SELF HOMOTOPY EQUIVALENCES WHICH INDUCE
THE IDENTITY ON HOMOLOGY, COHOMOLOGY
OR HOMOTOPY GROUPS
Martin ARKOWITZ and Ken-ichi MARUYAMA
Abstract. For a based, 1-connected, finite CW-complex X, we study the foll*
*owing sub-
groups of the group of homotopy classes of self homotopy equivalences of X*
*: E*(X), the
subgroup of homotopy classes which induce the identity on homology groups,*
* E*(X), the sub-
group of homotopy classes which induce the identity on cohomology groups a*
*nd Edim+r#(X),
the subgroup of homotopy classes which induce the identity on homotopy gro*
*ups in dimen-
sions dimX + r. We investigate these groups when X is a Moore space and w*
*hen X is a
co-Moore space. We give the structure of the groups in these cases and pro*
*vide examples of
spaces for which the groups differ. We also consider conditions on X such *
*that E*(X) = E*(X)
and obtain a class of spaces (including compact, oriented manifolds and H-*
*spaces) for which
this holds. Finally, we examine Edim+r#(X) for certain spaces X and comple*
*tely determine
the group when X = Sm x Sn and X = CPn _ S2n.
x1. Introduction
If X is a based topological space, let E(X) denote the set of homotopy class*
*es of self
homotopy equivalences of X. Then E(X) is a group with group operation given by *
*compo-
sition of homotopy classes. The group E(X) and certain natural subgroups are fu*
*ndamental
objects in homotopy theory and have been studied extensively. For a finite CW-c*
*omplex
X, these subgroups include E*(X), the subgroup of homotopy classes which induce*
* the
identity on the homology groups of X, E*(X), the subgroup of homotopy classes w*
*hich
induce the identity on the cohomology groups of X and Edim+r#(X), the subgroup *
*of ho-
motopy classes which induce the identity on the homotopy groups of X in dimensi*
*ons
dim X + r. For a survey of known results and applications of E(X), see [Ar], a*
*nd for a
list of references on the subgroups mentioned above, see [A-L, pp. 1-2].
In this paper we investigate the subgroups E*(X), E*(X) and Edim+r#(X). We *
*give
examples for which the groups are different and present some general results wh*
*ich show
when they are the same. In addition, we completely determine these groups in s*
*everal
specific cases.
We now briefly summarize the contents of this paper. Section 2 contains some*
* useful
results which will be needed in later sections. We first state several classic*
*al theorems
_____________
1991 Mathematics Subject Classification. Primary 55P10.
Key words and phrases. Homotopy equivalences, the group of homotopy equivale*
*nces, homotopy equiv-
alences which induce the identity, Moore spaces, co-Moore spaces, products of s*
*pheres.
Typeset by AM S-*
*TEX
1
2 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
such as the universal coefficient theorem for homotopy groups with coefficients*
* and the
Blakers-Massey theorem. We then consider Moore spaces, i.e., spaces with a sing*
*le non-
vanishing homology group, and give some elementary results about them. In Sect*
*ion 3
we study E*(X), E*(X) and Edim+r#(X) when X is a Moore space and determine these
groups completely. In Section 4 we examine co-Moore spaces Y , i.e., spaces wit*
*h a single
non-vanishing cohomology group, and calculate the groups E*(Y ), E*(Y ) and Edi*
*m+r#(Y )
for r = 0; 1. The results in Sections 3 and 4 provide us with many examples whi*
*ch show
the possibilities for the groups E*, E* and Edim+r#. In Section 5 we consider s*
*paces X such
that E*(X) = E*(X). We show that compact, oriented manifolds, H-spaces and spa*
*ces
with certain homological restrictions all have this property. In Section 6 we e*
*xamine the
groups Edim+r#(X) for specific spaces X in order to illustrate the varied behav*
*ior of these
groups. In particular, we completely determine Edim+r#(Sm x Sn) and Edim+r#(CP *
*n_ S2n).
For the remainder of this section we present our notation and conventions. *
*All topo-
logical spaces will be based and have the based homotopy type of a finite, 1-co*
*nnected
CW-complex. All maps and homotopies will preserve base points. If f : X ! Y *
*is a
map, then f*n : Hn(X) ! Hn(Y ), f#n : ssn(X) ! ssn(Y ) and f*n : Hn (Y ) ! Hn *
*(X)
denote respectively the induced homology, homotopy and cohomology homomorphism *
*in
dimension n. The subscript or superscript `n' will often be omitted. In this pa*
*per we do
not distinguish notationally between a map X ! Y and its homotopy class in [X; *
*Y ].
If G is an abelian group and n 3 an integer, then the Moore space M(G; n) i*
*s the
space, unique up to homotopy type, characterized by
ae
Hei(M(G; n)) = G i = n
0 i 6= n:
If G is free-abelian, M(G; n) is just a wedge of n-spheres. In general, the con*
*struction of
M(G; n) is as follows: Let G = F=R, where F is free-abelian and R F is a subgr*
*oup.
The inclusion R F induces a map j : M(R; n) ! M(F; n), and M(G; n) is the mapp*
*ing
cone of j. Thus we have the defining cofibre sequence of M(G; n):
M(F; n) -i! M(G; n) -q! M(R; n + 1):
Note that when G is finitely-generated, M(G; n) is a finite CW-complex of dimen*
*sion n if
G is free-abelian and of dimension n + 1 if G is not free-abelian. It is known *
*[Ba, p. 268-
269] that ssn+1(M(G; n)) = G Z2 for n 3 and that ssn+2(M(G; n)) is an extensi*
*on of
Tor(G; Z2) by GZ2 for n 4. Since M(G; n) is a double suspension, the set of ho*
*motopy
classes [M(G; n); X] can be given abelian group structure with binary operation*
* `+'. Then
ssn(G; X) = [M(G; n); X] is called the nth homotopy group of X with coefficient*
*s in G.
The group of self homotopy equivalences E(M(G; n)) of a Moore space has been st*
*udied
by Rutter in [Ru].
Finally, if A is an abelian group, we write
Mr
A = A . . .A (r summands ):
We also use `' to denote cartesian product of sets.
Acknowledgement. We would like to thank Gregory Lupton for having suggested Pro*
*po-
sition 6.3.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 3
x2 Preliminaries
We begin with some well-known results. The first is the universal coefficien*
*t theorem
for homotopy with coefficients [Hi, p. 30].
2.1 Theorem. There is a short exact sequence
0 ! Ext(G; ssn+1(X)) ! ssn(G; X) ! Hom (G; ssn(X)) ! 0;
where (f) = f#n : G ssn(M(G; n)) ! ssn(X).
We obtain the following corollary.
2.2 Corollary. Let f; g : M(G; n) ! M(G0; n) be maps and let G be free-abelian.*
* Then
f = g () f*n = g*n:
Next we have the Blakers-Massey theorem [Hi, p. 49].
2.3 Theorem. Given a cofibration X !i E !q Y with X (k - 1)-connected and Y (` *
*- 1)-
connected, k; ` 2. Then for any abelian group G and all m < k + `-2, there is *
*an exact
sequence
ssm (G; X) i#!ssm (G; E) q#!ssm (G; Y ) @!ssm-1 (G; X) ! . .:.
