Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I

0.22 MB·English

by  Petr M. Akhmet'ev

Geometri
approa
h towards stable homotopy
groups of spheres. The Steenrod-Hopf
invariant
8
0
0 ∗
2 Pyotr M. Akhmet'ev

n
a
J

9
Àííîòàöèÿ
]
T
Inthispaper ageometri
approa
h toward stablehomotopy groups
G
ofspheres,basedonthePontrjagin-Thom [P℄
onstru
tionisproposed.
.
h From this approa
h a new proof of Hopf1I5n,v3a1ri,a6n3t,1O2n7e Theorem by
t
a J.F.Adams [A1℄ for all dimnens>ion1s2e7x
ept is obtained.
m
It is proΠved that for in the stable homotopy group
[ of spheres n there is no elements with Hopf invariant one. The

1 new proof is based on geometri
topology methods. The Pontrjagin-
v Thom Theorem (in the form proposed by R.Wells [W℄) about the
2
representationofstablehomotopygroupsoftherealproje
tivein(cid:28)nite-
1
4 dimensional spa
e (this groups ismapped onto 2-
omponents of stable
1
homotopy groups of spheres by the Khan-Priddy Theorem [A2℄) by
.
1
obordism
lasses of immersions of
odimension 1 of
losed manifolds
0
(generally speaking, non-orientable) is
onsidered. The Hopf Invariant
8
0 is expressed as a
hara
teristi
number of the dihedral group for the
: self-interse
tion manifold of an immersed
odimension 1 manifold that
v
i represents the given element in the stable homotopy group. In the
X
new proof the Geometri
Control Prin
iple (by M.Gromov)[Gr℄ for
r
a immersions inagiven regularhomotopy
lassesbasedonSmale-Hirs
h
Immersion Theorem [H℄ is required.
f : Mn−1 # Rn n = 2l −1
Let , , be a smooth immersion of
odimension
1.
The
hara
teristi
number
(cid:10)wn−1(M);[Mn−1](cid:11) = h(f)
1
∗
This work was supported in part by the London Royal So
iety (1998-2000), RFBR
08-01-00663,INTAS 05-1000008-7805.
1
is
alled the stable Hopf invariant or the Steenrod-Hopf invaMrian−n1t. This

hara
teristi
number depends only on the immersed manifold itself.
The relationship with the de(cid:28)nition of the Steenrod-Hopf invariant in
algebrai
topology is
onsidered in [E℄,[K℄,[L℄.
Theorem (by J.F.Adams), [A℄
l ≥ 4 h(f) = 0
For , .
Skew-framed immersions
f : Mn−k # Rn k κ : E(κ) → Mn−k
Let beMann−ikmmersionΞof:
koκd→imeνn(sfi)on .Let
be a line bundle over and let be an isomorphism of the
f k
normal bundle of the immersion with the Whitney sum of
opies of the
κ
line bundle .
(f,κ,Ξ)
We shall
all tκhe∈trHip1l(eMn−k;Z/2)a sknew-framed immwer(sκio)n with
1

hara
teristi

lassMn−k . (If is odd then is the
orientation
lass of , see [A-E℄ for more details).
The Steenrod-Hopf invariants for skew-framed immersions
(cid:10)w (κ)n−k;[Mn−k](cid:11) = h(f,κ,Ξ)
1
The
hara
teristi

