SECONDARY BROWN-KERVAIRE QUADRATIC
FORMS AND ß-MANIFOLDS
FUQUAN FANG AND JIANZHONG PAN
Abstract. In this paper we assert that for each -oriented 2n-
manifold (c.f : Definition 1.1) M where n 4 and n 6= 3(mod4),
there is a well-defined quadratic function OEM : Hn-1(M, Z4) !
Q=Z, we call the secondary Brown-Kervaire quadratic forms, so
that
o OEM (x + y) = OEM (x) + OEM (y) + j(x [ Sq2y)[M],
o the Witt class of OEM is a homotopy invariant, if the Wu class
vn+2-2i( M ) = 0 for all i.
where j : Z2 ! Q=Z is the inclusion homomorphism and M the
stable normal bundle of M.
Among the applications we obtain a complete classification of
(n - 2)-connected 2n-dimensional ß-manifolds up to homeomor-
phism and homotopy equivalence, where n 4 and n + 2 6= 2i
for any i. In particular, we prove that the homotopy type of such
manifolds determine their homeomorphism type.
1. Introduction
Let M be a 2n-dimenisonal framed manifold (i.e. a ß-manifold with
a framing) where n = 1(mod2). The Kervaire invariant of M is the Arf
invariant of a Z2-valued Kervaire quadratic form of M
qM : Hn (M, Z2) ! Z2
satisfying
qM (x + y) = qM (x) + qM (y) + (x [ y)[M]2 (1.1)
It was invented by Kervaire to find the first example of non-smoothable
PL-manifold. Kervaire invariants and its various generalizations, e.g.
____________
Date: Dec. 20,1999.
The first author was supported in part by NSFC 1974002, Qiu-Shi Foundation
and CNPq and the second author is partially supported by the NSFC project
19701032 and ZD9603 of Chinese Academy of Science and Brain Pool program
of KOSEF .
1
2 FUQUAN FANG AND JIANZHONG PAN
the Brown-Kervaire invariants[4], play very important roles in geomet-
ric topology. Formally, qM is a üq adratic form" subject to the sym-
metric bilinear form
Hn (M, Z2) x Hn (M, Z2) ! Z2
(x, y) ! x [ y[M]2
For a Spin manifold of even dimension, there is another symmetric
bilinear form ~M studied by Landweber and Stong [16]:
~M : Hn-1 (M, Z2) x Hn-1 (M, Z2) ! Z2
(x, y) ! Sq2(x) [ y[M]2
A natural algebraic question to ask is whether there is an intrinsic üq a-
dratic formö f M subject to ~M . To answer this turns out to be the
main novelty of this paper. For a large family of Spin manifolds includ-
ing all ß-manifolds, the so called -oriented manifolds, we will define a
Q=Z-form subject to ~M , which resembles to the Brown-Kervaire qua-
dratic forms in the formulation. It has the most similar properties of
the Brown-Kervaire quadratic forms, e.g., the isomorphism class of the
form is a homotopy invariant if the manifold has vanishing Wu classes.
A bit surprising to us, this invariant applies to give a classification of
(n - 2)-connected 2n-dimensional ß-manifolds up to homotopy equiv-
alence and homeomorphism (n 4).
To state our main results, let us start with some notations.
Let {Yk}k2N be a connected spectrum with U 2 H0(Y ) ~= Z a gen-
erator so that i*U 2 H0(S0) a generator, where i : S0 ! Y is the
inclusion map of the spectrum.
Definition 1.1. (i) {Yk}k2N is called -orientable if Sq2U = 0, Ø(Sqn+2 )(U) =
0 and 0 2 (U), where is a secondary cohomology operator associ-
ated with the Adem relation (see Section 3 for the definition):
Ø(Sqn)Sq3 + Ø(Sqn+2 )Sq1 + Sq1Ø(Sqn+2 ) = 0 n = 2(mod4)
Ø(Sqn)Sq3 + Sq1Ø(Sqn+2 ) = 0 n = 0(mod4)
Ø(Sqn+1 )Sq2 + Sq1Ø(Sqn+2 ) = 0 n = 1(mod4)
where Ø : A2 ! A2 is the anti-automorphism of the Steenrod algebra
A2 [1].
A spherical fibration , (a manifold) is called -orientable if its Thom
spectrum T , (stable normal bundle M ) is. We define the universal -
orientable -spectrum fW(n) by setting fWk(n) to be the total space of
the following Postnikov tower:
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS 3
fWk (n)
# 2
Wk(n) -k2! Kk+n+2
# 1
Sq2xffl(Sqn+2)
K(Z, k) -! Kk+2 x Kk+n+2
where Ki = K(Z2, i), K(Z, i) are the Eilenberg-Maclane spaces, k2 2
( 1 *lk) and lk is the basic class.
Note that a spectrum Y is -orientable if and only if U 2 H0(Y ) can
be lifted to a map w : Y ! fW(n). We call such a lifting a -orientation
of Y . A -orientation of a manifold is understood as a -orientation
of its Thom spectrum.
Remark 1.2. The sphere spectrum S0 is -orientable. Thus stably
parallelizable manifolds are -orientable.
Our main results are:
Theorem 1.3. Let M be a -oriented manifold of dimension 2n, where
n 6= 3(mod 4). Then there is a function OEM : Hn-1 (M, Z4) ! Q=Z
such that, for all x, y 2 Hn-1 (M, Z4),
OEM (x + y) = OEM (x) + OEM (y) + j(x [ Sq2y)[M],
where j : Z2 ! Q=Z is the inclusion.
Remark 1.4. In general, OEM depends on the -orientation, just like the
Kervaire quadratic form depends on the framing of the manifold. We
will prove that OEM (x) depends only on the -oriented bordism class
[M, x].
Remark 1.5. If n = 3(mod 4), the analogous definition gives only a
linear function.
Let BSpinG be the classifying space for spherical Spin fibrations. By
Brown[4], a Wu orientation of a Spin spherical fibration , & M is a
lifting of the classifying map , : M ! BSpinG to BSpinG . A
Wu orientation of M , the stable normal bundle of M, is understood as
a Wu orientation of M, where BSpinG ! BSpinG is a principal
fibration with vn+2 2 Hn+2 (BSpinG , Z2) as the k-invariant.
We call quadratic forms OEMi : Hn-1 (Mi, Z4) ! Q=Z , i = 1, 2
Witt equiavlent if there exists an isomorphism ø : Hn-1 (M1, Z4) !
Hn-1 (M2, Z4) so that OEM2 (ø x) = OEM1 (x) for all x 2 Hn-1 (M1, Z4).
Theorem 1.6. Let M1 and M2 be -oriented 2n-manifolds. Suppose
that the Wu classes vn+2-2j( Mi ) = 0 for all 2j n + 2. If f : M1 !