If G is free-abelian, then the sequence is exact for m k + ` - 2:
We also will need the following proposition.
2.4 Proposition. If X is (k - 1)-connected and Y is (` - 1)-connected, k; ` 2*
*, and
dim P < k + `, then the projections X _ Y ! X and X _ Y ! Y induce a bijection
[P; X _ Y ] -! [P; X] [P; Y ]:
Proposition 2.4 is a consequence of [Sp2, p. 405] since the inclusion X _ Y *
*! X x Y is
a (k + ` - 1)-equivalence.
Now let M(G; n) and M(G0; n) be Moore spaces with defining cofibre sequences
M(F; n) -i! M(G; n) -q! M(R; n + 1) and
0 q0
M(F0; n) -i! M(G0; n) -! M(R0; n + 1);
where F and F0 are free-abelian, R F, R0 F0, G = F=R and G0= F0=R0:
2.5 Proposition. Let f : M(G; n) ! M(G0; n). Then f*n = 0 () there exists a :
M(R; n + 1) ! M(F0; n) such that f = i0aq:
Proof. The implication `(' is obvious, so we show `)'. Since (fi)* = 0 : Hn(M(F*
*; n)) !
Hn(M(G0; n)), fi = 0 by Corollary 2.2. Now consider the Barratt-Puppe exact seq*
*uence
of the defining cofibration of M(G; n)
* i*
[M(R; n + 1); M(G0; n)] q![M(G; n); M(G0; n)] ! [M(F; n); M(G0; n)]:
4 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
Since i*(f) = 0, there exists b : M(R; n + 1) ! M(G0; n) such that f = bq. The*
*refore
(q0b)* = 0; and so q0b = 0 by Corollary 2.2. Then Theorem 2.3 applied to the d*
*efining
cofibration of M(G0; n) yields the exact sequence
i0# 0 q0# 0
ssn+1(R; M(F0; n)) ! ssn+1(R; M(G ; n)) ! ssn+1(R; M(R ; n + 1)):
Since q0#(b) = 0, b = i0#(a) = i0a for some a 2 ssn+1(R; M(F0; n)): Hence f = b*
*q = i0aq:
Next we consider abelian groups G1 and G2 and Moore spaces M1 = M(G1; n1) and
M2 = M(G2; n2): Let X = M1_M2 = M(G1; n1)_M(G2; n2) and denote by ij : Mj ! X
the inclusions and by pj : X ! Mj the projections, j = 1; 2. If f : X ! X, then*
* define
fjk : Mk ! Mj by fjk = pjfik for j; k = 1; 2:
2.6 Proposition. The function which assigns to each f 2 [X; X], the 2 x 2 matr*
*ix
(f) = f11ff12 ;
21f22
where fjk 2 [Mk; Mj]; is a bijection. In addition,
L
(1) (f + g) = (f) + (g), so is an isomorphism [X; X] ! j;k=1;2[Mk; Mj]:
(2) (fg) = (f)(g), where fg denotes composition in [X; X] and (f)(g) denotes
matrix multiplication.
(3) Under the identification Hr(M1 _ M2) = Hr(M1) Hr(M2), we have f*r(x; y)*
* =
(f11*r(x) + f12*r(y); f21*r(x) + f22*r(y)), for x 2 Hr(M1) and y 2 Hr(M2*
*):
(4) Under the identification Hr(M1 _ M2) = Hr(M1) Hr(M2), we have f*r(x; y) =
(f*r11(x) + f*r21(y); f*r12(x) + f*r22(y)), for x 2 Hr(M1) and y 2 Hr(M2*
*).
(5) If ffr : ssr(M1)ssr(M2) ! ssr(M1_M2) and fir : ssr(M1_M2) ! ssr(M1) ssr(*
*M2)
are the homomorphisms induced by the inclusions and projections respecti*
*vely,
then firf# ffr(x; y) = (f11#r(x) + f12#r(y); f21#r(x) + f22#r(y)) for x *
*2 ssr(M1)
and y 2 ssr(M2).
Proof. Clearly [X; X] [M1; X] [M2; X]. But [Mj; X] [Mj; M1] [Mj; M2] by Pro*
*po-
sition 2.4 for j = 1; 2. The rest of the proof is straightforward and hence omi*
*tted.
Next let G be a finitely-generated abelian group and write G = F T , where *
*F is a
free-abelian group of finite rank and T is a finite group. Let i1 and i2 be the*
* inclusions of
F and T into G and let p1 and p2 be the projections of G onto F and T . If OE :*
* G ! G is
a homomorphism, set OE11 = p1OEi1 : F ! F , OE21 = p2OEi1 : F ! T , OE12 = p1OE*
*i2 : T ! F
and OE22 = p2OEi2 : T ! T . Then for x 2 F and y 2 T ,
OE(x; y) = (OE11(x); OE21(x) + OE22(y));
since OE12 2 Hom (T; F ) = 0. The proof of the next proposition is clear.
2.7 Proposition. With the above notation, OE : G ! G is an isomorphism () OE11 *
*and
OE22 are isomorphisms. Furthermore, OE = 1 () OE11 = 1, OE22 = 1 and OE21 = 0.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 5
x3 Moore Spaces
In this section and the next we study the self homotopy equivalences of Moor*
*e and
co-Moore spaces by means of 2 x 2 matrices. This method was used earlier by Sie*
*radski
to study the self homotopy equivalences of a cartesian product [Si]. Let G be a*
* finitely-
generated abelian group and write G = F T , where F is a free-abelian group of*
* rank r
and T is a finite abelian group. We consider the Moore space X = M(G; n) = M(F;*
* n) _
M(T; n) for n 3. We set M1 = M(F; n) which is a wedge of r n-spheres, and M2 =
M(T; n) which is a wedge of spaces of the form M(Zm ; n) = Sn [m en+1. Througho*
*ut this
section X will denote the Moore space M(G; n) = M1 _ M2.
We now let f 2 [X; X] and use the notation of x2 so that fjk = pjfik 2 [Mk; *
*Mj], for
j; k = 1; 2. Then f 2 E(X) , f*n is an isomorphism. By Propositions 2.6 and 2.7*
*, we can
identify f 2 E(X) with the 2 x 2 matrix
f11 f12
f21 f22
where f11 2 E(M1), f12 2 [M2; M1], f21 2 [M1; M2] and f22 2 E(M2). The group st*
*ructure
in E(X) is then given by matrix multiplication.
Now we investigate the subgroup E*(X) = {f 2 E(X)| f*i = 1, for all i} of E(*
*X).