lass is
alled the
(f,κ,Ξ)
Steenrod-Hopf invariant of the skew-framed immersion .
The Main Theorem 1
f : Mn−k # Rn,κ,Ξ) n = 2l − 1
Let ( be a skew-framed immersion, ,
dim(M) = n−k = n+1 +7 n ≥ 255 l ≥ 8
2 . Then for (i.e. for )
h(f,κ,Ξ) = (cid:10)w (κ)dim(M);[M](cid:11) = 0 (mod 2).
1
Corollary
n ≥ 255
Adams' Theorem for .
D
8
4
Let be the dihedral group of order ,
D = {a,b|a4 = b2 = e,[a,b] = a2}.
4
This is the group of symmetries of the two
oordinate axes in the plane. Let
I = {e,a,a2,a3},I = {e,b,a2,a2b},I = {e,ab,a2,a3b}
a b c
2
D Z/4 I
4 a
be the subgroups in of index 2. The
y
liπ
(cid:21)sZub/g2r⊕ouZp/2 is generateId
bIy rotation of the plane through the angle 2. The (cid:21)subgroup b
c
( ) is generated by symmetrZie/s2(or re(cid:29)e
tioIns)∩wIit=h r{ees,pae2
}t to the bise
tors
b c
of the
oordinate axes. The (cid:21)subgroup is generated by
the
entrfal :syMmmn−ektry#. Rn,κ,Ξ)
Let ( Nn−2k be a skew-fraRmnedg g:eNnenr−i2
k i#mmRenrsion. Its
double points mνa(ngi)fold is immersed into , Ψ : ν(g) = kη,∗and the
nη∗ormal bundle admits a
anoniN
anl−d2ek
ompoDsition , wheηre∗
4
is a two-dimensional bundle over E(Dw)i→th K(-Dstr,u1
)ture. The bundle
4 4
is a pηul:l-Nban
−k2ko→f thKe(uDniv,e1r)sal bundle , via the
lassifying
4
map . N¯n−2k → Nn−2k
Let us
onsider the
anoni
al 2-fold
overing over the
g
double poiInt⊂mDanifoIld=of{et,hae2,imabm,ae3rbs}ion . This
overing
orresponds to the
c 4 c
subgrouκ¯p ∈ H1(N¯,n−2k;Z/2) .
Let I → I be the
oIho=m{oelo,gay2 ≃
laasbs}
=orrZe/sp2onding to the
c d d
epimorphism witah3tbhe image κ¯ = i∗(κ) i : N¯n(−th2ke#kerMnenl−iks
generatedbytheelement ).Bythede(cid:28)nition ,
is the
anoni
al immersion of the double point
overing. Let us de(cid:28)ne the
following
hara
teristi
number
h¯(g,η,Ψ) = (cid:10)κ¯n−2k;[N¯n−2k](cid:11).
Lemma 2
¯
h(f,κ,Ξ) = h(g,η,Ψ).
Proof of Lemma 2
ImmediatefromHerbert's Theorem.(Con
erningHerbert's Theorem,see e.g.
[E-G℄.)
De(cid:28)nition (Cy
li
stru
ture for skew-framed immersions)
(f,κ,Ξ) Nn−2k
Let be a skew-framed immersion, be the (odd-dimensional)
f
double self-interse
tion point manifold of . A mapping
µ : Nn−2k → K(I ,1)
a
I = {e,a,a2,a3} f
a
( ) is
alled a
y
li
stru
ture for if
(cid:10)µ∗(t);[Nn−2k](cid:11) = h(f),
t ∈ Hn−2k(K(I ,1);Z/2)
a
where is the generator.
The following lemma is proved by an expli
it
al
ulation.
3
The Main Lemma 3 (jointly with P.J.E

les (1998))
n−k = n+1+7 n−2k = 15 n ≥ 31 µ : Nn−2k → K(D ,1)
Let 2 ( ), , 4 be a
y
li
f h(f,κ,Ξ) = 0 (mod2)
stru
ture for . Then .
Lemma 4. Cy
li
Stru
ture for skew-framed immersions
n ≥ 255 f : Mn−k # Rn n−k =
For , an arbitrary skew-framed immersion ,
n+1 +7
2 is regularly homotopi
to an immersion with a
y
li
stru
ture.
Proof
The proof is a
orollary of Lemma 5 and Proposition 6.
De(cid:28)nition (Cy
li
Stru
ture for generi
mappings of the standard
proje
tive spa
e)
g : RPn−k → Rn n ≥ 3k n = 2l −1
Let Nn−2kbe a generi
mappi(nng−, 2k−1) , with double
(p∂oNin)tnm−2akn−i1fo⊂ldRPn−k and
riti
alpoints -dimensionalsubmanifold
η : (Nn−2k,∂.N) → (K(D ,1),K(I ,1))
4 b
Let be the stru
tured mapping
g

