Xas defined by Hemmi-Morisugi-Ooshima. The conclusion is thatμθf and f μθeach has a fixed point.
Copyright © 2006 Martin Arkowitz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let G be a topological group and f :G→G a map (i.e., a continuous function). Let pk:G→Gbe thekth power map defined by pk(x)=xk. Recall that a fixed point of f is an elementx0∈G such that f(x0)=x0. In 1993 Duan Haibo proved the following interesting fixed point theorem.
Theorem1.1 [1]. IfGis a compact, connected topological group and f :G→Gis a map, then for anyk≥2, the mappkf :G→Ghas 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
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.
2. 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 as-sumption can be weakened). The cohomology of a space with coefficients in the additive group of rationals will be writtenH∗(X)= {Hn(X)}, so that cohomology will always be taken with rational coefficients. A map f :X→X induces a linear transformation
f∗n:Hn(X)→Hn(X). The Lefschetz number is defined by
L(f)=N
n=0
(−1)nTrf∗n, (2.1)
whereHi(X)=0, fori > N, and Tr denotes the trace. Lefschetz’s famous fixed point the-orem asserts that ifL(f)=0, then f has a fixed point [5].
Next we state some basic facts about H-spaces. An H-space consists of a spaceXand a mapm:X×X→X(called the multiplication) such thatmrestricted to each factor is homotopic to the identity map id. For an H-spaceX, the power mappk:X→X,k≥1, is inductively defined as follows:p1=id, andpkis the composition
X Δ X×X pk−1×id X×X m X, (2.2)
where Δ is the diagonal map. The multiplication m induces a homomorphism m∗: H∗(X)→H∗(X×X)≈H∗(X)⊗H∗(X). An elementx∈Hn(X) is called primitiveif m∗(x)=x⊗1 + 1⊗x. Ifx∈Hn(X) is primitive, then it follows immediately from the definitions that
p∗k(x)=kx. (2.3)
The H-spaceX is said to behomotopy-associative if the mapsm(m×id),m(id×m) : X×X×X→Xare homotopic. The Hopf-Leray-Samelson theorem ([6, page 268] and [7, Theorem 7.20]) asserts that ifXis a homotopy-associative H-space, then
H∗(X)=Λx1,x2,...,xr
, (2.4)
an exterior algebra on odd degree generatorsx1,x2,...,xrwhich are primitive.
Lemma2.1. IfAis ann×nmatrix of rationals andBis a diagonaln×nmatrix of rationals, A= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
a11 a12... a1n a21 a22... a2n
..
. .... .. ... an1 an2... ann
⎞ ⎟ ⎟ ⎟ ⎟
⎠, B= ⎛ ⎜ ⎜ ⎜ ⎜ ⎝
b1 0... 0 0 b2... 0
..
. .... .. ... 0 0... bn
⎞ ⎟ ⎟ ⎟ ⎟
⎠, (2.5)
thenTr(AB)=a11b1+a22b2+···+annbn=Tr(BA). We write the diagonal matrixBas diag(b1,b2,...,bn).
Theorem2.2 [1]. If X is a homotopy-associative H-space, f :X→X any map and pk: X→Xthekth power map,k≥2, thenpkf :X→Xhas a fixed point.
Proof. We show thatL(pkf)=0. For this we consider the trace of (pkf)∗n=f∗np∗n k
Hn(X) p∗nk
Hn(X) f∗n Hn(X). (2.6)
By the theorem of Hopf-Leray-Samelson,
H∗(X)=Λx1,x2,...,xr, (2.7)
where thexiare primitive elements of odd degree|xi| =mi. Ifn≥1 then a basis ofHn(X) consists of elements
yi1i2···il=xi1xi2···xil, (2.8)
wherel≥1, 1≤i1< i2<···< il≤randmi1+mi2+···+mil=n. We examine the matrix
ofp∗knand f∗nwith respect to this basis. Sincep∗
k(xi)=kxi, p∗n
k
yi1i2···il
=kly
i1i2···il. (2.9)
Now suppose that there areb(1n)basis elements inHn(X) of length one (i.e., those of the formyi),b(n)2 basis elements inHn(X) of length two (i.e., those of the formyi1i2,i1< i2),
etc., whereb(in)≥0. Then the matrixBofp∗n
k is diagonal,
B=diag
k,...,k b(1n)
,k2,...,k2
b(2n)
,...