3.1 Theorem. Let f 2 E(X) be given as
f = f11ff12
21f22
with f11 2 E(M1) and f22 2 E(M2). Then f 2 E*(X) () f11 = 1 and f22 2 E*(M2). If
f; g 2 E*(X), then
fg = f1 f12 1 g12 = 1 f12+ g12 :
21 f22 g21 g22 f21+ g21 f21g12+ f22g22
Proof. The cohomology of X occurs only in dimensions n and n + 1 and
Hn (X) = Hn (M1) Hn (M2) = Hn (M1) = F and
Hn+1 (X) = Hn+1 (M1) Hn+1 (M2) = Hn+1 (M2) = T:
By Proposition 2.6, f*(x; y) = (f*11(x); f*22(y)) for x 2 Hk(M1) and y 2 Hk(M2)*
*. Thus
f*k = 1 for k = n; n + 1 , f*n11= 1 and f*n+122= 1. But E*(M1) = 1 by Corollary*
* 2.2, so
f*n11= 1 , f11 = 1. Also f*n+122= 1 , f22*n= 1 by the universal coefficient the*
*orem for
cohomology. This proves the first assertion.
To establish the formula for the product fg it suffices to show (1) f21+f22g*
*21 = f21+g21,
i.e., f22g21 = g21, and (2) g12+ f12g22 = f12+ g12, i.e., f12g22 = f12.
For (1), note that (f22g21)* = g21* since f22* = 1. But this implies f22g21*
* = g21 by
Corollary 2.2.
6 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
For (2), let
0 q0
M(F0; n) i!M2 ! M(R0; n + 1)
be the defining cofibre sequence for M2 = M(T; n), where T = F0=R0 and F0 free-*
*abelian.
Since (g22- 1)* = 0 : T ! T , by Proposition 2.5 there exists a : M(R0; n + 1) *
*! M(F0; n)
such that g22 = 1 + i0aq0. Then
f12g22= f12(1 + i0aq0)
= f12+ f12i0aq0
= f12
since (f12i0)* = 0 and so f12i0= 0.
Next we consider E*(X) = {f 2 E(X)| f*n = 1}. Let T = T 0 P , where T 0is t*
*he
2-primary torsion subgroup of G and P is the sum of p-primary torsion subgroups*
* of G
for all primes p 6= 2. Thus T 0= Z2a1 . . .Z2as, where afi 1. We write M(T 0; n*
*) as
MT0, M(P; n) as MP and M(Z2afi; n) as Nfi, for fi = 1; :::; s. Then
M2 = M(T; n) = M(T 0; n) _ M(P; n) = MT0 _ MP and
MT0 = M(Z2a1; n) _ . ._.M(Z2as; n) = N1 _ . ._.Ns:
Let jff: Sn ! M1 be the inclusion of Sn onto the ffth sphere of M1 and let pff:*
* M1 ! Sn
be the projection onto the ffth sphere, ff = 1; :::; r. Similarly let kfi: Nfi*
*! M2 and
rfi: M2 ! Nfibe the fith inclusion and projection, respectively, for fi = 1; ::*
*:; s. Let
Sn !ifiNfiqfi!Sn+1 be the defining cofibration of Nfiand jn 2 ssn+1(Sn) the non*
*-trivial
class. Then we define ufffi: M2 ! M1 to be the composition
M2 rfi!Nfiqfi!Sn+1 jn!Sn jff!M1;
ff = 1; :::; r and fi = 1; :::; s and we define vflffi: M2 ! M2 to be the compo*
*sition
M2 rffi!Nffiqffi!Sn+1 jn!Sn ifl!Nflkfl!M2;
fl; ffi = 1; :::; s.
3.2 Theorem. Let f 2 E(X) be given by f = f11f f12 with f11 2 E(M1) and
21f22
f22 2 E(M2). Then f 2 E*(X) () f11 = 1, f21 = 0 and f22 2 E*(M2). If f; g 2 E*(*
*X),
then
fg = 10 f12f 1 g12 = 1 f12+ g12 :
220 g22 0 f22g22
Furthermore,
(r+s)sM
E*(X) Z2;
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 7
where r the rank of G and s is the number of 2-torsion summands in G. In addit*
*ion,
generators of E*(X) consist of all
"ufffi= 10 ufffi1 and "vflffi= 10 1 0+ v ;
flffi
where ff = 1; :::; r and fi; fl; ffi = 1; :::; s.
Proof. We have f 2 E*(X) , f*n = 1. By Proposition 2.7, this is equivalent to *
*f11 2
E*(M1) = 1, f22 2 E*(M2) and f21*n= 0 : F ! T . But by Corollary 2.2 the latter*
* implies
f21 = 0 : M1 ! M2. This proves the first assertion. Thus E*(X) E*(X) (which is*
* easily
proven directly) and so the formula for fg follows from Theorem 3.1.
Next E*(X) [M2; M1] E*(M2). But
[M2; M1] ssn(T ; M1)
Ext(T; ssn+1(M1))
Ext(T; F Z2)
Mr
Ext(Z2a1 . . .Z2as; Z2)
Mrs
Z2:
Now consider E*(M2). Let Z*(M2) [M2; M2] consists of all homotopy classes w*
*hich
are zero on homology. Then by Theorem 2.1 we have the short exact sequences
0 -! Z*(M2) -! [M2; M2] -H! Hom (T; T ) -! 0
flfl ?
fl ?y
0 -! Ext(T; ssn+1(M2))-! ssn(T ; M2)-! Hom (T; ssn(M2))-! 0;
s2L
where H(h) = h*n. Thus Z*(M2) Ext(T; ssn+1(M2)) Ext(T; T Z2) Z2.
We next see that E*(M2) Z*(M2). There is a bijection ae : Z*(M2) ! E*(M2) d*
*efined
by ae(g) = g +1, for g 2 Z*(M2). We show that ae is a homomorphism : ae(g +h) =*
* g +h+1
and ae(g)ae(h) = (g + 1)(h + 1) = gh + g + h + 1. Thus it suffices to show gh *
*= 0 for
0 q0
g; h 2 Z*(M2). Let M(F0; n) i! M2 ! M(R0; n + 1) be the defining cofibre seque*
*nce of
M2. By Proposition 2.5, there exist a; b : M(R0; n + 1) ! M(F0; n) such that
g = i0aq0 and h = i0bq0:
(r+*
*s)sL
Thus gh = i0aq0i0bq0= 0 since q0i0= 0. This completes the proof that E*(X) *
* Z2.
Finally, it is straightforward to verify that the elements described in Theo*
*rem 3.2 are
generators.
8 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
3.3 Remark. Theorem 3.2 shows that E*(X) is always abelian and is the trivial g*
*roup if
and only if G has no 2-torsion. We also note that the decomposition of E*(X) in*
*to a direct
sum of Z2's could have been obtained directly using the isomorphism E*(X) Z*(X*
*).
We next compare E*(X) and E*(X). Let : E*(X) ! E*(X) be the inclusion and d*
*efine
ae : E*(X) ! Hom (F; T ) by ae f1 f12 = (f21)*n : F ! T:
21 f22
3.4 Proposition. The following is a split short exact sequence
0 ! E*(X) ! E*(X) ae!Hom (F; T ) ! 0;
and so E*(X) is a semi-direct product of E*(X) by Hom (F; T ).
Proof. By Theorem 3.1, ae is a homomorphism. Furthermore, Kernelae = Image . *
*We
define a homomorphism oe : Hom (F; T ) ! E*(X) such that aeoe = 1. Let a : F ! *
*T be a
homomorphism and ^a: M1 ! M2 be the corresponding homotopy class. Then we set
oe(a) = 1^a01 :
3.5 Remarks. (1) The order of the group E*(X) is given by |E*(X)| = |T |r x 2(*
*r+s)s.
