Acta Mathematica Sinica, New Series
1996, Vol.12, No.2, pp. 162–166
Involutions Fixing the Disjoint Union
of Copies of Even Projective Space
Hou Duo & Bruce Torrence
Abstract.
We show that for any differentiable involution on an r-dimensional manifold (M, T )
whose fixed point set F is a disjoint union of real projective spaces of constant dimension 2n, we
have: if r = 4n then (M, T ) is bordant to (F × F, twist), if 2n < r = 4n then (M, T ) bounds.
Keywords.
Involution, Cobordism
Introduction
Suppose M r is a closed manifold and T : M → M is a differentiable involution on M . Let
(M, T ) denote the unoriented bordism class of this involution (see, for example, [2]). Let F
denote the set of points fixed by T . In this paper we will prove the following
Theorem. Given a closed r-dimensional manifold with differentiable involution (M, T ), with
fixed point set being any disjoint union F = ∪RP (2n), here n being fixed with 2n < r, then if
r = 4n, (M, T ) is equivariantly bordant to (F × F, twist); if r = 4n, then (M, T ) bounds.
Given any manifold N n with real vector bundle ξ c → N . Let (N, ξ) ∈ Nn (BO) denote the
bordism class of the stable bundle determined by the map
N → BO(c) → BO
classifying ξ, with n fixed. The classes (RP (n), ξ) as ξ ranges over all vector bundles form a
Z2 -subspace of Nn (BO), where the sum (RP (n), ξ) + (RP (n), η) represents the disjoint union.
According to [6], if λ → RP (n) denotes the canonical twisted line bundle and if n is even,
a basis for this subspace is {(RP (n), jλ)|0 ≤ j ≤ n}.
Proof of Theorem
Suppose we are given a closed manifold with differentiable involution (M, T ) with fixed
point set F = ∪RP (2n) where n ≥ 1 is fixed. In [2], Conner and Floyed showed that if Nrz2
denotes the group of unoriented bordism classes of involutions on r-manifolds, the composition
Nrz2 −→
r
Nj (BO(r − j)) −→
j=0
Received June 5, 1993. Revised February 23, 1994.
r
j=0
Nj (BO)
Hou Duo et al.
Involutions Fixing the Disjoint Union of Copies of Even Projective Space
163
is monic, where the first map assigns to each involution its fixed point data, and the second is
induced by the standard inclusion BO(.) → BO. Let γ → F denote the normal bundle of F in
M . Then either (F, γ) = 0 ∈ N2n (BO) and (M, T ) bounds, or we may write
(F, γ) =
(RP (2n), jλ),
j∈Q
where Q is a nonempty subset of {0, 1, 2, · · · , 2n}. We will show that the latter is impossible.
Let k = r − 2n be the codimension of RP (2n) in M r . Let γjk → RP (2n) denote the normal
bundle of the RP (2n), with fixed data (RP (2n), jλ). Then the Stiefel-Whitney class
j
Wj (γjk ) = Wj (jλ) =
aj = 0
j
since j ≤ 2n, and hence k ≥ j.
According to Kosniowski and Stong[3.p.309] , if f (x1 · · · xr ) is any symmetric polynomial over
Z2 in r variables of degree at most r, then
f (1 + y1 , 1 + y2 , · · · , 1 + yk , z1 , z2 , · · · , z2n )
[RP (2n)],
f (x1 · · · xr )[M ] =
(1 + y1 )(1 + y2 ) · · · (1 + yk )
j∈Q
where j is, as above, the indexing copies of RP (2n) in F , and where the expressions are evaluated
by replacing the elementary symmetric functions σi (x), σi (y) and σi (z) by the Stiefel-Whitney
classes Wi (M ), Wi (γjk ), and Wi (RP (2n)) respectively, and evaluating the resulting cohomology
class on the fundamental homology class of M r , or RP (2n) in our case, since j ∈ Q, j ≤ k.
The expression (1 + y1 )(1 + y2 ) · · · (1 + yk ) in each denominator on the right is equal to the
total class W (jλ) = (1 + a)j . Now we divide Q into three subsets:
E=
{j|j even},
O1 = j|j odd, and
O2 = j|j odd, and
j
2n + 1
=
,
2
2
j
2n + 1
=
.
2
2
Proposition 1. E = ∅.
∗
Proof. If E = ∅, let j= max{j|j ∈ E}. We choose the symmetric polynomial:
∗
2n−∗j j−1
k
k
1 + σ1 (x) +
f (x1 · · · xr ) = σ1 (x) −
,
1
1
2n
k
,
f (x1 · · · xr ) = σ1 (x) −
1
From [3] we have the formula
j> 0,
if
j= 0.
k
k
j
σ1 (1 + y, z) =
+ σ1 (y) + σ1 (z) =
+ 1+
a.
1
1
1
Then for each j ∈ Q, the formula is

