BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 78, Number 5, September 1972
ON THE HAEFLIGER KNOT GROUPS
BY R. JAMES MILGRAM 1
Communicated by E. H. Brown, February 3, 1972
ABSTRACT. Let Ckn be the group of isotopy classes of differentiable
embeddings S" c Sn+k. In [4J A. Haefliger established an isomorphism
C* - nn+l(G,SO, G*) where Gu is the set of (oriented) homotopy
equivalences of the sphere Sk~*. In this note, we indicate methods
which make the calculation of these groups feasible. In particular,
we determine the first seven nonzero groups in the metastable range.
We also develop connections between the composition operation in
homotopy and such geometric operations as spin-twisting knots.
1. The space of the Haefliger knot groups.
THEOREM
A. There is a space Fk which is 2k — 3-connected and
nn(Fk) s C*.
Indeed, Fk is the fiber in the map
SGk/SOk -* SG/SO,
induced by the usual inclusions BGk ^ BG, BSOk<± Bso. Alternately,
Fk is the fiber in the map
SO/SOk -+ G/Gk9
induced by the inclusions BSOk <•» BGk, Bso <± BG.
COROLLARY B. H^(Fk ; Z2) = E(- • • Aj • • •) where I runs over all sequences of integers (il9..., it\ satisfying
(i)0^i1^...^U^2),
(ii) i1 = 0 implies t = 2,
(iii) it^ fc-1.
Moreover, dim^j) is ix + 2i2 + 4i3 + ... + l~Ht - 1. {fiere E is
an exterior algebra on these stated generators.)
B follows from A on applying the results of [7]. In the same way, it
is possible to obtain partial information about ff#(Fk;Zp) for p odd.
Similarly, we can determine H*(Fk ;Z2) as a module over the Steenrod
algebra ^(2), and H*(Fk ; Zp) over <stf(p) in the range of dimensions less
than 3fc - 2.
In this range, H*(Fk;Zp) has one nonzero generator e^-i in each
dimension 4s— 1, and is zero otherwise. For general p, the <s/(p)-structure
AMS 1969 subject classifications. Primary 5520, 5570; Secondary 5540.
1
This research supported in part by the NSF.
Copyright © American Mathematical Society 1972
861
862
R. J. MILGRAM
[September
is involved. However, for p = 3 we have the concise formula
^ 4 s _ 1 ) = r s J(6 4s _ 1+4l .).
The structure of H*(Fk ; Z2) is considerably more complex.
COROLLARY C. In dimensions less than 3k — 2, H*(Fk;Z2) has generators of the form
el u {ej ® eO,
with] ^ k — 1, of dimension 2/ + i — 1 and
(Compare [8, §3] where a space very similar to this is studied.)
COROLLARY
D. (Fk+1 u cFk) ~ y£k+1Pk-i
in our range.
Here Pk-± is the truncated projective space p°°/pfc-2. Also, the map
Fk^> Fk+1 is the natural one which geometrically corresponds to
compositions Sk a Sn+k c Sn+k+i. D gives a homotopy-theoretical
proof of the second main result of [4], namely, the isomorphism
7i n (F,,G,)^7r n _, + 1(SO,SO,_0
for n ^ 3g —6.
Finally, passing to rational homology, the map H%(Q(G/SO)) -» H^(Fk)
is surjective, and we have
COROLLARY E.
r
~
.
-,A.
^
r g, i = 3(4),
0
i > 2k - 3,
*i(n) ® 6 = <
[ 0 otherwise.
In particular, for i = 3(4) and i > 2k-^3,7if(Fk) contains a Z-direct summand.
The identification of the Hurewicz image of these summands is a
major problem in further clarifying the homotopy type of the Fk. Some
results should be possible using the techniques of [3].
2. Some calculations. We use Corollary C to calculate the F2-term
of the Adams spectral sequence for Fk in dimensions less than 3 k - 3
at the prime two. (In the range in which the calculations are carried through,
the p-primary calculations p = 3,5,7... are direct.) The £2-terms for
p = 2 exhibit a type of periodicity
ExOf(2)(H*(Ffc+20,Z2) s
Ext%f{H*{Fk\Z2)
1972]
863
ON THE HAEFLIGER KNOT GROUPS
for t - s < 2r + 2(fe + 2r - 1). The author does not know if this periodicity has any geometric analogue, however.