(2) Proposition 3.4 gives examples of Moore spaces X = M(G; n) such that E*(*
*X) 6=
E*(X). Simply take any G = F T with F 6= 0 and T 6= 0.
(3) As a consequence we have the following partial realization result for E**
* : If A is any
finite abelian group without 2-torsion, there is a space X such that E*(*
*X) A.
Just set X = M(Z A; n).
We next determine the extension in Proposition 3.4. We use the notation of T*
*heorem
3.2 and Proposition 3.4. For ff = 1; :::; r and fi = 1; :::; s, we define wfiff*
*: M1 ! M2 as the
following composition
M1 pff!Sn ifi!Nfikfi!M2:
If ffiff2 Hom (F; T ) is wfiff*n, then oe(ffiff) = w 1 0 2 E*(X). Now l*
*et H
fiff 1
E*(X) be the subgroup generated by the generators "uff0fi0= 10 uff0fi01and"vf*
*lffi=
1 0 1 0 0 *
* 0
0 1 + vflffiofE*(X) and w"fiff= wfiff 1 , where ff; ff = 1; :::; r and f*
*i; fi ; fl; ffi =
1; :::; s. Then the action of Hom (F; T ) on E*(X) in Proposition 3.4 is given *
*by
(1) "w-1fiff"uff0fi0"wfiff= 10 1 +uff0fi0w and
fiffuff0fi0
(2) w"-1fiff"vflffi"wfiff= "vflffi:
In (1) if ff 6= ff0, wfiffuff0fi0= 0; if ff = ff0, wfiffuff0fi0= vfifi0. Furth*
*ermore, Hom (F; P )
clearly acts trivially on E*(X).
Recall that a group has nilpotency 2 if all commutators of length 3 are tr*
*ivial.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 9
Lr
3.6 Theorem. The group E*(X) is isomorphic to H P . Moreover, H is the semi-
direct product of E*(X) with Hom (F; T 0) with action of the latter group on th*
*e former
group given on generators by (1) and (2). In addition, nilE*(X) 2, and E*(X) i*
*s abelian
if and only if r = 0 or s = 0.
Proof. Only the last assertion requires proof. It easily follows from (1) and (*
*2) that the
commutator ae
1 i*
*fff 6= ff0
["uff0fi0; "wfiff] = "u-1ff0fi0"w-1fiff"uff0fi0"wfiff=
"vfifi0*
*ifff = ff0;
and commutators of all other generators are trivial. The theorem now follows.
We next give a simple illustration of Theorem 3.6 in the following corollary.
3.7 Corollary. E*(M(Z Z2; n)) D4, the dihedral group with 8 elements.
Proof. Let A = E*(M(Z Z2; n)). By Remark 3.5(1), A has 8 elements which are re*
*pre-
sented by matrices
1 f12
f21 f22 ;
where f21 2 ssn(M(Z2; n)), f12 2 ssn(Z2; Sn) and f22 2 E*(M(Z2; n)). Consider *
*the
0 q0
defining cofibre sequence Sn !i M(Z2; n) ! Sn+1 of M(Z2; n) and define x; y 2 A*
* by
0
x = 1i01 jnq+ i0j 0and y = 10 0 :
nq i 1
(In the notation of Theorem 3.6, x = 1w 01 10 u1 and y = 1w 01 .) Clear*
*ly x
has order 4, y has order 2 and yxy = x-1. Therefore A D4.
Now we turn to the group Edim+r#(X) = {f 2 E(X)| f#i = 1, for i dim X + r},*
* where
r 0.
3.8 Theorem. For the Moore space X = M(G; n),
(1) Edim#(X) = E*(X) and
(2) Edim+1#(X) = 1; ifn > 3.
Proof. If G is torsion-free, dim X = n and the theorem follows easily. Now assu*
*me T 6= 0
so that dim X = n + 1.
(1) If f : X ! X, then f#n = 1 , f*n = 1 by the Hurewicz theorem. Therefore
it suffices to show f 2 E*(X) ) f#n+1 = 1. Consider the defining cofibre seq*
*uence
M(F; n) !i X !q M(R; n + 1) of X. By Proposition 2.5, f = 1 + iaq for some a*
* :
M(R; n + 1) ! M(F; n). But f#n+1 = 1 + i#n+1 a#n+1 q#n+1 . However,
G Z2 = ssn+1(X) q#n+1-!ssn+1(M(R; n + 1)) = R;
and so q#n+1 = 0.
10 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
(2) We have Edim+1#(X) E*(X). First note that if G has no 2-torsion, then E*
*dim+1#(X)
= 1 since E*(X) = 1. Now suppose that G has s summands of 2-torsion, s 1. We w*
*rite
T = T 0 P , where T 0is the 2-torsion subgroup and P is the sum of all other p-*
*torsion
subgroups. Then T 0= Z2a1 . . .Z2as. If f 2 Edim+1#(X), then
f = 10 f12f ;
22
where f22 2 E*(M2). Also ssn+2(M1 _ M2) ssn+2(M1) ssn+2(M2) by Proposition 2.*
*4.
Hence by Proposition 2.6,
(x; y) = (x + f12#(y); f22#(y));
for every x 2 ssn+2(M1) and y 2 ssn+2(M2). Thus it suffices to prove
(i) (f12)#n+2 = 0 ) f12 = 0 and
(ii) (f22)#n+2 = 1 and f22 2 E*(M2) ) f22 = 1:
For (i), let MT0 = M(T 0; n) and MP = M(P; n) and so M2 = MT0 _ MP . Then
f12|MP 2 [MP ; M1] = ssn(P ; M1) = Ext(P; F Z2) = 0, and so f12|MP = 0. Now l*
*et MT0 =
N1 _ . ._.Ns, where Nff= M(Z2aff; n) and let kff: Nff! M2 be the inclusion. It *
*suffices
0 q0
to show f12kff= 0 for ff = 1; :::; s. Let M(F0; n) i! M2 ! M(R0; n + 1) be the*
* defining
cofibre sequence of M2 = M(T; n). By Proposition 2.5, there exists a : M(R0; n+*
*1) ! M1
such that f12 = aq0. Let jff: Sn+1 ! M(R0; n + 1) be the inclusion onto the fft*
*h sphere,
ff = 1; :::; s, and let aff= ajff: Sn+1 ! M1. Then aff= (ffl1jn; :::; fflrjn), *
*where ffli = 0 or 1.
Since the diagram
Nff -kff! M2
?? ?
yqff ?yq0 &f12
Sn+1 -jff!M(R0; n + 1)-a! M1
is commutative, it suffices to show that aff#n+2qff#n+2= 0 implies aff= 0, for *
*ff = 1; :::; s.
Since n > 3, the following sequence is exact by Theorem 2.3
0 -! ssn+2(Sn)-iff#-!ssn+2(Nff)qff#--!ssn+2(Sn+1-)!0:
flfl fl
fl flfl
Z2 Z2
Choose x 2 ssn+2(Nff) such that qff#(x) = jn+1. Then
0 = aff#qff#(x)
= aff#(jn+1)
= (ffl1jnjn+1; :::; fflrjnjn+1):
Since jnjn+1 2 ssn+2(Sn) in a generator of Z2, ffli= 0 for all i. Thus aff= 0.
