For 4k dimensional orientable manifolds, w4k−1 = 0
Aleksandar Milivojevic
What follows is a proof given by Massey [1], that w4k−1 (M 4k ) = 0 for an orientable 4k manifold M .
(Note that this encompasses the earlier result of Whitney that an orientable four-manifold has w3 = 0.)
We denote by wi and νi the Stiefel-Whitney and Wu classes of M , respectively. All calculations are
done mod 2.
We show that xw4k−1 = 0 for all x ∈ H 1 (M, Z2 ), which will imply w4k−1 = 0 by Poincare duality.
Observe that by the Wu formula w = Sq(ν) on total classes, we have w4k−1 = Sq2k−1 (ν2k ) for degree
reasons. So, we want to show that xSq2k−1 (ν2k ) = 0 for all x of degree one.
Since Sqi (x) = 0 for i > 1, the Cartan formula for Steenrod squaring gives us
Sq2k−1 (xν2k ) = xSq2k−1 (ν2k ) + Sq1 (x)Sq2k−2 (ν2k )
and so
xSq2k−1 (ν2k ) = Sq1 (x)Sq2k−2 (ν2k ) + Sq2k−1 (xν2k ) = x2 Sq2k−2 (ν2k ) + Sq2k−1 (xν2k ).
First let us show that the term Sq2k−1 (xν2k ) vanishes. For this we recall the Adem relations, which
state
b 2i c X
j−k−1
i
j
Sq Sq =
Sqi+j−k Sqk
i − 2k
k=0
i+1 0
for 0 < i < 2j. As a special case we obtain the relation Sq1 Sqi = i−2
Sq , which gives us
1 Sq
Sq1 Sqi = Sqi+1 if i is odd.
So, we have
Sq2k−1 (xν2k ) = Sq1 Sq2k−2 (xν2k ).
Since Sq2k−2 (xν2k ) is an element of degree 4k − 1, evaluating Sq1 on it corresponds to multiplying
with ν1 . Since ν1 = 0 (because M is orientable, and ν1 = w1 by the Wu formula), we get
Sq2k−1 (xν2k ) = 0.
So, for now we have
xw4k−1 = x2 Sq2k−2 (ν2k ).
Before we proceed, let us make the preliminary observation, using the Cartan formula, that
i
i i+j
j i
1
j i−j
Sq (x ) =
(Sq (x)) x
=
x .
j
j
If i is a power of 2, then ji = 0 for all j except j = 0, i, when it is equal to 1.
1
Now we consider x2 Sq2k−2 (ν2k ). By the Cartan formula and the above observation that only squares
of degree 0 or 2 survive when evaluating x2 , we have
Sq2k−1 (x2 ν2k ) = Sq2 (x2 )Sq2k−4 (ν2k ) + x2 Sq2k−2 (ν2k ),
and thus
x2 Sq2k−2 (ν2k ) = Sq2k−2 (x2 ν2k ) + x4 Sq2k−4 (ν2k ).
We show that Sq2k−2 (x2 ν2k ) = 0. Indeed, since x2 ν2k is of degree 2k + 2, we have Sq2k−2 (x2 ν2k ) =
ν2k−2 x2 ν2k , and now we see
Sq2k−2 (x2 ν2k ) = ν2k−2 x2 ν2k
= ν2k x2 ν2k−2
= Sq2k (x2 ν2k−2 )
= (x2 ν2k−2 )2
2
= x4 ν2k−2
.
Using our previous observation about squares of powers of x again, we have
2
Sq1 (ν2k−2
)=0
and so
2
2
= Sq1 (x3 ν2k−2
).
x4 ν2k−2
2
2
) = 0, and so Sq2k−2 (x2 ν2k ) = 0.
is of degree 4k − 1, Sq1 (x3 ν2k−2
Since x3 ν2k−2
Therefore,
x2 Sq2k−2 (ν2k−2 ) = x4 Sq2k−4 (ν2k ).
Iterating this calculation gives us
x2 Sq2k−2 (ν2k−2 ) = x4 Sq2k−4 (ν2k ) = x8 Sq2k−8 (ν2k ) = x16 Sq2k−16 (ν2k ) = · · ·
We can apply this calculation as long as the power of 2 appearing is less than or equal to 2k. Either
the highest such power of 2 is strictly less than 2k, in which case another iteration of the calculation
will give only the one term that we showed vanishes in the base case (and an analogous computation
shows it vanishes now as well), or n = 4k is a power of 2, in which case this reiteration terminates
with the equality
x2 Sq2k−2 (ν2k−2 ) = x2k Sq0 (ν2k ) = ν2k x2k = (x2k )2 = x4k .
However, observe that in this case we have
Sq1 (x4k−1 ) = (4k − 1)x4k = x4k
on the one hand, since 4k − 1 is odd. On the other hand, we already saw that Sq1 (x4k ) = 0. So,
x2 Sq2k−2 (ν2k−2 ) = 0, and we conclude that
xw4k−1 = 0
for all x of degree one, and so w4k−1 = 0.
Remark. Wu proved that w4k+1 (M 4k+2 ) = 0 for M of dimension 4k + 2. The above proof does not
cover this case since it relies heavily on Sqi evaluating to 0 on elements of degree n − i (where n is the
dimension of the manifold) for odd i.
Reference
[1] Massey, W. S. ”On the Stiefel-Whitney classes of a manifold.” American Journal of Mathematics
82.1 (1960): 92-102.
2