orresponding to . This stru
tured mapping is de(cid:28)ned analogously with the

ase of skew-framed immersions. Obviously, the restri
tion of the mapping
η
to the boundary of the douKbl(eIp,o1i)nt⊂s mKa(nDifo,ld1), i.e. to the
riti
al points
b 4
Isub⊂mIanifold, has the target . The standard in
lusion
d a
asthesubgroup oftheindex2iswell-de(cid:28)ned.We shall
alla mapping
µ : (Nn−2k,∂Nn−2k−1) → (K(I ,1),K(I ,1))
a d
g
a
y
li
stru
ture for , if the following
onditions hold:
(∗)
the homologi
al
ondition:
(cid:10)µ∗(t);[Nn−2k,∂Nn−2k−1](cid:11) = 1 (mod 2),
t ∈ Hn−2k(K(I ,1),K(I ,1);Z/2)
a d
where Hn−2k(K(I ,1)i,sK(tIhe);Zg/e2n)erator in the
a d

okernel of the homomorphism (remark: this
µrel(a[t∂iNven−
2hk−ar1a])
t∈erHisti
nu(mKb(eIr,1i;sZ/w2e)l)l-de(cid:28)ned, by expli
it
al
ulations
∗ n−2k−1 d
is trivial);
(∗∗)
the boundary
ondition :
p ◦η| : ∂Nn−2k−1 → K(I ,1) → K(I ,1),
c ∂N b d
I = {e,b,a2,ba2} p : K(I ,1) → K(I ,1)
b c b d
Id,=wh{eer,ea2 ≃ ba2} µis|∂tNhne−s2kt−a1ndard proje
tion
with the image ,
oin
ides with .
4
Lemma 5. Geometri
Control
Let (f,κ,Ξ), f : Mn−k # Rn
be a skew-framed immersion. Let us assume (see Proposition 6 below) that
there exists a generi
mapping
g : RPn−k → Rn, n ≥ 3k,
with a
y
li
stru
ture
µ : (Nn−2k,∂Nn−2k−1) → (K(I ,1),K(I ,1)).
a d
(f′,κ,Ξ′)
Thenthereexistsaskew-framedimmersion intheregularhomotopy
f

lass of with a
y
li
stru
ture.
The idea of the proof of Lemma 5.
Take the mapping g ◦κ : Mn−k → RPn−k → Rn.
f
By [Gr℄, 1.2.2, in the regular homotopy
lass of there exists a generi
immersion f′ : Mn−k # Rn
C0
de(cid:28)ned as a -
lose generi
regular perturbation ( arbitrary small) of a
(singular) mapping
g ◦κ.
A
onstru
tion by S.A.Melikhov (2004)
J (2l−4+1) = r Z/4
Let usSd7e/niotdeimby(J) =the2lj−o1in+o7f= n−k
opiePsLof the standarid : J ⊂- leRnns
J
spa
e , . There is a (cid:21)embedding
l ≥ 8
for p.′ : Sn−k → J r S7 → S7/i
pˆ :LSent−k/i → J be the joinof
opies of thep′stapnd:aRrPdn
−okve→r J ,
be the quotient mappπing: oRfPn−,k → Sn−k/i pbˆe the