. (2.10)
Next we consider the matrixA of f∗n with respect to this basis. Now f∗nis obtained by taking the homomorphism onintegraln-dimensional cohomology induced by f and tensoring it with the rationals. ThusAis a matrix of integers. Lete(1n)be the sum of the firstb(1n)diagonal entries ofA,e
(n)
2 the sum of the nextb (n)
2 diagonal entries, etc. Then by Lemma 2.1,
Thus Tr((pkf)∗n)≡0(modk) forn≥1, and soL(pkf)≡1(modk). Sincek≥2, we have
L(pkf)=0. This completes the proof.
Remark 2.3. There are some simple extensions ofTheorem 2.2:
(1) If the H-space X has a homotopy inverse, one can define pk for all integersk. Theorem 2.2then holds for all|k| ≥2.
(2)Theorem 2.2holds for the map f pk:X→Xsince it is well known thatL(f pk)=
L(pkf).
(3) Letkbe an integer such that|k| ≥2. Suppose thatXis a homotopy-associative H-space and there is a mapμ:X→Xsuch thatμ∗(xi)=aixi, whereai≡0 (modk), fori=1, 2,...,r. Then the previous proof shows that iff :X→Xis any map, then μ f and f μeach has a fixed point. We will return to this inSection 4.
We note the following immediate consequence of Duan’s theorem which appears in [5, Theorem 1, page 49].
Corollary2.4. LetGbe a compact, connected topological group,a∈Gandk≥2. Then there existsx0∈Gsuch thatxk
0=a.
Proof. LetLa:G→Gbe left multiplication bya. By Duan’s theorem,La−1pk+1has a fixed
pointx0.
3. 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 inSection 4.
LetYbe a space and consider the vector spaceI∗(H∗(Y))= {In(H∗(Y))}of
indecom-posablesofH∗(Y) defined by
I∗H∗(Y)= H+(Y)
H+(Y)·H+(Y), (3.1)
whereH+ denotes positive-dimensional cohomology. A map f :Y →Y induces f∗:
H∗(Y)→H∗(Y) and this induces a linear transformation I∗(f∗) :I∗(H∗(Y))→
I∗(H∗(Y)).
For the rest of this section we consider spacesXwhose cohomology has the following form
H∗(X)=Λx1,x2,...,xr
⊗Py1,y2,...,ys
/yn1
1 ,yn22,...,ynss
, (3.2)
whereΛ(x1,x2,...,xr) is an exterior algebra on odd dimensional generatorsx1,x2,...,xr, P(y1,y2,...,ys) is a polynomial algebra on even dimensional generatorsy1,y2,...,ysand
yn1
1 ,y2n2,...,ysnsis the ideal generated by the powersy1n1,y2n2,...,ynss . In short,H∗(X) is a tensor product of monogenic algebras.
Now letXbe a space satisfying (3.2) and f :X→Xa map. The vector space of inde-composablesI∗(H∗(X)) can be split into its odd and even degree parts
I∗H∗(X)=V⊕W, (3.3)
whereV =ioddIi(H∗(X)) and W=
ievenIi(H∗(X)). Then I∗(f∗) :I∗(H∗(X))→ I∗(H∗(X)) induces linear transformations
fV:V−→V, fW:W−→W. (3.4)
The following theorem will be proved inSection 5.
Theorem3.1. LetXbe a space satisfying (3.2) and f :X→X a map. Suppose that−1is not an eigenvalue of fW. Then
L(f)=0⇐⇒ fV has no eigenvalue equal to1. (3.5)
We make some remarks on the theorem.
Remarks 3.2. (1) The matrices of the linear transformations fVandfWcan be determined from the induced linear transformation f∗applied to the algebra generatorsx1,x2,..., xr,y1,y2,...,ysofH∗(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 applyingTheorem 3.1to showL(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 ifrors=0. In addition, the conclusionL(f)=0 holds without the hypothesis that fV has no eigenvalue equal to 1, provided allniare odd. This can be seen from the proof.
(3) A result similar toTheorem 3.1 has been proved by Lupton and Oprea [2]. In Remark 5.3we discuss the relation of their result to our work.
We next give some examples related toTheorem 3.1.
Examples 3.3. (1) We indicate one way (though not the only way) to construct spacesX satisfying (3.2). LetAbe a space such thatH∗(A)=Λ(x1,...,xr). For example,Acould be the product of any number of the following spaces: homotopy-associative H-spaces and odd dimensional spheres. LetBbe a space such thatH∗(B)=P(y1,...,ys)/yn1
1 ,...,ynss . For example,Bcould be the product of any number of the following spaces: projective spaces and even dimensional spheres. ThenX=A×Bis a space which satisfies (3.2).