The proof of (ii) is similar and hence omitted.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 11
x4 Co-Moore Spaces
Let G be a finitely-generated abelian group and write G = F T , where F is f*
*ree-abelian
of rank r and T is a finite group. Let n 3 and denote by C(G; n) the co-Moore *
*space of
type (G; n) defined by ae
eHi(C(G; n)) = G i = n
0 i 6= n:
We note that C(G; n) = M(F; n) _ M(T; n - 1). We adopt the following notation i*
*n this
section: Y denotes C(G; n) with n 4, M1 denotes M(F; n) and M02denotes M(T; n *
*- 1).
The prime in M02is to distinguish M02from M2 = M(T; n) of x3.
Given f 2 [Y; Y ] = [M1 _ M02; M1 _ M02] we obtain as in x3, fjk = pjfik , f*
*or j; k = 1; 2,
where f11 2 [M1; M1], f21 2 [M1; M02], f12 2 [M02; M1] and f22 2 [M02; M02]. By*
* Proposition
2.6 the identification of f with the 2 x 2 matrix
f11 f12
f21 f22
is a bijection compatible with multiplication (i.e., composition and matrix mul*
*tiplication).
In this section we investigate E*(Y ), E*(Y ) and Edim+t*(Y ). The discussi*
*on of E*(Y )
and E*(Y ) is completely analogous to that of E*(X) and E*(X) in x3. Therefore *
*we state
most of the results without proof. However, see Remark 4.9 for another approach*
* . Clearly
f 2 E(Y ) corresponds to the 2 x 2 matrix
f11 f12
f21 f22
where f11 2 E(M1), f21 2 [M1; M02], f12 2 [M02; M1] and f22 2 E(M02).
4.1 Theorem. f 2 E*(Y ) () f11 = 1 and f22 2 E*(M02) = E*(M02). Furthermore, if
f; g 2 E*(Y ),
fg = f1 f12 1 g12 = 1 f12+ g12 :
21 f22 g21 g22 f21+ g21 f21g12+ f22g22
4.2 Theorem. f 2 E*(Y ) () f11 = 1, f12 = 0 and f22 2 E*(M02). Furthermore, if
f; g 2 E*(Y ),
fg = f1 0 1 0 = 1 0 :
21 f22 g21 g22 f21+ g21 f22g22
In addition,
(r+s)sM
E*(Y ) Z2;
12 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
where r is the rank of G and s is the number of summands of the form Z2a in G.
Proof. We briefly comment on the last assertion. We have E*(Y ) [M1; M02] E*(*
*M02).
Lrs
By Theorem 3.2, E*(M02) Z2. Now [M1; M02] = ssn(F ; M02) = Hom (F; ssn(M02*
*)) =
rsL
Hom (F; T Z2) = Z2. The result follows.
In the discussion preceding Theorem 4.8 we indicate the generators of E*(Y )*
* in analogy
to Theorem 3.2. Also we discuss E*(C(G; 3)) and E*(C(G; 3)) in Remark 4.10.
Next we consider Edim#(Y ). Note that Y = M1 _ M02has dimension n and so Edi*
*m+t#(Y )
is the set of all f 2 E(Y ) such that f#i = 1 for i n + t.
4.3 Lemma. Edim#(Y ) E*(Y ).
Proof. Let f : Y ! Y with f#n-1 = 1 and f#n = 1. We have the commutative diagram
ssi(Y )-f#i---!ssi(Y )
?? ?
y hi ?yhi
Hi(Y ) --f*i--!Hi(Y );
where hi is the Hurewicz homomorphism. If i = n - 1, hi is an isomorphism, and*
* so
f*n-1 = 1. If i = n, hi is an epimorphism and so f*n = 1.
4.4 Proposition. Consider the element
f = f1 f12 2 E*(Y );
21 f22
where f22 2 E*(M02). Then f 2 Edim#(Y ) , f21 = 0. For f; g 2 Edim#(Y ), the pr*
*oduct is
given by
fg = 10 f12f 1 g12 = 1 f12+ g12 :
220 g22 0 f22g22
Proof. If f 2 E*(Y ), then
f#n (x; y) = (x + f12#(y); f21#(x) + f22#(y));
for x 2 ssn(M1) and y 2 ssn(M02) by Proposition 2.6. But f12# : ssn(M02) = T *
*Z2 !
ssn(M1) = F is zero. Moreover, f22 2 E*(M02) = Edim#(M02) by Theorem 3.8. The*
*refore
f22# = 1 and so
f#n (x; y) = (x; f21#(x) + y):
Hence f#n = 1 , (f21)#n = 0. Now we show (f21)#n = 0 , f21 = 0. Consider t*
*he
homomorphism of the universal coefficient theorem (2.1)
[M1; M02] = ssn(F ; M02) ! Hom (ssn(M1); ssn(M02)) = Hom (F; ssn(M02*
*));
where (g) = g#n . Since Ext(F; ssn+1(M02)) = 0, is an isomorphism. Thus f21#n *
*= 0 ,
f21 = 0. This completes the proof.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 13
Lr Ls2
4.5 Corollary. Edim#(Y ) T Z2 .
s2L
Proof. By Proposition 4.4, Edim#(Y ) [M02; M1] E*(M02). However E*(M02) *
*Z2 by
Theorem 3.2. Furthermore, [M02; M1] = ssn-1(T ; M1) Ext(T; ssn(M1)) Ext(T; F*
* )
Lr
T .
It follows that Edim#(Y ) and E*(Y ) are distinct groups in general. This d*
*iffers from
Moore spaces X since Edim#(X) = E*(X).
By Theorem 4.2, E*(Y ) E*(Y ) and by Lemma 4.3, Edim#(Y ) E*(Y ). We denot*
*e these
inclusion maps by : E*(Y ) ! E*(Y ) and : Edim#(Y ) ! E*(Y ). There are homom*
*orphisms
ae : E*(Y ) ! Hom (F; T ) and o : E*(Y ) ! Hom (F; T Z2) defined byae(f) = f*n*
*12: F ! T
and o(f) = f21#n : F ! ssn(M0T) = T Z2.
4.6 Proposition. The following are split short exact sequences
0 ! E*(Y ) ! E*(Y ) ae!Hom (F; T ) ! 0 and
0 ! Edim#(Y ) ! E*(Y ) o!Hom (F; T Z2) ! 0:
We make some remarks on the preceding results.
4.7 Remarks. (1) By Theorem 4.2, E*(Y ) is abelian, and E*(Y ) = 1 if and only*
* if G has
no 2-torsion.
(2) The order of the group E*(Y ) is given by |E*(Y )| = |T | x 2(r+s)s. Thi*
*s follows from
either exact sequence of Proposition 4.6.
(3) We easily obtain examples from Proposition 4.6 of co-Moore spaces Y with*
* E*(Y )
6= E*(Y ) and Edim#(Y ) 6= E*(Y ).
(4) We obtain a partial realization result from Proposition 4.6 : Given any *
*finite abelian
group A without 2-torsion, then E*(C(Z A; n)) Edim#(C(Z A; n)) A.
