FINITENESS OF SUBGROUPS OF SELF-HOMOTOPY EQUIVALENCES 11

Proof=?: Let v1

E

Vo

be a basis element. Then g(vj)- f(vj)

=

0 and dHi(vj)

+

Hid(v1)

=

Hd(v1)

=

H(v1)

=

0 by Lemma 4.3. Thus

Now let Wk

E

vl

be a basis element. Then

g(wk)

=

Ha(wk)

=

H(wk

+

r(wk)

+

Zk) by Lemma 3.2, where Zk E

(Vo)

=

f(wk)

+

H(id(wk)

+

di(wk)) since H(zk)

=

0 by Lemma 4.3

=

f(wk)

+

dHi(wk)

+

Hid(wk)·

~:

If Vj E Vo is a basis element,

Ha(v1)

=

H(vj

+

Vj)

=

f(vj)

+

Hdi(v1)

=

f(vj)

+

dHi(vj)

+

Hid(vj)

=

g(vj)·

Note that g( Vj)

=

f( Vj)

+

dHi( Vj) so that H( Vj)

=

0. If Wk

E

vl

is a basis element,

then

Thus

H

ends at

g.

with Zk

E

(Vo)

=

f(wk)

+

H(di(wk)

+

id(wk)) since H(zk)

=

0

=

f(wk)

+

dHi(wk)

+

Hid(wk)

=

g(wk)·

0

4.5 Corollary With the hypothesis of the previous proposition, if

H

is a homotopy

from

f

to

g,

then

0

We now show by example that the converse of Proposition 3.3(ii) is false. We

give examples of maps whose obstruction to being homotopic to the identity { does

not vanish, but which are homotopic to {.

4.6 Example

Let

M

=

A(v2n-l,v2n,W4n-l)

with subscripts denoting degrees,

d(v2n-d

=

0

=

d(v2n) and d(w4n-l)

=

v~n·

Note that M is the minimal model

of S

2n-l

X

S2n.

Then M is 2-stage with Vo generated by

V2n-l,

V2n and

vl

Proof=?: Let v1

E

Vo

be a basis element. Then g(vj)- f(vj)

=

0 and dHi(vj)

+

Hid(v1)

=

Hd(v1)

=

H(v1)

=

0 by Lemma 4.3. Thus

Now let Wk

E

vl

be a basis element. Then

g(wk)

=

Ha(wk)

=

H(wk

+

r(wk)

+

Zk) by Lemma 3.2, where Zk E

(Vo)

=

f(wk)

+

H(id(wk)

+

di(wk)) since H(zk)

=

0 by Lemma 4.3

=

f(wk)

+

dHi(wk)

+

Hid(wk)·

~:

If Vj E Vo is a basis element,

Ha(v1)

=

H(vj

+

Vj)

=

f(vj)

+

Hdi(v1)

=

f(vj)

+

dHi(vj)

+

Hid(vj)

=

g(vj)·

Note that g( Vj)

=

f( Vj)

+

dHi( Vj) so that H( Vj)

=

0. If Wk

E

vl

is a basis element,

then

Thus

H

ends at

g.

with Zk

E

(Vo)

=

f(wk)

+

H(di(wk)

+

id(wk)) since H(zk)

=

0

=

f(wk)

+

dHi(wk)

+

Hid(wk)

=

g(wk)·

0

4.5 Corollary With the hypothesis of the previous proposition, if

H

is a homotopy

from

f

to

g,

then

0

We now show by example that the converse of Proposition 3.3(ii) is false. We

give examples of maps whose obstruction to being homotopic to the identity { does

not vanish, but which are homotopic to {.

4.6 Example

Let

M

=

A(v2n-l,v2n,W4n-l)

with subscripts denoting degrees,

d(v2n-d

=

0

=

d(v2n) and d(w4n-l)

=

v~n·

Note that M is the minimal model

of S

2n-l

X

S2n.

Then M is 2-stage with Vo generated by

V2n-l,

V2n and

vl