(2) We next show that the hypothesis that−1 is not an eigenvalue of fW is necessary in general. LetX be the complex projective spaceCP2n+1and let f :X→X be a map of degree−1, that is, f∗2(u)= −ufor everyu∈H2(X). Then 1 is not an eigenvalue of f
V,
−1 is an eigenvalue of fWandL(f)=0.
(3) Here we show that the strict inequality 1< n1< n2<···< nr is necessary in Theorem 3.1. Define f :S2×S2→S2×S2 by f(x,y)=(y,−x). Let{u,v} ⊆H2(S2×S2) be the basis corresponding to the two 2-spheres. Then f∗2(u)=v, f∗2(v)= −u and
4. Theta spaces
In this section we will useTheorem 3.1to extend Duan’s theorem to spacesXwhich sat-isfy (3.2). In order to do this it is necessary to describe a mapX→Xwhich 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 useXto denote a space which satisfies (3.2) of Section 3and will useY to denote an arbitrary space (of the homotopy type of a finite, connected CW-complex).
Definition 4.1. LetYbe a space and{m1,m2,...,mt}a set of positive integers defined as follows:
ImH∗(Y)=0⇐⇒m=m
i, for somei=1, 2,...,t. (4.1)
Letθ:{m1,m2,...,mt} →Zbe an integer-valued function. Then aθ-structureonY is a mapμθ:Y→Ysuch that
Imiμ∗θ(y)=θmi(y), (4.2)
for everyy∈Imi(H∗(Y)). The pair (Y,μ
θ) (or justY) is called aθ-space. Aconstantθ -structureis one such thatθ(mi)=k, for alli, wherek∈Zis a fixed integer.
There is a long list ofθ-spaces in [3] and we mention some of them below. Allθ func-tions in the following list have the formθ(mi)=ke(mi), for some integerkand functione.
(i) H-spaces and co-H-spaces have constantθ-structure given by the power map. (ii) Semi-simple Lie groupsGand their classifying spacesBGhaveθ-structure given
by the unstable Adams operationsψkonBGandΩψkonΩBG=G, for certaink. (iii) Complex and Quaternionic Grassman manifoldsGp,qwith some restrictions on
pandqhaveθ-structure.
(iv) The Stiefel manifoldsU(2n+ 2)/U(2n) have constantθ-structurekif and only if k≡0, 1, 5 (mod 8).
In addition, the existence ofθ-structure on a large class of spaces is obtained from the following corollary of Theorem 1 in [3]:
IfX is a simply-connected space which satisfies (3.2) ofSection 3, then there exists infinitely manyθ-structures onX.
In Theorem 2 of [3] the authors consider self maps f :Y→Yof aθ-space, for certain θ, and show the existence of fixed points of f μθ andμθf. The restrictions onθare that θ(mi)=ke(mi), wheree(mi)=(b−a)mi+ 2a−b, forb≥a≥1 and|k| ≥2. We prove a similar theorem below (by different methods) which restricts the spaces to those satisfy-ing (3.2) but allows a much larger class of functionsθ.
Theorem 4.2. Let X be any space satisfying (3.2) and f :X →X any map. Let
{m1,m2,...,mt} be the set of degrees of the non-zero indecomposables of H∗(X)and let θ:{m1,m2,...,mt} →Z− {0,±1}be any function. Ifμθ:X→Xis aθ-structure onX, then
Proof. We applyTheorem 3.1toμθf :X→X. We decomposeI∗(H∗(X))=V⊕Winto odd and even parts and first consider I∗(f∗μ∗θ)|W=(μθf)W :W →W. Suppose w∈Wmiis an eigenvector of (μ
θf)Wwith eigenvalue−1. Then
fWθmi(w)=Imif∗μθ∗(w)=μθfW(w)= −w, (4.3)
and so fW(w)=(−1/θ(mi))w. Thus−1/θ(mi) is an eigenvalue of fW which is a ratio-nal number. But fW is induced by a map f :X→X and so, as noted in the proof of Theorem 2.2, with respect to some basis ofW, fW is represented by an integral matrix (see also [2,§3]). But the only rational eigenvalues of an integral matrix are integers [8, Theorem 4.16]. Therefore−1/θ(mi) cannot be an eigenvalue of fW sinceθ(mi)= ±1. Thus−1 is not an eigenvalue of fW. A similar argument shows that 1 is not an eigenvalue of fV. Therefore byTheorem 3.1,L(μθf)=0. An analogous argument holds for f μθ. Remark 4.3. We illustrate howTheorem 4.2can be used in some concrete examples. Let X=A×Bbe a space of the type discussed inExamples 3.3(1). Ifμθ is aθ-structure on Aandμθis aθ-structure onB, thenμθ×μθis aθ-structure onA×B. More specifically, supposeA is a homotopy-associative H-space with μθ the kth power map and B is a product of even dimensional spheres and projective spaces withμθ aθ-structure which is constant atl(for example, the product of maps of degreel). Ifkandlare both=0,±1, thenTheorem 4.2applies to theθ-structureμθ×μθonX.