Next we determine the extension of the first exact sequence of Proposition 4*
*.6 (the
second can be obtained similarly) in analogy with Theorem 3.6. We first define*
* certain
generators of E*(Y ). As in x3 we write T = T 0 P and set M0T0= M(T 0; n - 1) *
*and
M0P= M(P; n - 1). Furthermore, T 0= Z2a1 . . .Z2asand we set N0fi= M(Z2afi; n -*
* 1).
Now let xfiff: M1 ! M02and yflffi: M02! M02be defined as the respective composi*
*tions
M1 pff!Sn jn-1!Sn-1 ifi!N0fikfi!M02 and
M02rffi!N0ffiqffi!Sn jn-1!Sn-1 ifl!N0flkfl!M02
for ff = 1; :::; r and fi; fl; ffi = 1; :::; s. Note that the suspension of yfl*
*ffiis the map vflffiof x3.
Then E*(Y ) is generated by the elements
"xfiff= x1 0 and "yflffi= 1 0 :
fiff1 0 1 + yflffi
14 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
Now define zfffi: M02! M1 as the composition
M02rfi!N0fiqfi!Sn jff!M1;
for ff = 1; :::; r and fi = 1; :::; s. In the split exact sequence
0 ! E*(Y ) -! E*(Y ) -ae!-Hom (F; T ) ! 0;
oe
the 2-torsion part of Image oe is generated by all
"zfffi= 10 zfffi1:
Let H E*(Y ) be the subgroup generated by all "xfi0ff0, "yflffiand "zfffi. The*
*n the action of
Hom (F; T ) on E*(Y ) is given by
(1) "z-1fffi"xfi0ff0"zfffi= x1 0 and
fi0ff01 + xfi0ff0zfffi
(2) "z-1fffi"yflffi"zfffi= "yflffi:
In (1), if ff 6= ff0, xfi0ff0zfffi= 0; if ff = ff0, xfi0ff0zfffi= yfi0fi. Furth*
*ermore, Hom (F; P ) acts
trivially on E*(Y ).
Lr
4.8 Theorem. The group E*(Y ) is isomorphic to H P . Moreover, H is the sem*
*i-
direct product of E*(Y ) with Hom (F; T 0) with the action of the latter group *
*on the former
group given by (1) and (2). In addition, nilE*(Y ) 2 and E*(Y ) is abelian if *
*and only if
r = 0 or s = 0.
4.9 Remark. It follows from xx3 and 4 that for n > 3,
E*(M(G; n)) E*(C(G; n)) and
E*(M(G; n)) E*(C(G; n)):
We explain this by appealing to S-duality theory (for more details, see [Sp1] a*
*nd [Sp2,
pp. 462-63]). First note that under suspension
E*(M(G; 3)) E*(M(G; 4)) . . .and
E*(M(G; 3)) E*(M(G; 4)) . .:.
Thus we define stable groups of equivalences by Est*M(G) = E*(M(G; n)) and E*Ms*
*t(G) =
E*(M(G; n)), for all n 3. Similarly for co-Moore spaces C(G; n) with n 4, we *
*obtain
stable groups E*Cst(G) and Est*C(G). Now let X = M(G; n) be a Moore space and *
*X*
be the N-dual of X. Then X* is a co-Moore space C(G; N - n). Furthermore, for*
* any
spaces A and B, the group of stable homotopy classes {A; B} is isomorphic to th*
*e group
{B*; A*} under an isomorphism that reverses composition. From this we easily ob*
*tain that
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 15
Est*M(G) is anti-isomorphic to E*Cst(G). Thus we conclude that E*(M(G; n)) E*(*
*C(G; n)).
By starting with a co-Moore space and taking its N-dual, we similarly conclude *
*that
E*(C(G; n)) E*(M(G; n)).
4.10 Remark. We comment on E*(C(G; n)) and E*(C(G; n)) when n = 3. Then Y =
C(G; 3) = M1 _ M02, where M1 = M(F; 3) and M02= M(T; 2). Theorems 4.1 and 4.2 a*
*nd
Proposition 4.6 hold for Y = C(G; 3) with the exception of the direct sum decom*
*position
of E*(Y ) in Theorem 4.2. To obtain a decomposition for E*(Y ) we proceed as i*
*n the
proof of Theorem 4.2 and conclude that E*(Y ) Hom (F; ss3(M02)) Ext(T; ss3(M0*
*2)). But
ss3(M02) = (T ), where is Whitehead's quadratic functor (see [Wh] and [Ba, p. *
*268]).
Thus we have
iMr j
E*(Y ) (T ) Ext(T; (T )):
Next we consider Edim+1#(Y ). Note that [N0ff; Sn] = ssn-1(Z2aff; Sn) *
*Ext(Z2aff;
Z) Z2aff, where N0ff= M(Z2aff; n - 1), and a generator of this group is the pr*
*ojec-
tion qff: N0ff! Sn. Thus elements of [N0ff; Sn] can be represented as mffqff; *
* mff=
0; 1; :::; 2aff- 1.
4.11 Proposition. If n > 4, then the group Edim+1#(Y ) consists of all f = 10*
* f121 ,
where f12 : M02! M1 is such that pfif12kff: N0ff! Sn is mfffiqff, for mfffi= 0;*
* 2; :::; 2aff-2,
and ff = 1; :::; s and fi = 1; :::; r.
Proof. Let f = 10 f12f 2 Edim#(Y ), where f22 2 E*(M02). Then for x 2 ssn+1(M*
*1) and
22
y 2 ssn+1(M02),
f# (x; y) = (x + f12#(y); f22#(y)):
Thus f 2 Edim+1#(Y ) , f12#n+1 = 0 and f22#n+1 = 1. But the latter is equivale*
*nt to
f22 2 Edim+1#(M02). Therefore this is equivalent to f22 = 1 by Theorem 3.8 sin*
*ce n > 4.
Now consider f12 : M02! M1 and set h = f12. Then M02= M0T0_M0P= N01_. ._.N0s_M0P
and we consider h|M0P : M0P! M1. We have that ssn+1(M0P) = 0 since it is an ext*
*ension
of Tor(P; Z2) by P Z2 according to x1. Therefore (h|M0P)#n+1 = 0. Hence h#n+1 *
*= 0 if
and only if (pfihkff)#n+1 = 0,
N0ffkff!M02h!M1 pfi!Sn;
ff = 1; :::; s, fi = 1; :::; r. Now consider the defining cofibre sequence Sn-*
*1 !iffN0ffqff!Sn
and the corresponding exact sequence (2.3)
0 -! ssn+1(Sn-1 )iff#--!ssn+1(N0ff)qff#--!ssn+1(Sn)-!0:
flfl ? fl
fl ?y(pfihkff)# flfl
Z2 ssn+1(Sn) Z2
16 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
We have pfihkff= mfffiqfffor some mfffi= 0; 1; :::; 2aff- 1. It follows that (p*
*fihkff)# iff#=
mfffiqffiff= 0. Choose z 2 ssn+1(N0ff) such that qff#(z) = jn 2 ssn+1(Sn). Then
(pfihkff)# (z)= mfffiqff#(z)
= mfffijn:
Thus (pfihkff)# = 0 if and only if mfffiis even. Therefore those maps f12 : M*
*02! M1
such that (f12)#n+1 = 0 are precisely those such that pfif12kff = mfffiqfffor *
*mfffi=
0; 2; :::; 2aff- 2, where ff = 1; :::; s and fi = 1; :::; r.