4 FUQUAN FANG AND JIANZHONG PAN
M2 is a homotopy equivalence preserving the spin structure (resp. Wu
orientation) if n = 0, 1(mod4) (resp. n = 2(mod4)). Then
OEM1 (f*x) = OEM2 (x)
for all x 2 Hn-1 (M2, Z4).
Since the Wu class v0 = 1, the assumption in the above theorem
implies that n + 2 6= 2i for any integer i.
For framed manifolds, the Brown-Kervaire secondary quadratic forms
have the following property:
Proposition 1.7. If M is a framed manifold of dimension 2n, where
n 6= 3(mod4). Then OEM factors through Z4 Q=Z (resp. Z2 Q=Z ),
provided n = 2(mod4) (resp. n = 0, 1(mod4)).
To state the next results, we need some preliminaries.
Let H be a finitely generated abelian group, and
~ : Hom(H, Z2) Hom(H, Z2) ! Z2
be a symmetric bilinear form. We say that ~ is of diagonal_zero_if
~(x, x) = 0 for each x 2 Hom(H, Z2). A function OE : Hom(H, Z4) !
Q=Z is called quadratic_with respect to ~ if
OE(x + y) = OE(x) + OE(y) + j(~(x, y))
where j : Z2 ! Q=Z is the inclusion. This gives a triple (H, ~, OE).
We say triples (H1, ~1, OE1), and (H2, ~2, OE2) are isometric if there exists
an isomorphism ø : H1 ! H2 such that ~1(x, y) = ~2(ø x, ø y) and
OE1(x) = OE2(ø x) for all x, y. We denote by [H, ~, OE] the isometry class
of a triple.
Remark 1.8. Since the natural map Hom(Hn-1(M), Z2) ! Hn-1 (M, Z2)
is not an isomorphism in general, the notions of isometry associated
with ~ and ~M as above are different. They do agree however for (n-2)-
connected manifolds which we will assume in the later application. We
will use both of them when necessary.
Let i denote the maximal exponent of the 2-torsion subgroup of
Hn-1(M) and let Sq1i2 Hn (K(Z2i, n - 1), Z2) ~=Z2 be the unique gen-
erator. Considering Sq1ias a cohomology operation we get a function
qM (Sq1i) : Hn-1 (M, Z2i) ! Z2.
This gives a homomorphism since Sq1ix [ Sq1iy = Sq1i(x [ Sq1iy) = 0
for x, y 2 Hn-1 (M, Z2). We denote by [Hn-1(M), ~M , qM (Sq1i)] for the
isometry class of the triple. By [6], the Kervaire invariant of a smooth
framed manifold of dimension 2n, where n 6= 2i- 1, is zero. For i 5,
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS 5
there are smooth manifolds of dimension 2i+1- 2 of Kervaire invariant
1. It is still an open problem whether there is such a manifold for i 6.
The Kervaire invariant does not depend on the framings of the un-
derlying 2n-manifold if n 6= 1, 3, 7 and the manifold is highly connected,
e.g. (n - 2)-connected. Moreover, by [4] the Kervaire form is a homo-
topy invariant if n 6= 1, 3, 7 and (n - 2)-connected.
Let M be a (n - 2)-connected 2n-dimensional ß-manifold. Observe
that if n 3, there exists a (n - 2)-connected ß-manifold, N, so that
M = N#X and Hn(N, Q) = 0, where X is a (n - 1)-connected 2n-
manifold. Since the classification of (n - 1)-connected 2n-manifolds
is well understood [25 ], for convenience in the following theorem we
assume that Hn(M, Q) = 0. For such a manifold, consider the corre-
spondence
ß : M 7- ! [Hn-1(M), ~M , OEM ]( resp.[Hn-1(M), ~M , OEM , qM (Sq12i)])
if n = 0(mod2) (resp. n = 1(mod2)).
In the following theorem let ff(n + 2) be the number of 10s in the
binary expansion of n + 2.
Theorem 1.9. Suppose n 4 and ff(n + 2) 2. Then ß gives
a 1-1 correspodence between the homeomorphism types (resp. homo-
topy types) of (n - 2)-connected 2n-dimensional ß-manifolds M so that
Hn(M, Q) = 0 with the following algebraic data
(a) "n = {[H, ~, OE] : diag ~ = 0 and OE factors through j : Z4 ! Q=Z}
if n = 2(mod4) ,
(b) "n = {[H, ~, OE] : OE factors through j : Z2 ! Q=Z} if n = 0(mod4),
(c) "n = {[H, ~, OE, !] : ! 2 Hom(tor(H) Z2i, Z2), OE factors through
j : Z2 ! Q=Z} if n = 1(mod4),
(d) "n = {[H, ~, !] : ! 2 Hom(torH Z2i, Z2)} if n = 3(mod4).
where i is the highest exponent of the 2-cyclic subgroup of H and if
n = 1(mod2), the pairing ~(x, x) = 0 (resp. ffi!(x)) if x can be lifted to
a Z4 class with order 4 (resp. x is of order 2), ffi 2 {0, 1} is ambiguous.
Remark 1.10. The classification of (n-2)-connected 2n-manifolds with
torsion free homology groups has been given by Ishimoto[9][10]. But
his method does not work if the homology group has torsion.
The organization of this paper is as follows.
In x2 we define the secondary Brown-Kervaire form and state its
basic properties.
In x3, we set up the necessary foundations on the stable homotopy
theory of the Eilenberg-Maclane spaces.
In x4, we are addressed to show Theorems 1.3 and 1.6.
In x5, we prove Theorem 1.9.
6 FUQUAN FANG AND JIANZHONG PAN
2. A Q=Z-quadratic form of -oriented manifolds
Let us begin with some conventions. All homology/cohomology
groups will be with integral coefficients unless otherwise stated. All
spaces will have base points. Let
(i) [X, Y ] denote the set of homotopy classes of pointed maps from X
to Y .
(ii) {X, Y } = lim[SkX, SkY ].
(iii) ßs*(X) be its 2-localization to simplify the notation.
Let ~ : K(Z4, n - 1) x K(Z4, n - 1) ! K(Z4, n - 1) be the mul-
tiplication of K(Z4, n - 1) and let H(~) be the Hopf construction of
~.
Proposition 2.1. The homomorphism
H(~)* : ßs2n(K(Z4, n - 1) ^ K(Z4, n - 1)) ! ßs2n(K(Z4, n - 1))
is injective if n 6= 3(mod4), and zero if n = 3(mod4).
Remark 2.2. If Z4 is replaced by Z2, then H(~)* is trivial.
By Theorem 3.1 and the proof of it, we obtain
ßs2n(K(Z4, n - 1) ^ K(Z4, n - 1)) ~=Z2 if n 4,
ßs2n(K(Z4, n - 1)) ~=Z4 if n = 2(mod4).
Let ~0 be a generator of Im H(~)* if n 6= 2(mod4), and a specified
generator of ßs2n(K(Z4, n - 1)) ~= Z4 otherwise. For a given spectrum
Y , let
H*(K(Z4, n - 1); Y ) = lim ß*+k(K(Z4, n - 1) ^ Yk).