5. Proof ofTheorem 3.1
We state a special case of a theorem of Halperin which will be needed to proveTheorem 3.1. This requires the use of rational homotopy theory, in particular, Sullivan minimal models (see [9] and [10]). For a spaceXwhich satisfies (3.2), one can construct the min-imal modelᏹofX. This has the following properties:ᏹis a free-commutative, graded, differential algebra with generatorsx1,...,xr,y1,...,ys(which are in one-one correspon-dence with the generators ofH∗(X) and have the same degree) and generatorsz1,...,zs with|zi| = |yi|ni−1. Then
ᏹ=Λx1,...,xr,z1,...,zs⊗Py1,...,ys, (5.1)
with|xi|and|zi|odd and|yi|even. Note that a vector space basis forᏹconsists of all xη1
1 ···xηrr yλ11···ysλsz1τ1···zsτs, where 0≤ηi,τi≤1 and 0≤λi. The differentialdonᏹis defined by:dxi=0,dyi=0 anddzi=yini. ClearlyH∗(ᏹ,d)=H∗(X). We split the vector spaceI∗(ᏹ) of indecomposables ofᏹinto the direct sum of an odd degree partOand an even degree partE. We identifyO= x1,...,xr,z1,...,zsandE= y1,...,ys. A map f : X→X induces a homomorphismφ:ᏹ→ᏹ. This determinesI∗(φ) :I∗(ᏹ)→I∗(ᏹ) and thenceφO:O→OandφE:E→E. We now state a special case of Halperin’s theorem for spaces which satisfy (3.2).
Using this theorem, we now proveTheorem 3.1.
Proof. We fixl, 1≤l≤s, and writeφ(zl) as a linear combination of basis elements in the vector spaceᏹ,
φ(zl)=
i
alizi+ j
bl jxj+l, (5.2)
wherelis decomposable and|zl| = |zi| = |xj|for alliand jin the above sums. We let I= {i| |zi| = |zl|}and applydto both sides of (5.2) to obtain
φdzl=
i∈I
aliyini+dl. (5.3)
Butφ(dzl)=φ(ylnl)=(φ(yl))nl, and so (5.3) yields
φylnl=
i∈I
aliynii +dl. (5.4)
Next we writeφ(yl) as a linear combination of basis elements
φyl=
k∈K
clkyk+δl, (5.5)
whereK= {k| |yk| = |yl|}andδl is decomposable. Sincen1<···< ns, it follows that I∩K= {l}. Then we obtain from (5.4) and (5.5),
k∈K
clkyk+δl nl
=
i∈I
aliyini+dl. (5.6)
Consider anytwith 1≤t≤s. We will equate the terms which are linear combinations of ya
t for alla >0 on the left side of (5.6) with those on the right side of (5.6). For this it is necessary to analyze the elementsdlandδlin terms of the vector space basis above, noting thatl andδl are decomposable and that|dl| = |zl|+ 1 and|δl| = |yl|. Nowδl may contain a term of the formuayat, whereua is a rational anda≥2. Thus the only possible terms on the left side of (5.6) which are powers ofytarecnlltytnlandunlaytanl. For the right side of (5.6) note thatlmay contain a term of the formvbytbzt, whereb >0 andvbis a rational. Thendlwill containvbytnt+b. Thus the only possible terms which are powers ofyton the right side of (5.6) arealtytntandvbytnt+b. Now supposet∈Kandt=l. Thent /∈Iand (5.6) yields
cltnlynlt =vbyntt +b. (5.7)
Thus ift > l, thennl< nt, and soclt=0. Next supposet∈I andt=l. Thent /∈K and (5.6) yields
unl
Ifl > t, thenanl> nt, and soalt=0. Finally, ifl=t, thencnlll =all. Putting this information into (5.2) and (5.5), we have
φOzl=
i≥l
alizi+ j
bl jxj, (5.9)
φEyl=
k≤l
clkyk, (5.10)
whereall=cllnl. NowE= y1,...,ysandφE:E→E. From (5.10), the eigenvalues ofφE (in degree|yl|) are rational numbers of the formcii. These are the same as the eigenvalues of fW. By hypothesis, none of these eigenvalues equals−1. ClearlyI∗(φ) :x1,...,xr →