We have T = T 0 P , where T 0= Z2a1 . . .Z2as, and we denote by 2T 0the subg*
*roup
of T 0which is represented in each component by an even integer.
4.12 Corollary. For n > 4,
M r M r
Edim+1#(Y ) (2T 0 P ) (Z2a1-1 . . .Z2as-1 P ):
T dim+t
For any space X, let E1#(X) = E# (X). Then the following question is s*
*uggested
t0
by Corollary 4.12: Is E1#(Y ) = 1 for every co-Moore space Y ?
Finally, we consider the co-Moore space C(Z Z2; n) to illustrate many of th*
*e results
of this section.
4.13 Corollary. If Y = C(ZZ2; n), then E*(Y ) D4, the dihedral group of 8 elem*
*ents,
E*(Y ) Z2Z2, Edim#(Y ) Z2Z2 and Edim+1#(Y ) = 1. Furthermore, E*(Y ) and Edim*
*#(Y )
are distinct subgroups of E*(Y ).
x5 Spaces with E* = E*
Recall that a finite CW-complex X is called a Poincare complex of dimension *
*n if
(i)Hi(X) = 0 for i > n.
(ii)Hn(X) Z generated by .
(iii)The homomorphism P : Hq(X) ! Hn-q(X) defined by P (x) = x \ is an isom*
*or-
phism for all 0 q n.
5.1 Proposition. If X is a Poincare complex, then E*(X) = E*(X).
Proof. If f : X ! X, then f*(f*(x) \ ) = x \ f*(). Now let f 2 E*(X). Then
P (f*(x))= f*(x) \
= x \ f*()
= P (x)
and so f*(x) = x. Therefore f 2 E*(X) and so, E*(X) E*(X). For the opposite
inclusion, we conclude from the universal coefficient theorem [Sp2, p. 248],
0 ! Ext(Hn+1 (X); Z) ! Hn(X) ! Hom (Hn (X); Z) ! 0;
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 17
that Hn(X) Hom (Hn (X); Z). Thus if f : X ! X and f*n = 1, then f*() = . Now
let f 2 E*(X). Then
f*(P (x))= f*(x \ )
= f*(f*(x) \ )
= x \ f*()
= P (x):
Thus f* = 1, and so E*(X) E*(X).
5.2 Corollary. (1) If X is an H-space, then E*(X) = E*(X).
(2) If X is a compact, oriented manifold, then E*(X) = E*(X).
Proof. (1) Browder proved that X is a Poincare complex [Ka, p. 31].
(2) By the Poincare duality theorem [Mas, p. 365], X is a Poincare complex.
Now let X be a space and for every n > 0 write
Hn(X) = Fn Tn;
where Fn is a free-abelian group and Tn is a finite group.
5.3 Proposition. (1) If for every n > 0, Hom (Fn; Tn-1) = 0, then E*(X) E*(X).
(2) If for every n > 0, Hom (Fn; Tn) = 0, then E*(X) E*(X).
Proof. (1) We have Hom (Hn(X); Z) Fn and Ext(Hn-1(X); Z) Tn-1. Thus the uni-
versal coefficient theorem for cohomology yields the short exact sequence
0 ! Tn-1 ! Hn (X) ss!Fn ! 0:
If f 2 E*(X), we have the commutative diagram with exact rows
0 -! Tn-1 -! Hn (X) -ss!Fn -! 0
?? ? ?
y1 ?yf* ?y1
0 -! Tn-1 -! Hn (X) -ss!Fn -! 0:
Then the difference f* - 1 = ss, for some homomorphism : Fn ! Tn-1. Thus if
Hom (Fn; Tn-1) = 0, f* = 1, and so E*(X) E*(X).
(2) This is proved as in (1) using the universal coefficient theorem which e*
*xpresses
homology in terms of cohomology (see the proof of Proposition 5.1).
The following corollary gives two extreme cases of Proposition 5.3.
5.4 Corollary. (1) If Hi(X) has no torsion for all i > 0, then E*(X) = E*(X).
(2) If Hi(X) is a finite group for all i > 0, then E*(X) = E*(X):
18 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
x6 Homotopy Equivalences Which Induce the Identity on Homotopy Groups
From earlierTsections it is not clear if there exist spaces X with E1#(X) 6=*
* 1, where
E1#(X) = Edim+t#(X). We show that such examples exist.
t0
6.1 Proposition. If X = Sm x Sn, with n > m 2, then the group Edim#(X) is abel*
*ian
and equals Edim+t#(X) for all t > 0.
Proof. We use the exact sequence of [Sa, p. 71]
ssm+n (X) ! E(X) ae!E(Sm _ Sn);
where ae is given by restriction to the n-skeleton and is defined as follows: *
*Consider the
cofibre sequence
Sm _ Sn !i X !q Sm ^ Sn = Sm+n ;
and the corresponding pinching map ` : X ! X _ Sm+n . Then if z 2 ssm+n (X), (z*
*) is
the composition
X !` X _ Sm+n 1_z!X _ X r! X;
where r is the folding map. We first show ae(Edim#(X)) = 1: Let f 2 Edim#(X) a*
*nd let
i1 : Sm ! Sm _ Sn and i2 : Sn ! Sm _ Sn be the inclusions. Then it suffices t*
*o show
ae(f)i1 = i1 and ae(f)i2 = i2. But
i# (ae(f)i1)= f# (ii1)
= ii1
= i# (i1):
Since i# : ssn(Sm _ Sn) ! ssn(X) is an isomorphism, ae(f)i1 = i1. Similarly ae(*
*f)i2 = i2.
Thus
Edim#(X) Kernelae = Image :
Therefore any element of Edim#(X) is of the form (z) = r(1 _ z)`, for some z 2 *
*ssm+n (X).
To complete the proof it suffices to show (z)#k = 1 : ssk(X) ! ssk(X) for all k*
*. It is
known that the following diagram commutes
Sm _ Sn ---i-! X
?? ?
yi ?yj1
X ---`-! X _ Sm+n ;
where j1 is the inclusion. Furthermore i# : ssk(Sm _ Sn) ! ssk(X) is onto for a*
*ll k. Thus
if y 2 ssk(X), y = i# (x) for some x 2 ssk(Sm _ Sn), and so
(z)# (y)= r# (1 _ z)# `# i# (x)
= r# (1 _ z)# j1# i# (x)
= i# (x)
= y:
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 19
The proof shows that Image = Edim+t#(Sm x Sn) for all t 0. From [Sa, Thm. *
*2.6,
Lem. 4.1 and (5.1)] we obtain
6.2 Corollary. If n > m 2, then
E1#(Sm x Sn) Edim#(Sm x Sn)
(ssm+n (Sm )=[m ; ssn+1(Sm )])(ssm+n (Sn)=[n; ssm+1;(Sn)])
where k 2 ssk(Sk) is the identity map and the brackets denote Whitehead product*
*s. In
particular,
E1#(S2 x Sn) ssn+2(S2) Z2 and
E1#(S3 x Sn) ssn+3(S3) ssn+3(Sn):
Next we show that Edim#(X) can be infinite while Edim+1#(X) is trivial.
