In this chapter, we consider a Riemannian manifoldMn and an isometricTr-action
on M fixing a point x ∈ M. We assume r is bounded below by a logarithmic function ofn. We use the Griesmer bound from the theory of error-correcting codes to prove an estimate on the codimensions of fixed-point sets of involutions in Tr.
This estimate will be used in the next section.
Proposition 4.1. Let n ≥4, and assume Tr acts effectively by isometries on Nn
with fixed point x. Let c be a nonnegative integer.
1. If r > log2n + 2c + 1.5, then there exists an involution σ ∈ Tr satisfying codF(σ)≡0 mod 4 and codF(σ)≤ n−c−1
2. Suppose σ ∈ Tr has been chosen as in the previous part such that codF(σ) is minimal. If r > log2n+ c2 + 2.5, then there exists an involution τ ∈ Tr
satisfying F(τ) 6⊆ F(σ), codF(τ) ≡ 0 mod 4, codF(σ)∩F(τ) ≡ 0 mod 4, and codF(τ)≤ n−c−1
2 .
Note that, by the connectedness theorem, a totally geodesic inclusionNn−k,→
Mn with k ≤ n−c−1
2 is (c+ 2)-connected. In particular, π1(N) ∼= π1(M), so N is
simply connected if M is.
We proceed to the proof of Proposition 4.1. Choose a basis of TxM such that
the image of Zr2 ⊆Tr under the isotropy representation Tr ,→ SO(TxM) lies in a
copy of Zm 2 ⊆ Tm ⊆ SO(TxM) where m= n 2 . Endow Zm
2 with a Z2-vector space
structure, and consider now the linear embedding ι:Zr
2 ,→Zm2 .
Consider the first statement. The bound on r implies r ≥ 2. Consider the composition of Zr
2 → Zm2 → Z2, where the last map takes τ = (τ1, . . . , τm) to
P
τi mod 2. Clearly there exists Zr2−1 inside the kernel, and for each σ ∈ Z
r−1 2 ,
codF(σ) is twice the even weight ofι(σ) = τ. Hence it suffices to prove the existence of σ ∈Zr−1
2 with codF(σ)≤ n
−c−1 2 .
Equivalently, it suffices to prove that the weight of the σ ∈ Zr−1
2 whose image
in Zm2 has minimal (positive) Hamming weight is at most n−4c−1. If this is not the case, then the image of thatσ has weight at least n−4c. We now apply the Griesmer bound from the theory of error-correcting codes:
Theorem 4.2. (Griesmer bound) If Zu2 ,→Zm
every element in the image has weight at least w, then m≥ u−1 X i=0 lw 2i m .
This bound implies n 2 ≥ jn 2 k ≥ r−2 X i=0 n−c 2i+2 ≥ r−3−dc 2e X i=0 n−c 2i+2 + r−2 X i=r−2−dc 2e 1.
Observe that the lower bound onrimpliesr−3−c
2
≥0, so the second inequality is justified. Computing the geometric sum, canceling like terms, and rearranging yields
2r−dc2e−1 ≤n−c≤n, a contradiction to the assumed lower bound on r.
We now prove the second statement of Proposition 4.1. First, observe that the lower bound on rimpliesr ≥4. Let σ ∈Zr
2 be as in the statement. We define three
linear maps Zr
2 →Z2. For the first, fix a componentisuch that thei-th component
of ι(σ) is 1 (which corresponds to a normal direction of F(σ)), and define the map Zr2 → Z2 as the projection onto the i-th component of ι(τ). For the second
map, assign τ ∈ Zr
2 to the sum of the components of ι(τ) (as we did previously to
choose σ). And for the third map, let I be the subset of indices i where the i-th component of ι(σ) is 0, and define the map by assigning τ ∈ Zr
2 to the sum over
i ∈ I of the i-th componets of ι(τ). The intersection of the kernels of these three maps contains a Zr2−3, and every τ ∈ Zr−3
2 satisfies F(τ) 6⊆ F(σ), codF(τ) ≡4 0,
i-component of Zm2 , hence we have an (injective) linear code Z2r−3 → Zm2 −1 where twice the weight of the image of τ ∈Zr−3
2 is equal to codF(τ).
It therefore suffices to prove that the element inZr2−3 whose image has minimal weight has weight at most n−2c−1. The Griesmer bound implies
n 2 −1≥ r−4 X i=0 n−c 2i+2 .
Splitting the sum on the right-hand side into the first r−4−c
2
terms plus the last c
2
+ 1 terms and estimating as in the previous case yields a contradiction to the lower bound on r.
Chapter 5
Theorem P
In this chapter, we state and prove Theorem P. The “P” is for periodicity. The assumptions involve a positively curved manifold with symmetry rank depending on a constant c, and the conclusion is that the first c rational cohomology groups ofM are 4-periodic. The following chapter will show that Theorem P easily implies Theorem B as well.
We proceed to the statement of Theorem P. For an integer m, let δ(m) be 0 if m is even and 1 if m is odd.