Theorem 2.3. Suppose that {Yk}k2N is a -orientable spectrum. Then
there exists a homomorphism
h : H2n(K(Z4, n - 1); Y ) ! Q=Z
such that h(~) = 1_4(resp. 1_2) if n = 2(mod4) (resp. n = 0, 1(mod4)),
where ~ = i*(~0) and i* : H2n(K(Z4, n-1); S0) ! H2n(K(Z4, n-1); Y )
is induced by the inclusion.
Definition 2.4. A Poincar'e triple (M, ,, ff) of dimension 2n consists
of
(i) A CW complex M with finitely generated homology.
(ii) A fibration , over M with fiber homotopy equivalent to Sk-1, k
large.
(iii) ff 2 ß2n+k(T ,) such that an (2n + k) Spanier-Whitehead S-duality
is given by
S2n+k -ff!T , -! T , ^ M+
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS 7
where T , is the Thom complex of , and is the diagonal map.
Let Aff: {M+ , K(Z4, n - 1)} ! {S2n+k, T , ^ K(Z4, n - 1)} be the
S-duality map.
Definition 2.5. Let (M, ,, ff) be a Poincar'e triple and w is a -
orientation of the Thom spectrum T ,. For a homomorphism h in
Theorem 2.3, let
OEw,h : Hn-1 (M, Z4) ! Q=Z
be defined by setting
OEw,h(x) = h([(w ^ id)Aff(x)]).
Theorem 2.6. Let OEw,h be defined as above. Then for all x, y 2
Hn-1 (M, Z4),
(i) If n 6= 3(mod 4), the function is quadratic, i.e.
OEw,h(x + y) = OEw,h(x) + OEw,h(y) + j(x [ Sq2y)[M]
where j : Z2 ! Q=Z is the inclusion;
(ii) If n = 3(mod4), OEw,h is linear, i.e.
OEw,h(x + y) = OEw,h(x) + OEw,h(y).
Now we want to study how the function OEw,h depends on the choice
of the orientation of the Thom spectrum T ,.
Let wi, i = 1, 2 are orientations of the Thom spectrum T ,. Let
d1(w1, w2) 2 H1(T ,) Hn+1 (T ,)
denote the difference of the composition maps 2w1 and 2w2, where
2 is as in the definition of the universal -spectrum fW (n). Clearly,
w1 and w2 are homotopy if and only if d1(w1, w2) = 0 and a secondary
obstruction vanishes. The following theorem shows that the secondary
obstruction does not affect our quadratic function OEw,h.
Theorem 2.7. Let OEwi,hbe the quadratic forms associated with (wi, h),
i = 1, 2. If d1(w1, w2) = 0, then OEw1,h(x) = OEw2,h(x) for all x 2
Hn-1 (M, Z4).
In general, the quadratic form OEw,h does depend on the choice of w
and h. In order to obtain a well-defined invariant of the -oriented
manifold, we now choose certain type of -orientations of the Thom
spectrum T , in an universal way and then define the Brown-Kervaire
secondary quadratic forms to be the quadratic functions associated to
those -orientations.
8 FUQUAN FANG AND JIANZHONG PAN
Let fl & BSpinG be the universal Spin spherical fibration and U 2
H0(MSpinG , Z2) the universal Thom class. Note that
Ø(Sqn+2 )U = Ø(Sqn+1 )Sq1U = 0 if n is odd
Ø(Sqn+2 )U = Ø(Sqn)Sq2U = 0 if n = 0(mod4) ,
Thus U lifts to a map f : MSpinG ! W (n). By the Thom iso-
morphism, f*k2 gives an element of k~2 2 Hn+2 (BSpinG , Z2). Let
ß : BSpinG ! BSpinG be the principal fibration with k-invariant
~k2.
If n = 2(mod4), we get a similar principal fibration ß : BSpinG !
BSpinG , where BSpinG ! BSpinG is the fibration with
fibre Kn+1 and k-invariant vn+2.
It is easy to see that the fibration ß*fl is -orientable. Clearly the
classifying map of every -orientable stable spherical fibration lifts to
BSpinG .
Definition 2.8. The fibration ß*fl is called the universal -orientable
spherical Spin fibration. Its Thom spectrum, MSpinG , is called the
universal -orientable Thom spectrum.
For a closed -orientable manifold M2n, there is a Poincar'e triple
(M, M , ff) where M is the stable normal bundle and ff 2 ß2n+k(T M )
is the normal invariant of M (obtained by the Thom-Pontryagin con-
struction.)
Definition 2.9. Fix a connected spectral map u : MSpinG !
fW(n) and a homomorphism h in Theorem 2.3. For a -orientable
manifold M, let
OEM = OEw,h
where w = u O T (v) and T (v) is the Thom map of a classifying bundle
map of the stable bundle M .
Now we prove Theorem 1.6 assuming Theorem 2.7.
Proof of Theorem 1.6. Let ,i = Mi be the stable normal bundle of Mi
and ffi 2 ß2n+k(T ,i) be the normal invariant, i = 1, 2. By the definition,
OEMi = OEwi,hwhere wi = u O T (vi) and T (vi) : T (,i) ! MSpinG <~k2> the
Thom map.
Let ~f: f*,2 ! ,2 be a bundle map over the homotopy equivalence f.
Let ff3 = T (f~)-1*ff2, where T (f~) is the Thom map of f~. The Poincar'e
triple (M1, f*,2, ff3) together with the -orientation w2 O T (f~) gives a
quadratic form OE3, where w2 = u O T (v2) is a -orientation of M2. By
2.5 we get that
OE3(f*x) = OEM2 (x)
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS 9
for all x 2 Hn-1 (M2, Z4).
To prove the desired result, it suffices to prove OE3 = OEM1 .
Note that f*,2 and ,1 are stably equivalent as spherical fibration
since f is a homotopy equivalence. Thus we can regard f*,2 and ,1 as
the the same and so get two orientations for ,1, (u O T (v1), h) and (u O
T (v2)OT (f~), h). Since f preserves the Spin structures/Wu orientations,
ß O v2 O f ' ß O v1, where ß : BSpinG ! BSpinG /BSpinG is
the principal fibration as above. This clearly implies that there exists
a fibre automorphism g 2 Aut(,1) over the identity such that
T (ß O v2 O ~f) ' T (ß O v1) O T (g).
Notice that g gives a unique element g0 2 [M1, Gk], where Gk is the
space of self homotopy equivalences of Sk. By a formula in Brown
[4], the (nP+ 1)-dimensional component of d1(u O T (v1) O T (g), u O
T (v1)) is vn+2-2i [ g*0u2i-1, where u2i-1 is the transgression of w2i2
H2i(BGk, Z2). By assumption, it must vanish since the Wu classes van-
ish. On the other hand, the 1-dimensional component of d1(u O T (v1) O
T (g), u O T (v1)) is determined by the Spin structures and so it vanishes
since f preserves the Spin structures. By Theorem 2.7 it follows that
OEM1 = OE4,
the quadratic form associated with the Poincar'e triple (M1, ,1, ff1) and
the -orientation w2 O T (f~).