6.3 Proposition. If X = CP n_ S2n, then Edim#(X) Z and Edim+1#(X) = 1.
Proof. Let i1 : CP n! X and i2 : S2n ! X be the inclusions and p1 : X ! CP nand
p2 : X ! S2n the projections. If f 2 [X; X], we write frt = prfit, for r; t = *
*1; 2. Let
u 2 H2(CP n) = Z and v 2 H2n(S2n) = Z be generators and let x = p*1(u) 2 H2(X) *
*and
s = p*2(v) 2 H2n(X). Then H2n(X) = ZZ and is generated by xn and s. If f 2 Edim*
*#(X),
then f*(s) = kxn + `s for k; ` 2 Z. But f#2n = 1 and so f22 = 1. Thus ` = 1 and*
* hence
f*(s) = kxn + s:
Define : Edim#(X) ! Z by (f) = k. Note that if f 2 Edim#(X), f*(x) = x since *
*f#2 = 1.
We now show is a homomorphism: Given f; f0 2 Edim#(X), where f0*(s) = k0xn + *
*s.
Then
(ff0)*(s)= f0*(kxn + s)
= kxn + (k0xn + s)
= (k + k0)xn + s:
Thus (ff0) = (f) + (f0). Now we show that is a monomorphism: By Proposition
2.4 and the fact that [S2n; CP n] = 0, we have that f; f0 2 [X; X] are equal , *
*frt= f0rt
for (r; t) = (1; 1); (2; 1); (2; 2). But if f; f0 2 Edim#(X), then f22 = 1 = f0*
*22. Furthermore,
f*(x) = x = f0*(x) and so f11 = f011. Thus f; f0 2 Edim#(X) are equal , f21 = *
*f0212
[CP n; S2n]. But the latter holds if and only if f*21(v) = f0*21(v). Now f*21(v*
*) = (p2fi1)*(v) =
i*1f*(s) = i*1(kxn + s) = kun and similarly f0*21(v) = k0un. Therefore (f) = k *
*= k0= (f0)
implies f = f0. Finally we show that is an epimorphism: Given k 2 Z, let g 2 [*
*CP n; S2n]
be such that g*(v) = kun. Then define f 2 Edim#(X) by f11 = 1, f12 = 0, f21 = *
*g and
f22 = 1. Clearly (f) = k. Thus Edim#(X) Z.
20 MARTIN ARKOWITZ AND KEN-ICHI MARUYAMA
To see that Edim+1#(X) = 1, we use the Sullivan minimal model M of X [G-M]. *
*Let
M(k) M be the subminimal algebra of M generated by all free algebra generators
of degree k. Then Ek#(M) is the group of homotopy classes of differential gra*
*ded
algebra automorphisms OE : M(k) ! M(k) such that for every free algebra generat*
*or x
of M(k), OE(x) = x + O, where O is a decomposable element of M(k). With subscr*
*ipts
denoting degree, the free algebra generators of M(2n + 1) are u2, v2n+1, x2n an*
*d y2n+1
with differential d given by du = 0, dx = 0, dv = un+1 and dy = xu. Then each e*
*lement of
E2n+1#(M) is represented by a differential graded algebra automorphism OEq : M(*
*2n + 1) !
M(2n + 1) which is the identity on u, v and y and such that OEq(x) = x + qun fo*
*r some
rational q 2 Q. Hence
xu = dy
= d(OEq(y))
= OEqd(y)
= (x + qun)u
= xu + qun+1
Thus q = 0, and so E2n+1#(M) = 1. But by [Mar], there is a rationalization homo*
*morphism
Edim+1#(X) ! E2n+1#(M). It follows that Edim+1#(X) is a finite group. Since Edi*
*m+1#(X)
Edim#(X) = Z, we have that Edim+1#(X) = 1.
We mention another example, taken directly from [A-L, Prop. 6.3].
6.4 Example. If U(`) denotes the unitary group and X is the homogeneous space
U(n)=(U(n1) x . .x.U(nk));
where n1 . . .nk and n - (n1+ . .+.nk) 2, then Edim+t#(X) is infinite for eve*
*ry t 0.
We close with a conjecture regarding the stability of the groups Edim+t#(X).
6.5 Conjecture. If X is a finite CW-complex, then there exists an integer N suc*
*h that
Edim+t#(X) = Edim+N#(X)
for all t N.
Clearly this is true if Edim+s#(X) is finite for some s.
References
[Ar] M. Arkowitz, The Group of Self-Homotopy Equivalences - A Survey, Lecture *
*Notes in Mathematics,
Volume 1425 (R. Piccinini, ed.), Springer-Verlag, New York, 1990, pp. 170*
*-203.
[A-L] M. Arkowitz and G. Lupton, On Finiteness of Subgroups of Self-Homotopy Eq*
*uivalences, Contem-
porary Mathematics, Volume 181, American Mathematical Society, 1995, pp. *
*1-25.
[Ba] H. Baues, Algebraic Homotopy, Cambridge Studies in Advanced Mathematics 1*
*5, Cambridge Uni-
versity Press, Cambridge, 1989.
SELF EQUIVALENCES WHICH INDUCE THE IDENTITY 21
[G-M] Griffiths, P. and Morgan, J., Rational Homotopy Theory and Differential F*
*orms, Birkh"auser,
Boston, 1981.
[Hi] P. Hilton, Homotopy Theory and Duality, Gordon and Breach, New York, 1965.
[Ka] R. Kane, The Homology of Hopf Spaces, North Holland Mathematics Library, *
*Elsevier Science
Publishers, Amsterdam, 1988.
[Mar] Maruyama, K., Localization of a Certain Subgroup of Self-Homotopy Equival*
*ences, Pac. J. of Math.
136 (1989), 293-301.
[Mas] W. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathem*
*atics 127, Springer-
Verlag, New York, 1991.
[Ru] J. Rutter, The Group of Homotopy Self Equivalence Classes of CW-Complexes*
*, Math. Proc. Camb.
Phil. Soc. 93 (1983), 275-293.
[Sa] N. Sawashita, On the Group of Self-Equivalences of the Product of Spheres*
*, Hiroshima Math. J. 5
(1975), 69-86.
[Si] A. Sieradski, Twisted Self-Homotopy Equivalences, Pac. J. of Math. 34 (19*
*70), 789-802.
[Sp1] E. Spanier, Function Spaces and Duality, Annals of Math. 70 (1959), 338-3*
*78.
[Sp2] E. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
[Wh] J. Whitehead, A Certain Exact Sequence, Annals of Math. 52 (1950), 51-110.
Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA
E-mail address: Martin.Arkowitz@dartmouth.edu
Department of Mathematics, Faculty of Education, Chiba University, Yayoicho,*
* Chiba,
Japan
E-mail address: maruyama@cue.e.chiba-u.ac.jp