Yoshihiro,Saito

Osaka City University Repository

Jou rn al of th e In stitu te  of P olytechnics, 
Osaka C ity U n iv ersity ,
V ol. 6, No, I, Series A
On the homotopy groups of Stiefel manifolds
Yoshihiro S a ito  
(Received March 19, 1955)
J. H. C. Whitehead [3] gave a cellular decomposition of the Stiefel manifold 
Vk+m, m of m-frames in Euclidian (^ +  m)-space, and he and Baratt-Peacher [ I ]  
determined the homotopy groups izh-vj (Ffe+m, m) for i  2, 3.
In the present paper, by making use of the above J. H. C. Whitehead’s result and 
the Steenrod square, we shall gave a reduced cell complex of Pl-\ and determine 
for y < 5  (^ > y  +  2).
The basic tools used are following:
i)  {W hitehead'S theorem) [3 theorem 3]. If r<C2 k, then TVr m) (P l- i) ,  
I =  Min (r +  1, n - 1 ) ,  k ^ n - m ,  where is a space obtained from the /-dimensionul 
projective space by shrinking its (^ - 1 )-dimensional hyperplane to a point.
ii)  (Squaring formula'). If we denote by u^  the generator of ^ 2), 
we have  ^y j for / + 7 ^ / ,  where j is the binomal coefficient with the 
usual conventions.
I am deeply grateful to Prof. H. Toda for his kind advices during the prepara­
tion of this paper.
I. Notations.
We shall use the following notations throughout this paper.
: the space obtained from the /-dimensional projective space by shrinking its 
( ^ - l)-dimensional hyperplane to a point.
We denote (P?z-i), 7r^ +r (PVX, by tt?, respectively.
Let K  =  L y   ^ be a complex such that e^ '^  ^ is attached to L by a mapping / .
A map^g'"'*'^: -> (K, L) is defined as follows, where ^ is a map
to S^\ge^'^^ maps in e^ ~^  ^ by the suspension of g, in K  by f^g,
Now if f - g  is a nullhomotopic in L, we denote by ge'"^ '^ : ^  K
the following map: maps E^ '^^  in e^^  ^ by the suspension of g, and
ge'^ ' '^^ \E^ '  ^ is a null homotopy of f-g .
{ge'^^^}q, {ge'^ '^ }^q are cyclic subgroups of order q of L), 7T^+i(K) which
are generated by {ge^^^} whose representatives are ge^'^\ ge'^ '^  ^ respectively.
We denote the generators and these representatives of tt^+i (S^), tt^ + 2  (S^)^ 
7t^ ^ 3  by the same letters 7^ , s, v respectively.
We denote the w-dimensional cell of P«_i and the generator of {Pk-iyZz) 
by the same letter e"*.
2. Reduced complex of Pnt{,
Let /^-dimensional CW-complexes K, L  be of the same homotopy type. Consider 
two complexes =  and U  =  where a  and /3 are characteristica 13
mappings of and respectively. If / - a  is homotopic to /9 in L, then /3 is 
homotopic to a  in K, and U, W  are of the same homotopy type. Therefore if we 
can construct a reduced complex of , then L :=  K ^  ^ reduced complex
J n + 1
of P'nZi^ '^  where is a map representing an element of n^+i (i^ ). Now Pll-i is 
an /^-sphere. Hence by the determination of the homotopy class of the characteristic 
map for each i, we can determine the homotopy type of PlHi •
Throughout this paper we use the notation P^tl to denote a reduced complex of
JDn+f.r  n^l ,
3. The fiomotopy type and homotopy groups of PvLi,
We consider the following diagram
40 Yoshihiro S a i t o
j[^
In this diagram, the sequence--------> ^  - --- > jtr-i (Jr-i------> • * • are
exact.
Divide the following 4 cases.
Case I :  n=4=l.
In this diagram we have — (S^) and a^r^n^^r-i By a pro­
perty of the projective space, is attached to by a mapping of degree 0 . 
Therefore we have P?,ll =  ^ tc\ =  {e'^ -^ }^oo+ a n d  n\~ {7}e^^^}2+{ee'^)2-
Since and S q V = O , covers with a mapping of degree 2
and does not cover e .^ Hence where suspended space
of the projective plane whose homotopy groups are studied by H. Toda.^^
I) See [2], p. 79.