omposition iof◦thpˆe:sStna−nkd/aird→pJro→je
tRionn gˆ : Swn−itkh/i →. TRhne
J

ompoεsition is well-de(cid:28)ine◦d.pˆLedt: RPn−k → Rn
J
be an -small generi
alternation of the mapping , be
ε ε << ε gˆ◦π
1 1
de(cid:28)ned by a -small generi
alternation, , of the
omposition .
5
1
Proposition 6
d : RPn−k → Rn
The Melikhov map is equipped with a
y
li
stru
ture.
The rest of the paper
on
erns the proof of this result. This Proposition with
Lemma 5 implies Lemma 4 and Lemma 4 with the Main Lemma 3 implies
Lemma 2 and the main Theorem 1.
The beginning of the proof of the Proposition 6
Γ RPn−k
0
Let be the delated produ
t of the standard proje
tive spa
e
Γ = (RPn−k ×RPn−k \∆ )/T′,
0 diag
where the quotient is determined with respe
t to the free involution
T′ : RPn−k ×RPn−k \∆ → RPn−k ×RPn−k \∆
diag diag
′
T (x,y) = (y,x).
The
lassifying map η : Γ → K(D ,1)
Γ0 0 4
π (Γ ) = D T′
1 0 4
is well-de(cid:28)ned. (NIote⊂thDat ∆ an⊂d tΓhe involution
orresponds
c 4 antidiag 0
to the subgroup .) Let be a subspa
e,
alled the
antidiagonal, de(cid:28)ned as
∆ = {(x,y) ∈ (RPn−k ×RPn−k/T′)|T(x) = y},
antidiag
where T : RPn−k → RPn−k
is the standard involution on the
overing
RPn−k → Sn−k/i.
Γ
Let us denote by the spa
e
Γ \(U(∆ )∪U(∆ )),
0 antidiag diag
U(∆ ) ∆ U(∆
antidiag antidiag diag
where is a small regular neighborhoods of , is a
Γ ∆
0 diag
small regular neighborhood of the end of near the deleted diagonal .
1
The author was developed this proof following
onversations with Prof. O.Saeki and
Dr. R.R.Sadykov (2006))
6
ε
(The radius of the regular neighborhoods depends on a
onstant of an
approximation in the Melikhov
onstru
tion.)
Γ
The spa
e is a manifold with boundary. The involution
T : RPn−k → RPn−k
indu
es the free involution
T : Γ → Γ.
Γ
A polyhedron
Σ = {[(x,y)] ∈ Γ ,p(x) = p(y)}, Σ ⊂ Γ
0 0 0 0
of double points of the mapping
p : RPn−k → J
is
alled the singular set or the singular polyhedron.
The mapping η : Σ → K(D ,1)
Σ0 0 4
is well-de(cid:28)ned as the restri
tion of the mapping
η | .
Γ0 Σ0
The subpolyhedron
Σ ⊂ Γ
0 0
de
omposes as
Σ = Σ ∪K, K ⊂ Γ,
0 antidiag
where
Σ = Σ ∩U(∆ ).
antidiag 0 antidiag
η | η : K → K(D ,1)
The restri
tion Γ0 K will be denoted by K 4 .
η
K
Boundary
onditions of
K
The diagonal and antidiagonal boundary
omponents of will be denoted
by
Q = K ∩∂U(∆ ), Q = K ∩∂U(∆ ).
diag diag antidiag antidiag
η | : Q → K(D ,1)
The restri
tion K Qantydiag antidiag 4 is de
omposes as
i ◦η : Q → K(I ,1) ⊂ K(D ,1).
a antidiag antidiag a 4
η | : Q → K(D ,1)
The restri
tion K Qdiag diag 4 is de
omposed as
i ◦η : Q → K(I ,1) ⊂ K(D ,1).
b diag antidiag b 4
7
RK
The resolution spa
e .
RK K
Let us
onstru
t the spa
e
alled the resolution spa
e for . This spa
e
is in
luded into the diagram
K(I ,1) ←φ− RK −p→r K.
a
pr−1(Q ) RQ pr−1(Q ) RQ
diag diag diag antidiag
Let us denote by , by . The
Q
antidiag
boundary
onditions on are:
pr
RQ −→ Q
antidiag antidiag
φ ց ւ η
antidiag
K(I ,1).
a
Q
diag
The boundary
onditions on are:
pr
RQ −→ Q
diag diag
φ ↓ ↓ η
diag
K(I ,1) ←pb− K(I ,1).
d b
The diagrams above are in
luded into the following diagram:
K(I ,1)
a
↑ φ տ
RK ←− RQ ∪RQ
diag antidiag
↓ ↓
K ⊃ Q ∪Q
diag antidiag
↓ η ւ
K(D ,1).
4
d : RPn−k → Rn
Let us
onsider Melikhov's mappiin◦gp◦π : RPn−k → Sn−(kt/hiis→isJa⊂sRmnall
geneNri
n−a2lkternation of the
omposition d ).
NLent−2k ⊂ Γbe thedoublepointmanifold(withNbno−u2nkdary)of ,theembedding
0
is well-de(cid:28)ned. The manifold is de
omposed into two
manifolds (with boundary) along the
ommon
omponent of the boundary
Nn−2k = N ∪N ,
antidiag d
N = Nn−2k ∩U(∆ ),
antidiag antidiag
N = Nn−2k ∩Γ.
d
8
Lemma 7
res : N → RK
d
There exists a mappµin:gN → K(I ,1)
alled the resolution mapping that
d a
indu
es a mapping in
luded into the following diagram:
K(I ,1) = K(I ,1)
a a
↑ φ ↑ µ տ
res
RK ←− N ⊃ W ∪W
d diag antidiag
↓ η ↓ η ւ
K(D ,1) = K(D ,1),
4 4
W
antidiag
with boundary
onditions on :
µ| = i ◦η : W −→ K(I ,1) −i→a K(D ,1)
Wantydiag⊂Nd a antidiag antidiag a 4 ,
W
diag
and with boundary
onditions on :
µ = i ◦p ◦η : W → K(I ,1) → K(I ,1) → K(I ,1).
a b diag diag b d a
Lemma 8
The mapping
µ = η| ∪µ : Nn−2k = N ∪N → K(I ,1)
a Nantidiag antidiag d a
d
determines a
y
li
stru
ture for .
Proof of Lemma 8
∗
We have to prove the equality in the De(cid:28)nition of the Cy
li
Stru
ture for
generi
mappings.
T : Γ → Γ Γ/T
Γ Γ
Let us
onsider the fπree(Γi/nTvo)lution E and the quotient .
1 Γ
DThe fundamental group c ∈ E,\dDenotce2d=bya2 , is a quadratic
extension of
4 4
by means of an element , D .⊂ThEe element of order 4 is
4