Note that in the definitions of OE3 and OE4 the only different ingradients
are the normal invariants, after identifying ,1 with f*,2. By Theorem
2.7 once again OE3 = OE4. This implies the desired result.
Now we prove Proposition 1.7.
Proof of Proposition 1.7.Since M is a framed manifold, the stable nor-
mal bundle is trivial, i.e. the classifying map of M factors through a
point. Choose a -orientation w = u O T (v) : M with v the bun-
dle map of M to the trivial k-bundle on a point, then OEM (x) factors
through the stable homotopy group ßs2n(K(Z4, n - 1)). By Theorem
3.1 ßs2n(K(Z4, n - 1)) ~= Z4 if n = 2(mod4) and the order of elements
in ßs2n(K(Z4, n - 1) is at most 2 if n = 0, 1(mod4). On the other hand,
by Theorem 1.6 the definition of OEM does not depend on the choice of
the -orientations since M is a framed manifold. This completes the
proof.
10 FUQUAN FANG AND JIANZHONG PAN
3. Some preliminaries on stable homotopy theory
In this section we calculate the stable homotopy groups ßs2n(K(ß, n-
1)) (see Theorem 3.1). We will also introduce some 2-stage Postnikov
tower which will give the secondary cohomology operation used in
Section x1.
Theorem 3.1. The 2n-th stable homotopy group of K(ß, n - 1) for
n __4_is_as_follows:__________________________________________________________
|____________n___4_|_0(mod4)____|___1(mod4)___|_____2(mod4)_____|_3(mod4)__|_*
*|s|2(t+k)+s+p|t+2k+s+p|t+ks+p|k+s+p|
|_ß2n(K(ß,_n_-_1))_|(Z2)__________|(Z2)________(|Z4)______(Z2)____|(Z2)_____ |
t+k+s
where p = 2 and ß = G0 x Ztx Z2i1x . .x.Z2ikx Zs2, ij 2 if
1 j k and G0 Z2 = 0.
When ß = Z, Theorem 3.1 follows from [17 ].
Proof. It is easy to know that(since we are computing the 2-localization)
ßs2n(K(ß, n - 1)) = ßs2n(K(ß=G0, n - 1))
L
Assume G0 = 0 from now on. If ß = ß1 ß2 with ß1 nontrivial and
ß2 a nontrivial cyclic group, then
K(ß, n - 1) = K(ß1, n - 1) x K(ß2, n - 1)
and we have by a result in [3] that
M M
ßs2n(K(ß, n-1)) = ßs2n(K(ßi, n-1)) ßs2n(K(ß1, n-1)^K(ß2, n-1))
i=1,2
M M
= ßs2n(K(ßi, n - 1)) Hn+1(K(ß1, n - 1), ß2)
i=1,2
An easy calculation shows that Hn+1(K(G1, n - 1), G2) = Z2 if G1, G2
are nontrivial cyclic groups and thus Hn+1(K(ß1, n-1), ß2) = Zt+k+s-12.
On the other hand we know groups ßs2n(K(Z, n-1)) and ßs2n(K(Z2, n-
1)) by results in [20 ],[17 ]. To complete the proof it remains to calculate
ßs2n(K(Z2i, n - 1)) for 2 i < 1 which will be given in the following
results.
Recall that for each locally finite connected CW complex X we can
define a space
qX = Sq-1 /T X^X = Sq-1x(X^X)={(x, y, z) ~ (-x, z, y); (x, *) ~ *}
for every q 2 Z+ . By [20 ] Theorem 1.11, for a (n - 2)-connected space
X, qX is (2n - 3)-connected. Moreover, if X = K(ß, n - 1), we have
a fibration
Gq ! qK(ß, n - 1) ! K(ß, q + n - 1)
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS11
where Gq ' q q(K(ß, n - 1)) through dimension (3n + q - 3). Thus
ßsi(K(ß, n - 1)) ~=ßsi( q(K(ß, n - 1)) for n < i < 3n - 3.
When q = 1, qX = X ^ X .The corresponding sequence is :
H(~)
Fn-1(ß) ! K(ß, n - 1) ! K(ß, n)
where Fn-1(ß) = K(ß, n-1)^K(ß, n-1). After q-1 time suspensions
we get a fibration sequence at least in dimensions less than 3n + q - 4
q-1H(~) q q-1
qFn-1(ß) ! K(ß, n - 1) ! K(ß, n)
Let q be large enough so that we are always in the stable range and let
r = q + 2n ,then we have an exact sequence
. .!.ßs2n+2(K(ß, n)) !@ ßr( qFn-1(ß)) !
! ßs2n(K(ß, n - 1)) ! ßs2n+1(K(ß, n)) !@ . . .
Since we know that ßr( qFn-1(ß)) = Z2 for ß = Z and Z2i , we can
determine inductively ßs2n(K(ß, n - 1)) up to extension if we know the
map @.
Lemma 3.2. For ß = Z or Z2i, a homotopy class [g] 2 ßs2n+2(K(ß, n))
has @g 6= 0 iff g*( q-1('[Sq2')) 6= 0 where g : Sr+1 ! q-1K(ß, n),' 2
Hn (K(ß, n), Z2) is the generator and g* : H*( q-1K(ß, n), Z2) !
H*(Sr+1, Z2)
The proof is similar to that of Lemma 1.3 in [17 ]. The key points
are the followings:
o g* can be nonzero only on element q-1(' [ Sq2')
o the Hurewicz homomorphism H : Z2 ~=ßr( qFn-1(ß)) ! Hr( qFn-1(ß))
is nonzero
The first statement is clear while the second is an easy consequence of
the Whitehead exact sequence(c.f. [26 ], page 555)
With the lemma above we can now prove the Proposition2.1.
Proof of Proposition2.1.It suffices to prove that @ is trivial if n =
0, 1, 2(mod4) and nontrivial if n = 3(mod4).
For n = 0, 1, 2(mod4) there is no g : Sr+1 ! q-1K(Z, n) such
that g*( q-1(' [ Sq2')) 6= 0 since q-1(' [ Sq2') is detected by the
secondary cohomology operation 'n in [18 ]. Thus there is no g : Sr+1 !
q-1K(Z2i, n) such that g*( q-1(' [ Sq2')) 6= 0 by the naturality of
secondary cohomology operation and the fact that æ2i : K(Z, n) !
K(Z2i, n) corresponding to mod 2i reduction induces a homomorphism
sending q-1(' [ Sq2') to the corresponding element. It follows that
@ = 0
12 FUQUAN FANG AND JIANZHONG PAN
When n = 3(mod4),there is a map g : Sr+1 ! q-1K(Z, n) such
that g*( q-1(' [ Sq2')) 6= 0 since otherwise ßs2n(K(Z, n - 1)) 6= 0.