∗
∗

 a2n−j (1 + a)j−1 ,
f (1 + y, z) = a2n ,


0,
∗
if j is even, j> 0,
∗
if j is even, j= 0,
if j is odd.
∗
if
∗
164
Acta Mathematica Sinica, New Series
∗
Vol.12 No.2
∗
Since the degree of f (x) is (2n− j) + (j −1) = 2n − 1 < r, or 2n < r, one has 0 = f (x)[M ]. On
the other hand,
∗
Case 1. j> 0
∗
∗
a2n−j (1 + a)j−1
f (1 + y, z)
[RP (2n)] =
j [RP (2n)] =
j
(1 + a)
(1 + a)
j∈Q
j∈E
j∈E
∗
∗
∗
2N − 1+ j −j
∗
j
,
∗
2N > 2n.
∗
When j −j > 0, there will be j>j −j − 1 > 0. Hence there is a power of dyadic expansion of j
∗
which does not appear in the dyadic expansion of j −j − 1. Hence
∗
∗
2N − 1+ j −j
j −j − 1
=
= 0.
∗
∗
j
j
∗
∗
When j −j = 0, each power in the dyadic expansion of j appears in the dyadic expansion of
∗
2N − 1+ j −j, (= 1 + 2 + · · · + 2N − 1). Hence
∗
2N − 1+ j −j
∗
j
= 1.
∗
Case 2. j= 0
f (1 + y, z)
[RP (2n)] = a2n [RP (2n)] = 1.
j
(1
+
a)
j∈Q
This is a contradiction. As a result Proposition 1 is established.
Proposition 2. O1 = ∅
∗
Proof. If O1 = ∅, let j= max{j|j ∈ 01 }. Let the symmetric polynomial

k


, if k even,
 σ3 (x) + σ2 (x) −
2
F (x1 · · · xr ) =
k


,
if k odd.
 σ3 (x) −
3
By [3] and because of j being odd,

k
j
2n + 1


, if
=
,
 a2 + a +
2
2 2 σ2 (1 + y, z) =
k
j
2n + 1


,
if
=
,
a +
2
2
2
 2
if k is even,
a + a,




j
2n + 1
 a2 + k ,
if k is odd and
=
,
2 2 σ3 (1 + y, z) = 3
 k

j
2n + 1


,
if k is odd and
=
.