We have Image /1 =  0, o^ -=^  and Image 6*|:=0.
These imply that ttI is isomorphic to ti\.
We consider the characteristic map of Since does not
cover And we may suppose that does not cover This fact is proved
as follows: Since the fibre bundle ^^+4 , 4/F„+s, 3 has a cross-section for
n ^ A ip  we have (PJ?!?) =  (F .+b , 3) ^
+  {ee'^)2 . If covers then {ee^} is a nullhomotopic map in Tr^ . This is a
contradiction. Then we have Plt\ ^  ^ v 5 ^4-3^
We have n l ~  {7}e^ '^ }^2-^  r^f == Image
0| =  O and Kernel 01 . These implies that Tri =  {^^">24.
Since Sq^e '^^^4^0 and SqV+^ 4^0, covers with a mapping of degree 2
and covers by 77.
Consider the following two cases.
i ) / =  O mod 2. Since Sq^e  ^=  0, covers by ki> with even k. However 
the homotopy type of Pnt\ depends only on k mod 2. This is proved as follows: 
Let yw-2 V5 w+3) ^  e^ '^ \ where covers with a mapping of degree
2, by Ty and by ku, Let K' =  ^ v ^  covers 
^/«4-3 ^ mapping of degree 2 , by 7^  and by with k' 2a, Consider 
a map / :  (S« v yn+2 vs^-i-3) _  (5 m v y^«4-2 vs^«+3) such that i) /I maps in with 
degree I, ii) f\ maps in with degree I, iii) /m aps the upper hemi-sphere 
E%'^  ^ of in with degree I and maps the lower hemi-sphere in by 
ai^ . Consider also a m a p f :  ^ 7 ^ ^ + 2  v 5 «^+3) ^  (5 « v y ^^+2 v s^+^) g^^h that 
/m aps are mapped in S'", with degree I respectively and
is mapped in by — av. Then f - f\  f - f  are homotopic to identity maps of 
Pn-\, P'nt\ respectively. On the other hand, /  and f '  are extended to mappings 
g : K —^K\ g ' :K ' - ^ K  such that g'g\ g'-g  are homotopic to identity mappings of 
K, respectively. Thus we may suppose that does not cover e .^
Since Image dt ^  , we have ;ri =: K + ^ jg + {^^^!24.
ii)  / =  1  mod 2. Since Sq"^ e”#=0, covers e” by kv with odd k. By the same 
reason as in the case / =  O ( 2 ), the homotopy type of Pn-i depends only on k mod 2 . 
Thus we may suppose k =  l. Hence we have Image ^ 3  and 
7Tt=K+^}48+{^^"-^^}4.
We have ~  {ve^+^}24, (7§ =  , Image ^ 5  =  2vg^+\ „  |-g«+2| y|
onto-homomorphism. Thus we have nl =  {ee '^^ }^2-\-
Since Image ^s =  O, a\— j\ is an onto-homomorphism and 2{y^ e^ '^ }^ =O  
in 7t2 , we have nl =  {y^ e'^ -^ )^2+  [ee^^^]2+
We have  ^ Image and Image j t  =  0. These imply that
7rt =  {77^"-^n2+{v^"^^}2.
On the homotopy groups o f  Stiefel manifolds 41
2) For example, see Steenrod’s book; The topology of bibre bundles, p. 142,
42 Yoshihiro Sa it o
Since and =  0 for / =  0, covers by ti and by kv
with even k. However is nullhomotopic in n\, and so we may suppose that
does not cover
Since SqV+^4=0 and SqV+^4=0 for / ^ 1 , we may suppose that ^ ^  covers 
by Tj and by v.
Therefore we have ttI =  for both cases.
Since TTs =  O and Kernel y| =  we have ttI =
We have also Image ^i =  O, 6%  ^{se^ ^^ } and j l  is an onto-homomorphism. These 
imply that nl =  { 12 i>e^+^2-
Since Image o’! =  and Image j t  =  0, we have ttI =
We have Image =  and Kernel =  {2^^+^}. These imply
that nl^{2e^-^^)oo ior I =  ,^ ^{2e^^^ +  ve^ -^ }^oo for / =  1 , where / =  (2 ^^ +  ^+  v^^+"): 
5 «+5 _^pw+5 ^ mapping such that the upper hemi-sphere of is mapped in 
^ «+ 5  degree 2  and the lower hemi-shere E !I"  ^of in by the suspension of v.