x1,...,xr. By examining the matrix ofφO, we see that the eigenvalues ofφOconsist of those of the formaii=ciini together with the eigenvalues ofφO|x1,...,xr. Nowaii=1 if and only ifcii=1. Thus the number of eigenvalues ofφOwhich are 1 equals the number of eigenvalues ofφEwhich are 1 plus the number of eigenvalues ofφO|x1,...,xrwhich are 1. But the eigenvalues ofφO|x1,...,xrare just the eigenvalues of fV. By Halperin’s theorem,L(f)=0 if and only if no eigenvalue of fV equals 1. Remark 5.2. Halperin’s theorem as stated and proved in [4] is more general in two distinct ways than what we have stated above. First of all, the theorem applies to elliptic spaces. These are spaces whose (rational) cohomology and rational homotopy groups (i.e., ho-motopy groups tensored with the rationals) vanish in all sufficiently high dimensions. The spaces which satisfy (3.2) are elliptic spaces. Secondly, the theorem gives a formula for the Lefschetz number of a map f in terms of the eigenvalues of fV and fW. Since we are interested in fixed points of maps, we have only considered the case whereL(f)=0. This has led to a simplified statement of the theorem.
Remark 5.3. Lupton and Oprea consider an elliptic spaceXwhose minimal modelᏹis oddly graded, that is,ᏹ=Λ(x1,...,xr), an exterior algebra on odd dimensional gener-ators. It is not assumed that the differentiald=0. If f :X→X is a map such that the induced mapφO=I∗(φ) :I∗(ᏹ)→I∗(ᏹ) does not have 1 as an eigenvalue, then the main result of [2,§5] asserts thatL(f)=0. It is possible to modify the statement and proof ofTheorem 3.1slightly so as to include this result. One assumes that the minimal modelᏹofXhas the form
ᏹ=Λx1,...,xr,z1,...,zs
⊗Py1,...,ys
, (5.11)
with|xi|and|zi|odd and|yi|even. Furthermore,dxi∈Λ(x1,...,xr),dzi=ynianddyi= 0 withn1<···< ns. We assume thatφE:E→Edoes not have−1 as an eigenvalue. Then the modified version ofTheorem 3.1asserts thatL(f)=0 if and only ifφOdoes not have 1 as an eigenvalue. For the proof it is only necessary to show thatφO(xi)∈ x1,...,xrwhich requires straightforward arguments similar to those given in the proof ofTheorem 3.1.
Acknowledgment
References
[1] D. Haibao,A characteristic polynomial for self-maps ofH-spaces, The Quarterly Journal of Math-ematics. Oxford. Second Series (2)44(1993), no. 175, 315–325.
[2] G. Lupton and J. Oprea,Fixed points and powers of self-maps ofH-spaces, Proceedings of the American Mathematical Society124(1996), no. 10, 3235–3239.
[3] Y. Hemmi, K. Morisugi, and H. Ooshima,Self maps of spaces, Journal of the Mathematical Soci-ety of Japan49(1997), no. 3, 438–453.
[4] S. Halperin,Spaces whose rational homology and de Rham homotopy are both finite-dimensional, Algebraic Homotopy and Local Algebra (Luminy, 1982), Ast´erisque, vol. 113-114, Soc. Math. France, Paris, 1984, pp. 198–205.
[5] R. F. Brown,The Lefschetz Fixed Point Theorem, Scott, Foresman, Illinois, 1971. [6] E. H. Spanier,Algebraic Topology, McGraw-Hill, New York, 1966.
[7] J. W. Milnor and J. C. Moore,On the structure of Hopf algebras, Annals of Mathematics. Second Series (2)81(1965), 211–264.
[8] T. Hungerford,Abstract Algebra: An Introduction, Saunders college, Pennsylvania, 1990. [9] P. A. Griffiths and J. W. Morgan,Rational Homotopy Theory and Differential Forms, Progress in
Mathematics, vol. 16, Birkh¨auser, Massachusetts, 1981.
[10] Y. F´elix, S. Halperin, and J.-C. Thomas,Rational Homotopy Theory, Graduate Texts in Mathe-matics, vol. 205, Springer, New York, 2001.