By the above fact on map æ2i it is easy to see that there is a map
h : Sr+1 ! q-1K(Z2i, n) such that h*( q-1(' [ Sq2')) 6= 0.It follows
from the lemma above that @h 6= 0.
With the help of Proposition2.1 and the known results about ßs2n+j(K(ß, n))
for j = 0, 1, we can now determine the group ßs2n(K(Z2i, n - 1)).
Assume i 2 in the following unless otherwise stated.
Proposition 3.3. If n = 0(mod2),then
æ2i*: ßs2n(K(Z, n - 1)) ! ßs2n(K(Z2i, n - 1))
is an isomorphism.
Before the proof of the Proposition, let's give two remarks which are
clear from the proof of the Proposition.
Remark 3.4. If i = 1, æ2i*is onto.
Remark 3.5. If n = 0(mod4), then the spherical cohomology class in
ßs2n(K(Z, n - 1)) does not belongs to the image of the natural map:
ßr( qFn-1(Z)) ! ßs2n(K(Z, n - 1)).
Proof. Note that we have a commutative diagram
ßr( qFn-1(Z)) -- - ! ßs2n(K(Z, n - 1)) --- ! ßs2n+1(K(Z, n))
? ? ?
j2i*?y j2i*?y j2i*?y
ßr( qFn-1(Z2i)) -- - ! ßs2n(K(Z2i, n - 1)) --- ! ßs2n+1(K(Z2i, n))
In the above diagram,the two left horizontal maps are injective by
Lemma3.2, the left vertical map is obviously an isomorphism while
the fact that the right vertical one is also an isomorphism follows
by comparing the Whitehead exact sequences of q-1(K(Z, n)) and
q-1(K(Z2i, n)).On the other hand ,the fact that the right horizontal
map on the bottom line is onto follows from the long exact sequence
and the known results about ßs2n+j(K(Z2i, n)) for j = 0, 1.
PropositionL3.6. For n = 1(mod2), ßs2n(K(Z2i, n-1)) = ßs2n(K(Z, n-
1)) Z2.
Proof. The relevant commutative diagram in this case is
ßr( qFn-1(Z)) --- ! ßs2n(K(Z, n - 1)) -- - ! ßs2n+1(K(Z, n)) = 0
? ? ?
j2i*?y j2i*?y j2i*?y
ßr( qFn-1(Z2i)) --- ! ßs2n(K(Z2i, n - 1))-- - ! ßs2n+1(K(Z2i, n)) -- @-!
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS13
By the same argument as in the last Proposition , we know the map
@ is onto. If n = 3(mod4),the two left horizontal maps are trivial by
Lemma3.2, thus ßs2n(K(Z2i, n - 1)) ~=coker@ ~=Z2.
If n = 1(mod4), what we can get is an exact sequence
0 ! ßs2n(K(Z, n - 1)) ! ßs2n(K(Z2i, n - 1)) ! Z2 ! 0
To complete the proof, it suffices to prove the last map in the above
sequence has a section.
To do this we need another diagram
ßs2n(Kn-1) --- ! ßs2n+1(Kn) --@-! ßr-1( qFn-1(Z2))
? ? ?
j*?y j*?y j*?y
ßs2n(K(Z2i, n - 1)) --- ! ßs2n+1(K(Z2i, n)) --@-! ßr-1( qFn-1(Z2i))
where j : Z2 ! Z2i is the natural inclusion.
The same argument as above combined with the proof of Theorem
10.9 in[20 ] shows that the two @'s are onto and j* induces an isomor-
phism between kernels of two @'s. Finally we get the following diagram
which gives the desired section.
~=
ßs2n(Kn-1) --- ! Z2
? ?
j*?y ~=?y
ßs2n(K(Z2i, n - 1)) --- ! Z2
Lemma 3.7. If n is odd and Sq1i2 Hn (K(Z2i, n - 1), Z2) is a gener-
ator. Then
(Sq1i)* : ßs2n(K(Z2i, n - 1)) ! ßs2n(K(Z2, n)) ~=Z2
is an epimorphism.
Proof. It suffices to prove that the following map Sq1 : K(Z2, n - 1) !
K(Z2, n) induces an isomorphism on 2n-th stable homotopy group. By
the calculation in Milgram's book[20N], the first group is generatedNby
the class corresponding to Sq1(t) Sq1(t) and the second by s s
where s, t are the fundamental classes of the corresponding groups.
Now what we wantNfollows from theNfact that Sq1 induces a homomor-
phism mapping s s to Sq1(t) Sq1(t).
Proposition 3.8. Let eEn+q(q large) be the following 2-stage Postnikov
tower. Then there is a map f : qK(Z4, n - 1) ! eEn+q such that the
qj4 q
composite ( qFn-1(Z) !) qK(Z, n-1) ! K(Z4, n-1) ! eEn+qif
14 FUQUAN FANG AND JIANZHONG PAN
n = 1, 2(mod4)(or, n = 0(mod4)) induces an isomorphism on ßr where
r = q + 2n as above.
(1). n = 2(mod4)
Kr - i2! eEn+q
# 2
Kr-2 x Kr - i1! En+q -!2! Kr+1
# 1
qln-1 SqnxSqn+2
qK(Z4, n - 1) - ! K(Z4, q + n - 1) -! Kr-1 x Kr+1
where i*1(!2) = Sq2Sq1lr-2 + Sq1lr.
(2). n = 0(mod4)
Kr -i2! Een+q
# 2
Kr-2 -i1! En+q -!2! Kr+1
# 1
qln-1 Sqn
qK(Z4, n - 1) -! K(Z4, q + n - 1) -! Kr-1
where i*1(!2) = Sq2Sq1lr-2.
(3). n = 1(mod4)
Kr -i2! eEn+q
# 2
Kr -i1! En+q -!2! Kr+1
# 1
qln-1 Sqn+1
qK(Z4, n - 1) -! K(Z4, q + n - 1) -! Kr
where i*1(!2) = Sq2lr-1 .
Proof. Denote the tower in the Proposition by Een+q(Z4). Denote by
eEn+q(Z) a similar tower in which K(Z4, n+q-1) is replaced by K(Z, n+
q - 1). By Remark 3.5, it is easy to see that there is a map from
qFn-1(Z) to eEn+q(Z) which induces an isomorphism on ßr when n =
0(mod4). On the other hand it is not difficult to see that there is a
map from the tower Een+q(Z) to the tower Een+q(Z4) which induces an
isomorphism on ßr. It remains to prove that the natural map q'n-1 :
qK(Z4, n - 1) ! K(Z4, n + q - 1) can be lifted to eEn+q(Z4) and the
lifting is compatible to that of the map q'n-1 : qK(Z, n - 1) !