3
2
2

j
2n + 1


if
=
,
 a2 ,
2
2 F (1 + y, z) =
j
2n + 1


if
=
.
 0,
2
2
Hou Duo et al.
Involutions Fixing the Disjoint Union of Copies of Even Projective Space
We divide this into two cases and discuss them separately. In the first case,
∗
165
∗
j
2c1
= 0.
Let 2n = 2c1 + 2c2 + · · · + 2ch , c1 > c2 > · · · > ch. Apparently we’ll have j ≤j≤ k. Now we
∗
choose the symmetric polynomial, for 2n > 2 (j≥ 1),
∗
f (x1 · · · xr ) = (F (x) + 1)(j−1)/2 · F (x).
∗
∗
∗
And the degree of f (x) is 3 + (3/2) · (j −1) = 1/2(j −3)+ j< 2n + k = r. Therefore,
∗
2N − 1+ j∗ −j (1 + a)j−1 · aα
[RP
(2n)]
=
O = f (x)[M ] =
= 1.
(1 + a)j
2n − 2
j∈O1
j∈O1
For 2n = 2 (01 = {1}, O2 = ∅), f (x1 , · · · , xr ) = 1, we can obtain
O = f (x)[M ] =
In the second case,
∗
j
2c1
1
[P R(2)] = 1.
1+a
=1
∗
i.e 2c1 ≤j≤ k.
We choose the symmetric polynomial,
∗
∗
f (x1 · · · xr ) = (1 + F (x))(j−1)/2 · F (x)(2n−j+1)/2 .
∗
∗
Since j ∈ O1 , hence f (1 + y, z) = (1 + a)j−1 · a2n−j+1 and the degree of f (x) is
∗
∗
3/2((j −1) + (2n− j +1)) = 2n + n < 2n + 2c1 ≤ r.
Similarly, we have
∗
∗
(1 + a)j−1 a2n−j+1
O = f (x)[M ] =
[RP
(2n)]
=
(1 + a)j
j∈O
j∈O
1
1
∗
2N − 1+ j −j
∗
j −1
= 1.
According to Case 1 and Case 2, there is a contradiction. This completes the proof.
Suppose the subset O2 = ∅, now let A1 = {aj |aj = 2n − j + 1, j ∈ O2 }, 4 ≤ aj ≤ 2n. And
we form a sequence of subsets of A1 by induction:
If Ap ⊂ Ap−1 , (2 ≤ p) exists. Let
aq
ap
dp
x
=
, ap , aq ∈ Ap .
2 = min 2 |
2x
2x
(For example, if ap = 25 + 2, aq = 26 + 24 + 2 then
ap
ap
aq
aq
ap
ap
aq
= 1,
=
= 0,
=
= 1,
=
=
5
4
6
6
2
2
2
2
2
2
24
aq
ap
min 2x |
=
= min{24 , 25 , 26 }.)
2x
2x
aq
25
= 0,
166
Acta Mathematica Sinica, New Series
Vol.12 No.2
ai
= 1, ai ∈ Ap .
Let Ap+1 = ai |
2dp
Then we obtain a sequence of subsets as follows:
∗
A1 ⊃ A2 ⊃ · · · ⊃ Ah = {a},
∗
∗
∗
where a= 2n− j +1, j∈ O2 .
Proposition 3. When r < 4n, i.e k < 2n, then O2 = ∅.
Proof. If O2 = ∅, we choose the symmetric polynomial,
f (x1 · · · xr ) =
∗
2n−a∗
k
σ2 (x) −
.
2
∗
The degree of f (x) is 2(2n− a) = 2(j −1) ≤ 2k < 2n + k = r, and we have
∗
∗
2N + 2n− a∗ −j a2n−a (1 + a)2n−a
[RP (2n)] =
O = f (x)[M ] =
.
∗
a
(1 + a)j
j∈O
j∈O
2
∗
2
∗
∗
∗
Noting that 2N + 2n− a −j = 2N − 1 + (2n − j + 1)− a= 2N − 1 + aj − a, when aj =a, there
∗
exists a power 2dp , 1 ≤ p ≤ h − 1, |aj − a | = 2dp +higher degree powers. The power 2dp does
∗
not appear in the dyadic expansion of 2N − 1 + aj − a, but does appear in the dyadic expansion
∗
of a . Hence,
N
∗
2 − 1+ a −aj
= 0.
∗
a
∗
When a= aj , we have ( ) = 1. Thus we have a contradiction, and we conclude that when
r < 4n, (M, T ) is a bounding involution. If r = 4n, [3, p320] gives that (M, T ) is bordant to
(F × F , twist). If r > 4n, [3, p320] gives that the involution (M, T ) is a boundary.
References
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Royster D C, Involutions fixing the disjoint union of two projective spaces, Indiana Math J, 1980, 29:
267–276.
[5] Stong R E, Involutions fixing projective spaces, Michigan Math J, 1966, 13: 445–457.
[6] Torrence B F, Bordism classes of vector bundles over real projective spaces, Proceeding Amer Math Soc,
1993, 118:963–969.
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[1]
[2]
[3]
[4]
Hou Duo
Department of Mathematics
Hebei Teachers University
Shijiazhuang, 050016
China
Bruce Torrence
Department of Mathematics
Georgetown University
Washington, D. C. 20057
U.S.A.