Since SqV+^H=O, is attached to P ^ l  by a generator of Tif. Thus we have 
Ttt =  O for both cases.
Summing up the above, the homotopy type of P^lf is described as follows:
n^+2
2
, i V +2 \2pn+5
Case 2: n — Al+1,
In this case PV-\ is of the same homotopy type as the complex K  obtained by 
shrinking e^ ~~^  to a point in PJ^i-
Therefore the homotopy type of Pn-i is described as follows:
gW + l
gW+5
2
W^ + 4
I =  I
ii' 2 i
On the homotopy groups o f  Stiefel manifolds 43
Then we have:
=  7ri:=::{^^"+n2+{e^"^^}2+{^^n2, 4 = { 7 ^ e ^ + % + , 7rl={i^e-h,
7Tl =  { 1 2 ^e^+^}2 ,^  ^ 7r| =  {£g^+2} 2 + { 1 2 v6^+i}2 , ttI=  {ee^+^}^, =  {2e^-^^}^ for / ^ 0 , 
=  +  for / ^ 1 ,  7tI=:0, TT^ ==O, Tii =  {^^^+^24, 7ri =  {v6^+n2 , t t I^ tc I^ t iI .
Since Sq^e^^^ =  0 for / =  O and 4=^ 0 for / = 1 ,  is attached to by ki^ , 
where k is even for / =  O and k is odd for / =  1. However =O  in Trf, thus
we have the following, the homotopy type depends only on k mod 2 , Trf =: 
for / =  O and Trf =  O for / =  1 .
Case 3: w =  4/ +  2 .
We obtain P^\  from Pl-\ by the same manner in the preceding section. The 
homotopy tope PJ^f is described as follows:
=  O I = I
e,W +  4 gW +  4
I
2
^ n + 2
e>W, +  3 V
g W + 2
I
Tl 2 V Tl 2
/ ' \ \
,77, ,g?7. +  l  gW +  5
I I
qH
I
i ,n+l.
Then we have:
7Ti =  K+^}oo+{^^n2, 7rl={;y^"-^^},+ {£^n2, 7tl^{y}e^^^)2. Tli=O,
n\ =  {e ^ " -^ ^ } 2 +  Tri =  { v ^ n i 2 ,  ^ri =  { 2^ " + n c o +  { v ^ " } i2 ,  ^rf =  12,
nl {i^ e^ '^ }^ 2i y Trl =  2 , 7r|;=5^  7r J ^ 7r|, 7r| =  for / =  0 , Trf =  O for 1 =  1,
nl =  0, Tri =  { 1 2 v^^+2} 2 , 7il=:{~ee^+^ +  6ue^-^^},, n l =  +  where g=,(^e^+^
+  E T ^ ^  P^lt is a mapping such that g\ET^:=^e^+^ and g\E^^^
=  3ue^ '^ .^ We have g\E%^  ^=  {y^ , 2, 7^ } \ where {y},2,7i) is a Toda’s construction 
and is known { 7^ , 2 , yf\ =  db6 v (See [2, Chap. 5]), and so g  represents an element of 
TT. + 5  C m D , Tri =  K + ^ . +  te }8 for / ^ 0 , Tr| =  {2 "^^  ^+  v "^+ }^oo+ { ^ } 8  for / ^ 1 .
Since Sq^e'^+^^^O and SqV+^ 4=0, is attached to by a mapping of degree
2 and to by Thus we have Trf =  for / =  O and Trf =  {2e^+  ^+  v^ ” + ^ }8
for / =  1 .
Case 4: n = 4 / + 3 .