K(Z, n + q - 1) to eEn+q(Z).
We will give a proof only for n = 2(mod4) , the other cases are
similar. Consider the fiber inclusion map h : q q ! qK(Z4, n - 1),
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS15
we have the following Peterson-Stein formula
Sq2Sq1Sqnh( qln-1)+Sq1Sqn+2h( qln-1) = h* ( qln-1) 2 Hr+1( q q, Z2)=Q
where Q = Sq2Sq1(Imh*) + Sq1(Imh*) = Sq1(Imh*).
By Theorem 4.6 [20 ] and a familiar diagram chase argument as in
the proof of Proposition 2 in Chap.16 [19 ](see also [23 ]) , we have
q(` `) 2 Sqnh( qln-1) and qe2 [ (` `)2 Sqn+2h( qln-1). It follows
easily that h* ( qln-1) = 0 2 Hr+1( q q, Z2)=Q. It is not difficult to
see from this and a simple computation that ( qln-1) = 0 and a
lifting can be chosen such that !2 lies in its kernel.
To complete the proof, note that , as mentioned before , there is a
commutative diagram up to homotopy
Een+q(Z) --j4-! eEn+q(Z4)
? ?
? ?
y y
j4
En+q(Z) -- - ! En+q(Z4)
? ?
? ?
y y
j4
K(Z, n + q - 1) -- - ! K(Z4, n + q - 1)
x x
qln-1?? qln-1??
j4 q
qK(Z, n - 1) -- - ! K(Z4, n - 1)
The lifting from qK(Z, n-1) of qln-1 and the lifting from qK(Z4, n-
1) of qln-1 can be made compatible by a modification of the lifting
from qK(Z4, n - 1) of qln-1. The same way the liftings to eEn+q can
also be made compatible. Thus we have the following commutative
diagram up to homotopy which completes the proof.
eEn+q(Z) --j4-! Een+q(Z4)
x x
? ?
? ?
j4 q
qK(Z, n - 1) -- - ! K(Z4, n - 1)
Remark 3.9. The 2nd k-invariant !2 in the Postnikov tower above gives
an unique secondary cohomology operator (with Z4-coefficients) as-
sociated with the Adem relation
Sq2Sq1Sqn + Sq1Sqn+2 = 0 n = 2(mod4)
Sq2Sq1Sqn = 0 n = 0(mod4)
Sq2Sqn+1 = 0 n = 1(mod4)
16 FUQUAN FANG AND JIANZHONG PAN
Note that En+q is the universal example of the operator .
By Peterson-Stein[21 ], there are operators which are S-dual to
(which is uniquely determined by ) so it is a secondary operator
associated with the Adem relations:
Ø(Sqn)Sq3 + Ø(Sqn+2 )Sq1 + Sq1Ø(Sqn+2 ) = 0 n = 2(mod 4)
Ø(Sqn)Sq3 + Sq1Ø(Sqn+2 ) = 0 n = 0(mod 4)
Ø(Sqn+1 )Sq2 + Sq1Ø(Sqn+2 ) = 0 n = 1(mod 4)
as we stated in x1.
4. Proofs of Theorems 2.3, 2.6 and 2.7
Proof of Theorem 2.3. First note that it suffices to show this for the
universal spectrum fW(n) since the map i : S0 ! fW(n) factors through
i : S0 ! Y . Notice that Hi(fWk(n)=Sk) = 0 for i k + 2. Thus in
the following proof, we may assume that Yk=Sk satisfies the same for
k large. Assuming k large, without loss of generality we can assume
that Yk is a finite complex. Write Yk* for the m S-dual of Yk and
g : Yk*! Sm-k for the S-dual of the inclusion i : Sk ! Yk. Note that
g*(&Sm-k ) 6= 0, where &Sm-k is the cohomology fundamental class of the
sphere. By the S-duality we get a commutative diagram
{S2n+k, Sk ^ K(Z4, n - 1)} -i*! {S2n+k, Yk ^ K(Z4, n - 1)}
# ~= # ~=
g* 2n+k * m
{S2n+m , Sm ^ K(Z4, n - 1)} -! {S ^ Yk , S ^ K(Z4, n - 1)}
# # q2*
g* 2n+k *
[S2n+m , eEn+m] -! [S ^ Yk , eEn+m]
where eEm+n is the tower in Proposition3.8 and q2 : Sm ^K(Z4, n-1) !
eEn+m is a lifting of m ln-1. From the diagram above and Proposition
3.8,it suffices to show that the homomorphism g* at the bottom line is
injective.From now on we will restrict to the case n 2(mod4). The
other cases are similar. Let i0 : F ! eEn+m be the fibre of the composite
1 O 2. Note that F can be viewed as a fibration over K2n+m-2 with
fibre K(Z4, 2n + m) and k-invariant j*(Sq2Sq1)(l); where
j* : Hm+2n+1 (-, Z2) ! Hm+2n+1 (-, Z4)
is the homomorphism induced by the inclusion Z2 Z4 and l is the
basic class of Km+2n-2 .
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS17
Consider the following commutative diagrams
~=*
[S2n+m , F ] -! [S2n+m , eEn+m]
# J := g* # g*
i1* 2n+k * i0* 2n+k *
[S2n+k ^ Yk*, K(Z4, n + m - 2)] - ! [S ^ Yk , F ]-! [S ^ Yk , eEn*
*+m]
and
~=
[S2n+m , K(Z4, 2n + m)] -! [S2n+m ,*
* F ]
# g* # J
j*(Sq2Sq1)2n+k * ~= 2n+k *
* *
[S2n+k ^ Yk*, K2n+m-3 ] -! [S ^ Yk , K(Z4, 2n + m)] -! [S ^ Y*
*k , F ]
where i1 : K(Z4, n+m-2) ! F is the homotopy fibre of i0. j*(Sq2Sq1)
in the second diagram above is zero since Sq3Uk = 0 and thus by du-
ality Ø(Sq3)Hm-k-3 (Yk*) = Sq2Sq1Hm-k-3 (Yk*) = 0. Thus the second
diagram implies that J is a monomorphism. To complete the proof, it
suffices to show Ker(i0)* = Im(i1)* = 0 in the first diagram above.
Let q = m-n-k-1, if x 2 Hq-1(Yk*, Z4),then Sqn(x) 2 Hn+q-1 (Yk*, Z2) ~=
(Hk+2 (Yk, Z2))* = 0. On the other hand, by duality Ø(Sqn+2 )Uk = 0
implies that Sqn+2 Hq-1(Yk*, Z2) = 0. Thus
x 2 Ker Sqn \ KerSqn+2
Since Yk is -orientable, i.e, 0 2 (Uk). By [21 ] that 0 2 (x). Thus x
can be lifted to eEq-1and so (i1)*(x) = 0. This completes the proof.
For simplicity,denote by Fn-1(Z4) the space K(Z4, n-1)^K(Z4, n-
1) as before in the following proof.