These Ext groups are calculated by use of the tables in [6] and the
spectral sequence techniques in [8, §9]. After evaluating all the differentials we can obtain
THEOREM F. The first seven two-primary components of the C\ for
k > 7 are
(a) for k ES 2(8),
0
1
z2
r*
2
3
4
5
Z 2 Z®(Z2)2;
z2 z 2
0 Z®Z2
^2fc-3+j
(b)/orfc = 3(8),
0
rk
^2k-3+j
1
2
3
4
z z4 z2
0
z®z2
Z4 © Z 2 or
Z2 © Z2 or;
Z2 © Z 16 or
(c) for fc = 4(8),
0
4
3
1
z 2 Z2®Z2
z2 0
5
6
7
Z8©Z4
Z®Z2
Z2;
(d)/orfc = 5(8),
0
z
r*
1
2
(Z 2 ) 2
5
(Z 2 ) 3
Z16 e Z8 or Z © (Z 2 )
2
6
z. z 2 ;
z 16 e z 4
(e)/or fc = 6(8),
;
r*
0
1
2
3
z2
0
z@z2
0
1
z
Z4
41 5
z2 z2
6
0 Z®Z2;
(f) /or fc = 7(8),
j
k
r
2
3 4
z2 0 1Z
5
6
5ï 8 w z :i®ZAor
Z 4 or Z2 ® Z2 or
%2
%l\
(g) /or k = 0(8), t/ie groups are isomorphic to those for k = 4(8);
(h) for k = 1(8), the groups are isomorphic to those for k = 5(8), except
possibly for j = 3, where, however, the same two groups are the only possibilités.
864
R. J. MILGRAM
[September
Taking into account the 3-primary structure, we obtain
G. Ck = 0 for n < 2/c + 4 if and only if
(a) k = 0(2) and n = 2fc-2,
(b) k = 3(4), k # 1(3), andn = 2fc,
(c) k = 6(8),fc# 0(3), and n = 2k + 2.
COROLLARY
Thus, in these dimensions, a smooth embedding
f:Mn - (point) c Sw+k
implies the existence of a smooth embedding
f\Mn
c S w+k+1 .
REMARK. The ambiguities in F for k = 5(8) and ; = 3 would be
resolved if we knew the image of the Hurewicz map in dimension 2fc.
3. Some operations on the Ck. The map Ck>-+ Ck+1 is obtained by
passing to homotopy in the map h:Fk-+ Fk+l.
PROPOSITION
H. ft* is injective on H\Fk+1\Z^
for i < 3fc-2. In
particular,
h*(er u ej ® ej)k+ x = (er u e> (g) e*)*.
This enables us to obtain information on how far back a particular
embedding desuspends. For example, let g be the generator [3] of Ck2k-$
with k odd. Then
THEOREM
I. (a) Ifk = 1(4), g is not the suspension of any knot S2fc_3c->
S3fc-4
(b) If k = 3(4), g is the suspension of a knot g:S2k~3c-+ S3k~4 bttf is
not the double suspension of any knot s2k~4c+ S3k~5.
Composition defines an action of 7t^(S°), the stable homotopy of
spheres, on the Ck for k > jn. For example, if rj e n^S0) — Z 2 represents
the nontrivial element, then rj0 constructs from a knot /:Swc-> Sn+k a
new knot
fn:Sn+x^Sn+k+1.
On the other hand, Artin-Zeeman [2], [9] defined the notion of
twist-spinning knots, and Hsiang-Sanderson [5] generalized the twistspinning construction considerably.
THEOREM J. (a) Composition with rj corresponds to the Artin-Zeeman
twist-spinning operation.
(b) Composition with J(a) corresponds to the Hsiang-Sanderson construction for y = [1, a]. (Here, J : n^(SO) -> n^(S°) is the Whitehead
J-homomorphism [1].)
1972]
ON THE HAEFLIGER KNOT GROUPS
865
COROLLARY K. (a) Iterating the Artin-Zeeman twist-spinning operation
four times (for k > 2) always results in a trivial knot.
(b) The Artin-Zeeman twist-spinning of gk is never zero. The iteration
is nonzero if and only ifk= 1(4).
BIBLIOGRAPHY
1. J. F. Adams, On the groups J(X), I, Topology 2 (1963), 181-195. MR 28 #2553.
2. E. Artin, Zur Isotopie zweidimensionalen Flâchen im R4, Abh. Math. Sem» Univ.
Hamburg 4 (1926), 174-177.
3. A. Haefliger, Knotted (4k-l)-speres in 6k-space, Ann. of Math. (2) 75 (1962), 452-466.
MR 26 #3070.
4.
, Differentiate embeddings of Sn in Sn+q for q > 2, Ann. of Math. (2) 83 (1966),
402-436. MR 34 #2024.
5. W. C. Hsiang and B. J. Sanderson, Twist-spinning spheres in spheres, Illinois J. Math. 9
(1965), 651-659. MR 32 #6474.
6. M. Mahowald, The metastable homotopy ofSn, Mem. Amer. Math. Soc. No. 72 (1967).
MR 38 #5216.
7. R. J. Milgram, The mod 2 spherical characteristic classes, Ann. of Math. (2) 92 (1970),
238-261. MR 41 #7705.
8.
, Unstable homotopy from the stable point of view, Stanford University, Stanford,
Calif., 1971. (mimeograph).
9. E. G Zeeman, Twisting spun knots, Trans. Amer. Math. Soc. 115 (1965), 471-495.
MR 33 #3290.
DEPARTMENT OF MATHEMATICS, STANFORD UNIVERSITY, STANFORD, CALIFORNIA 94305