ommutes with all elements in the subgroup . The following diagram
is well-de(cid:28)ned:
9
N −→ RK −→ K ⊂ Γ
d
↓ ↓ ↓ ↓
N /T −→ RK/T −→ K/T ⊂ G/T
d
↓ ↓ ↓ ↓
K(I ,1) = K(I ,1) K(E,1) = K(E,1).
a a
RK → RK/T → K(I ,1) φ c ∈
a
The
omposition
oin
ides with (The
π (RK/T)\π (RK) π (RK) ⊂
1 1 1
π (RK/T)
ommutes with all eleRmKen/tTs i→n thKe(sIu,b1g)roup
1 a
of the index 2 andc the mapping is well-Ide(cid:28)ned.
a
The image of the element is the generator of the
y
li
group .) The
N → N /T →
K
o(mIp,o1s)ition of the left verti
al arroµw:sNin→theKd(Iia,g1ra)m d (Ndn−2Nkd,µ )
a d a a

oin
ides with t(hNe′nm−a2kp,pµin′)g N′n−2k = .NT′nh−e2pka∪irN′n−2k
is
obordant to a pair a , where cycl d , and
N′n−2k N′n−2k
cycl is a
losed manifold. The manifold (with boundary) cycl is the
N′n−2k/T
double
overing over the orientedmaniHfold(w(Kith(Ib,o1u)n,dKa(rIy),1)c;yZcl) Nd.The
n−2k a d
base of the
over µre′pr(e[sNen′nt−s2ak,
∂yN
l′e])in∈ H (K(I ,1),K(I ,1)) . Therefore
therelative
y
le a,∗ d n−2k a d istrivialand
(cid:10)µ∗(t);[Nn−2k,∂Nn−2k−1](cid:11) = (cid:10)µ′∗(t);[N′n−2k](cid:11).
a a cycl
The last
hara
teristi
number
oin
ides with
(cid:10)κn−k;[N¯n−k](cid:11) = 1.
Lemma 8 is proved.
K
A natural strati(cid:28)
ation of the polyhedron . Proof of Lemma 7
J (S7/i) j = 1...,r J
j
Let be the join of lens spa
es , . The spa
e admits
J(k ,...,k )
1 s
a natural strati(cid:28)
ation de(cid:28)ned by the
olle
tion of subjoins
0 < k < ··· < k < r
1 s
generated by lenses with numbers .
The preimage p−1(J(k ,...,k )) ⊂ RPn−k,
1 s
p : RPn−k → J R(k ,...,k )
1 s
, is denoted by .
A point x ∈ R(k ,...,k ) ⊂ RPn−k
1 s
is determined by the
olle
tion of
oordinates
(x ,...,x ,λ)
k1 ks
s x ∈
(up to the antipodal transformation of the (cid:28)rst
oordinates), where kj
S7 λ (s−1)
j
, and is a bary
entri

oordinate on the standard -simplex.
10