Proof of Theorem 2.6. For x 2 Hn-1 (M, Z4), let f(x) = (w^id)Aff(x) 2
H2n(K(Z4, n - 1); fW(n)). For k large, f(x + y) is the following com-
position of maps
id^w^(xxy)
S1 ^ S2n+k id^-ff!S1 ^ T , ^ M+ -!
! S1 ^ fW(n)k ^ (K(Z4, n - 1) x K(Z4, n - 1)) =
= fW(n)k^S1^(K(Z4, n-1)xK(Z4, n-1)) id^~-!fW(n)k^S1^K(Z4, n-
1),
where ~*(l) = l 1+1 l for the basic class l 2 Hn-1 (K(Z4, n-1), Z4).
Identifying fW(n)k ^ S1 ^ (K(Z4, n - 1) x K(Z4, n - 1)) with
{fW (n)k ^ S1 ^ K(Z4, n - 1)} _ {fW (n)k ^ S1 ^ K(Z4, n - 1)}_
_{fW (n)k ^ S1 ^ Fn-1(Z4)}.
It is readily to see that f(x + y) = f(x) + f(y) + g, here g is the
composition
18 FUQUAN FANG AND JIANZHONG PAN
S2n+k+1 id^-ff!S1 ^ T , ^ M+ id^w^-!S1 ^ fW(n)k ^ M+ ^ M+ id^x^y-!
fW(n)k ^ S1 ^ K(Z4, n - 1) ^ K(Z4, n - 1) id^H(~)-!fW(n)k ^ S1 ^ K(Z4, n - 1),
where H(~) is the Hopf constuction of ~.
Now the cofibration
Sk+1^Fn-1(Z4) i^id!S1^fW (n)k^Fn-1(Z4) ! S1^(fW (n)k=Sk)^Fn-1(Z4)
is also a fibration at least in the stable range. It follows immediately
that
( i^id)* : ß2n+k+1(Sk+1 ^Fn-1(Z4)) ! ß2n+k+1(S1^fW (n)k^Fn-1(Z4))
is surjective. On the other hand, it is easy to know that the generator
fi 2 ßs2n(Fn-1(Z4)) ~= Z2 satisfies fi*(l Sq2l) 6= 0. Thus, for the in-
clusion map i, the composition ( i ^ id) O fi 2 ß2n+k+1(S1 ^ fW(n)k ^
Fn-1(Z4)) induces a nontrivial homomorphism on the (2n + k)-th ho-
mology and thus ( i ^ id)* is an isomorphism. Moreover, the generator
g0 2 ßs2n(fW (n)k ^ Fn-1(Z4)) satisfies that g*0(Uk ^ Sq2ln-1 ^ ln-1) 6= 0.
Thus the composition (id ^ x ^ y)(w ^ )( ff) is null homotopy if and
only if = 0. By Proposition 2.1, the proof now follows
by the commutative diagram
Sk ^ Fn-1(Z4) -i^id! fW (n)k ^ Fn-1(Z4)
# id ^ H(~) # id ^ H(~)
Sk ^ K(Z4, n - 1) -i^id!fW(n)k ^ K(Z4, n - 1).
Proof of Theorem 2.7. Let ~ : Kn+k+1 x fWk(n) ! fWk(n) denote the
fiber multiplication. Since d1(w1, w2) = 0, w2 is the composition
w1xvUk ~
T , -! T , x T , - ! fWk(n) x Kn+k+1 - ! fWk(n),
where vUk 2 Hk+n+1 (T ,, Z2) is the second difference of w1 and w2, i.e,
the secondary obstruction to deform w1 to w2. Consider the commu-
tative diagram:
0 a
S2n+k -ff! (T , ^ M+ )T_ (T , ^ M+ ) - ! Wfk (n) ^ K(Z4, n - 1)
k k
S2n+k -ff! (T , x T ,) ^ M+ - b!Wfk (n) ^ K(Z4, n - 1)
where ff0 is a lifting of ff, b = ~(w1 x vUk) ^ x, a = (w1 ^ x) _ c, and
c = i(vUk) ^ x, i : Kn+k+1 ! fWk(n) the inclusion of the fibre. Write
ff0 = ff1 + ff2, here ff1 and ff2 are the factors of the wedge. Note that
OE2(x) = h(b O ff) = h(aff1) + h(aff2) = OE1(x) + h(aff2).
We are going to show h(aff2) = 0.
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS19
As aff2 factors through the map i ^ id : Kn+k+1 ^ K(Z4, n - 1) !
fWk(n) ^ K(Z4, n - 1), it suffices to prove that
(i ^ id)* : ß2n+k(Kn+k+1 ^ K(Z4, n - 1)) ! ß2n+k(fWk(n) ^ K(Z4, n - 1))
is zero. Note the homomorphism
(Sq1^id)* : ß2n+k(Kn+k ^K(Z4, n-1)) ! ß2n+k(Kn+k+1 ^K(Z4, n-1)) ~=Z2
is an isomorphism as it induces an ismomorphism on the (2n + k)-th
Sq1 i
homology groups. The composition Kn+k - ! Kn+k+1 -! fWk(n) is
null homotopy. Thus (i ^ id)* = 0. This completes the proof.
5. Proof of Theorem 1.9
In this section we prove Theorem 1.9. We first study the properties
of the invariants ~M and qM (Sq1i) defined in x1.
Lemma 5.1. Let M be a framed manifold of dimension 2n with n
odd. Let qM : Hn (M, Z2) ! Z2 be the Kervaire quadratic form. For
x 2 Hn-1 (M, Z2i),
(i) n = 3(mod4), [M, x] is reduced bordant to zero iff qM (Sq1i)x = 0.
(ii) n = 1(mod4), [M, x] is reduced bordant to [M0, x0] where x0 2
Hn-1 (M0) iff qM (Sq1ix) = 0.
Proof. Identify the reduced framed bordism group efr2n(-) with the
stable homotopy group ßs2n(-). Recall that ßs2n(K(Z2, n)) = Z2. By
[4] it is easy to see that the homomorphism
(Sq1i)* : ßs2n(K(Z2i, n - 1) ! ßs2n(K(Z2, n))
is identified with the following geometrically defined homomorphism
efr2n(K(Z2i, n - 1))! Z2
[M, x] ! qM (Sq1i)x
By Theorem 3.1 and Lemma 3.7 (i) follows since (Sq1i)* is an iso-
morphism. To prove (ii), note that there is an exact sequence by
Proposition3.6 and Lemma3.7
(Sq1i)*s
ßs2n(K(Z, n - 1)) ! ßs2n(K(Z2i, n - 1)) - ! ß2n(Kn).
This completes the proof.