The homotopy type of Plt\  is described as follows:
44
I =  I
Yoshihiro Sa it o
I
for
nl =  {'ee”^^  + 6ye”+^}„ nl =  +  4 =  K + ‘}^ +
2 IUi / = 0 ,
+ 3
The we have:
7ri =  {ve^}2. TTf==O, ; r i : = { 2 ^ - + n c + , 7rl={ve--^%, nl={pe^},
+ ttI==
7Ti =  {12ue^ ^^ }„^ ^
+ 3v^^+i}s for / =  0, Kt =  + + for / =  1, nl =  {e^+^}i, for
/~0, nl =  {2e^^  ^+ ue^^ }^s for /^1, nk =  0, nl =  , nl =  {ee -^^ }^,,
nt =  {se^ ^^ }^  + tzI =  {y]e^ ^^ }^ .
Since S q V + ^ O  and Sq^ '^^+^^O for / =  0, is attached to P l-I  by 
Since SqV'“^ 4^=0 for 1 =  1, is attached to P ^ i by Thus we
have Trf =: for both cases.
4. The homotopy groups of the Stiefel manifolds.
From the Whitehead’s theorem, we have the tables of the homotopy groups of 
the Stiefel manifolds.
Theorem I. The table o f  the homotopy groups nk+2 (Vk^m, m) is  the
follow ing  :
I 2 3 4 ^  m
4 I ^ 2 + ^ 2 Z 2  +  Z 2 Z 2  +  Z 2
4 /  +  I z . I Z o o  +  Z 4 Z s
4 /  +  2 Z , ^ 2  +  ^ 2 O
4 /  +  3 Z ^ Z , Z o o  - H  Z 2 Z 2
Theorem 2. The table o f  the homotopy groups nk+s (_Vk+m, m) the
follow ing:
k
^  m
I 2 3 4 5 ^  m
4 I
V , -
0
1
Z 2 4 Z 2  + Z 2 4 Z 4  +  Z 2 4 Z 0 0 - I - Z 4 + Z 2 4
Z s  + Z 2 4
I  Z 4  4-  Z 4 8
4 I +  I Z 2 4 Z 2  + Z 2 Z 2 + - Z g H - Z g Z 2  + Z g Z 2
4 I H -  2 Z 2 4 Z 2  + Z 2 4 Z g  +  Z 1 2 Z o o  + Z 1 2 Z i g
4 /  +  3 {!: 01 Z 2 4 Z 2  - f - Z 2 Z g  +  Z g Z 2  + Z g f  Z g  +  Z g  I  Z g
On the homotopy groups of Stiefel manifolds 45
Theorems.  The table of the homotopy groups 71^ +4 ( m) ( ^ > 6) is the 
following:
k
m I 2 3 4 5 6 ^  m
4 I
4 I + I
0
0
^ 2 4 Z 2  +  Z 2  
Z2 4 *  Z2
Z 2 + Z 2 + Z 2
Z2
Z2 Z2
Z o o
^ 2
O
4 /  +  2 jr / ^  O L / ^  I 0 ^ 2 4 Z2 Z2 Z2 Z ^  I  O
4 /  +  3 ■r / =  OL / =  I 0 Z2 z. Z^ Z o o  +  Z s r 1^6I  ^ 8
Theorem 4. The table of the homotopy groups ( ^ > 7 )  is the 
following:
m I 2 3 4 5 6 m<7
4 I
4 /  +  I I 
4 I 4- 
4 / -h
I =  O
1 =  I
2 
3
O
O
O
O
O
O
O
O
Z2
-^ 24
Z2
Z24
Z2 +  Z2 
Z2
z,
Z2 -f" Z2
Z2
Z2
Z2-HZ2-HZ2
Zoo
Z2
Zoo-h Zs 
Z2 +  Z2
O
J ^ 2  
I O
1^6
Z2
References
[ 1 ]  M. G. Baratt and G. F. Peacher, A note on Ttr{.Vn,m), Proc. Nat. Acad. Sci, U. S. A. 38 
(1952) 119-121.
[ 2 ]  H. Toda, Generalized Whitehead products and homotopy groups of spheres, Jour, of the 
Inst, of Polytechnics, Osaka City Univ., 3 (1952).
[  3 ]  J. H. C. Whitehead, On the group T t V n , m) ccrid shere bundles, Proc. London Math. Soc., 
48 (1944).


On the homotopy groups of Stiefel manifold

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On the homotopy groups of Stiefel manifold

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