20 FUQUAN FANG AND JIANZHONG PAN
Now we want to study which bilinear forms ~ can be realized by (n-
2)-connected 2n-dimensional ß-manifolds. Note that a sphere bundle
over Sn+1 with fiber Sn-1 is a ß-manifold if the characteristic map of
the bundle, ` 2 ßn(SO(n)), belongs to the kernel of the stablization
homomorphism S* : ßn(SO(n)) ! ßn(SO). Recall that the homotopy
groups of ßn(SO(n)) are as follows (c.f: [11]):
ßn(SO(n)), n 3, 6= 6
_____________________________________________________________________________
|__n___3,_6=_6_|_8s______|8s_+_1__|8s_+_28|s_+_3_|8s_+_4_8|s_+_58|s_+_68|s_+_*
*7 |
|_ßn(SO(n))_|Z2___Z2___Z2Z|2__Z2_|_Z4___|__Z___|Z2___Z2_|_Z2___|_Z4___|__Z___|
and ß6(SO(6)) = 0.
Let ß : SO(n) ! Sn-1 be the canoincal SO(n - 1)-fiberation. For a
Sn-1 -bundle over Sn+1 with characteristic map ` 2 ßn(SO(n)), say M`, it
is easy to see that Sq2 : Hn-1 (M`, Z2) ! Hn+1 (M`, Z2) is an isomorphism
if and only if ß*(`) 2 ßn(Sn-1 ) = Z2 is nonzero. By duality this implies
that z [ Sq2z = 0 for all z 2 Hn-1 (M`, Z2) if and only if ß*(`) = 0. The
latter is equivalent to the fact of that the bundle has a section.
Lemma 5.2. Let M be a ß-manifold of dimension 2n. Then
(i) ~M (x, x) = 0, 8x 2 Hn-1 (M, Z2) if n = 2(mod4).
(ii) ~M (x, x) = 0, 8x 2 Im(æ2 : T Hn-1 (M, Z4) ! Hn-1 (M, Z2)),
if n is odd where T is the set of elements of order 4.
(iii) If n = 0(mod4), then there is a Sn-1 -bundle over Sn+1 , M, so that
~M (x, x) 6= 0, where x 2 Hn-1 (M, Z2) is a generator.
Proof.For each x 2 Hn-1 (M, Z2), consider the reduced bordism class [M, x] 2
e fr2n(Kn-1) ~=Z2. It is easy to see that x [ Sq2x[M] is a bordism invariant.
One verifies the following map defines a homomorphism
e fr2n(Kn-1)! Z2
[M, x] ! x [ Sq2x[M]
By Remark 3.4 the reduction homomorphism
e fr2n(K(Z, n - 1)) ! e fr2n(Kn-1)
is surjective if n is even.
If n = 2(mod4), let ` 2 ßn(SO(n)) be a generator. By the tables (I)(II) of
[11] it follows that ` lies in the image of the inclusion map ßn(SO(n - 1)) !
ßn(SO(n)). By the remark above this implies that the sphere bundle M`
has a section. Therefore z [ Sq2z = 0 for all z 2 Hn-1 (M`, Z2). On
the other hand, one can verify that [M`, z] 2 e fr2n(Kn-1) is a generator if
z 2 Hn-1 (M`, Z2) is nonzero. This proves (i).
If n = 0(mod4), by [11] there is an element fi 2 kerS* : ßn(SO(n)) !
ßn(SO) so that ß*(fi) is nonzero. This proves (iii).
SECONDARY BROWN-KERVAIRE QUADRATIC FORMS AND ß-MANIFOLDS21
If n is odd, by Lemma 5.1 the homomorphism
q(Sq1) : e fr2n(Kn-1) ! Z2
[M, x] ! qM (Sq1x)
is an isomorphism. Thus there is a ffi 2 Z2 so that ffiqM (Sq1x) = x[Sq2x[M]
for all [M, x]. In particular, if x can be lifted to the Z4-coefficient class w*
*ith
order 4, Sq1x = 0 and so x [ Sq2x = 0. This completes the proof.
Now we are ready to prove Theorem 1.9.
Proof of Theorem 1.9.By Theorem 1.6 the data of invariants are homotopy
invariants of the manifolds. Thus the homotopy and homeomorphism clas-
sification of such manifolds are the same.
There is an isomorphism
e fr2n(K(H, n - 1)) ~=ßs2n(K(H, n - 1)).
Therefore from Theorem 3.1 there is a reduced framed bordism class [M, f] 2
fr2n(K(H, n - 1)) corresponding to the given algebraic data [H, ~, OE] (resp.
[H, ~, OE, !] ) if n is even (resp. odd). This together with Lemmas 5.1 and
5.2 implies this is an 1-1 correspondence.
Add some Sn-1 x Sn+1 to M if necessary so that f* : Hn-1(M) ! H is
surjective. By surgery on M we may assume further that f* : Hn-1(M) ! H
is an isomorphism and Hn(M, Q) = 0. Therefore the data can be realized by
a (n - 2)-connected 2n-dimensional ß-manifold, M, so that Hn(M, Q) = 0
and ß(M) = [H, ~, OE] (resp. [H, ~, OE, !].
Now it suffices to prove that the map ß is injective.
Suppose that Mi, i = 1, 2, are two framed smooth manifolds with the
same data (for TOP manifold, the similar argument works identically). Note
that the Kervaire invariants of Mi must vanish since Hn(Mi, Q) = 0. By
the assumption there are maps fi : Mi ! K(H, n - 1), so that (M1, f1)
and (M2, f2) are reduced framed bordant, where fi induces an isomorphism
on the (n - 1)-th homology groups. Since both Mi framed cobordant to
some homtotopy spheres, there is a framed homotopy sphere, , so that
(M1, f1) and (M2# , f2) are framed bordant. By Freedman [8] or Kreck
[14] it follows that M1 and M2# are diffeomorphic since Hn(Mi, Q) = 0.
Therefore M1 and M2 are almost diffeomorphic. The same argument as
above applies to show that M1 and M2 are homeomorphic to each other.
This completes the proof.
Acknowledgements: This paper is a revised version of [7]. This work
began during first author's studys and visits at Jilin University, Nankai
Institute of Mathematics, Universität Mainz, Universität Bielefeld, the Max-
Planck-Institut für Mathematik and I.H.E.S. He would like to express his
sincere thanks to all of those Institutions and to Yifeng Sun and Xueguang
Zhou for their encouragements and supports, to Matthias Kreck for teaching
him his surgery theory [14]. The second author joins the project at the later
22 FUQUAN FANG AND JIANZHONG PAN
part of the work mainly for clarifying the argument. Part of the work was
done during his visit to Korea University. He would like to thank Prof.Woo
Mooha and Department of Mathematics Education for the hospitality.
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Nankai Institute of Mathematics, Nankai University, Tianjin 300071,
P.R.C
E-mail address: ffang@sun.nankai.edu.cn
Institute of Math.,Academia Sinica ,Beijing 100080 ,China and
Department of Mathematics Education , Korea University , Seoul ,
Korea
E-mail address: pjz62@hotmail.com