Theorem P. Letm ≥c≥0. Assume Mm is a closed, simply connected, positively
curved manifold with an effective, isometric Ts-action satisfying s≥2 log
2m+
c
2−
δ(m). If F is a component of the fixed-point set of Ts, then there exists H ⊆ Ts with F-component P such that P ,→M is c-connected and H∗(P;Q)is 4-periodic. Moreover, P may be chosen to satisfy dimP ≡m mod 4 and dimP ≥c.
Using Theorem 3.1, an immediate corollary is the following:
Corollary 5.1. Letm≥c≥0, and letMm be a closed, simply connected, positively curved manifold with an effective, isometricTs-action. Ifs≥2 log
2m+
c
2, then there
exists a c-connected inclusion P ,→M such that H∗(P;Q) is 4-periodic.
We now set up the proof of Theorem P. To simplify various statements in the proof, we make the following definitions. SupposeMmis a closed, simply connected,
positively curved manifold, and suppose T is a torus acting almost effectively by isometries on M with fixed-point component F. Finally, define δ to be 0 if m is even and 1 if m is odd, and suppose that cis a nonnegative integer.
For an element or subgroup H of T, let F(H) denote the F-component of MH. Since T is abelian and connected, T acts on F(H). In general, if T acts almost effectively on S and restricts to an action on N ⊆ S, let dkS(N) denote
the dimension of the kernel of the action on N. Recall that we called this quantity dim ker(T|N) in Chapter 3. Equivalently, if T0 is a maximal subtorus of T acting
almost effectively on N, then
dkS(N) = dimT −dimT0.
For example, if T itself acts almost effectively on N, then dkS(N) = 0, and if (at
least) a codimension one subtorus acts almost effectively, then dkS(N)≤1.
Next, we will write conM(N) ≥ c when N ,→ M is at least c-connected. De-
M whose dimensions are at least c and congruent to m modulo four and whose inclusions into M are c-connected:
C ={F(H) | codMF(H)≡4 0, dimF(H)≥c, and conMF(H)≥c+ 2}.
We now define three properties that M might satisfy. The first with S = M is the assumption of Theorem P, the last is the conclusion, and the intermediate property is one which occurs in the course of the proof that S impliesP. With the notation above, we define the following properties:
Property S: There exists S ∈ C and T0 ⊆ T acting almost effectively on S with dim(T0)≥2 log2(dimS) + c2 −δ.
Property I: Property S holds, and there exist N ∈ C and an involution σ ∈ T0 with codNF(σ|N)≡4 0 and 0<codNF(σ|N)<dim(S)/(3·2dkS(N)).
Property P: There exists P ∈ C such thatH∗(P;Q) is 4-periodic.
Theorem P can now be stated as “S with dim(S) = m implies P.” The basic strategy is to use double induction as follows. Given that S holds for some S, we consider the situation with dim(S) minimal. The first step is to show that this implies I for some dim(N) ≤ dim(S). Now take N with minimal dimension such that I holds. We then show that P holds by using our choice ofS and N.
With the setup complete, we proceed to the proof of Theorem P. Roughly speaking, the first step is to show that S impliesI:
Lemma 5.2. If S ∈ C has minimal dimension such that Property S holds, then Property P holds or I holds for some N with dim(N)≤dim(S).
Proof. First, by dividing T0 by the kernel of its action on S, we may assume that T0 acts effectively. If dim(S)≤5, thenS is trivially 4-periodic. HenceP holds with P =S.
Suppose therefore that dim(S)≥6. Using Property S, we conclude dim(T0) ≥ 2 log2(dimS) + c
2−δ
≥ log2(dimS) + c
2+ log2(6)−1 > log2(dimS) + c
2+ 1.5.
By Proposition 4.1, there exists σ ∈ T0 with codSF(σ|S) ≤ 12(dimS−c−1) and
codSF (σ|S) ≡ 0 mod 4. Choose the σ satisfying these properties which has the
minimal codimension inS. Since codS≡0 mod 4, we have codF (σ|S)≡0 mod 4.
In addition, the connectedness theorem and our assumption onSimpliesF (σ|S),→
S ,→M is (c+2)-connected. Finally, codSF(σ|S)≤(dimS−c−1)/2 and dimS ≥c
imply dimF(σ|S)≥(dimS+c+ 1)/2≥c. Hence F (σ|S)∈ C.
We claim that dkSF (σ|S) ≤ 2. If not, then N is a fixed by a 3-torus acting
effectively on S. Since one of the involutions, say σ0, inside the subgroup of invo- lutions has codF (σ0|S)≡ 0 mod 4 and σ0 6∈ hσi, we obtain a contradiction to the
minimality of codF (σ|S). Hence we have dkSF (σ|S)≤2.
that a 2-torus acting effectively on S fixes F(σ|S). Then there exists an involution
σ0 with F(σ|S) ⊆ F(σ0|S) ⊆ S with all inclusions strict. It follows that F(σ|S) is
the transverse intersection ofF(σ0|S) andF(σσ0|S). If dimF(σ|S)≤ 12dim(S), then
S holds for F(σ|S) in place of S, a contradiction to the minimality of dim(S). On
the other hand, dimF(σ|S)≥ 12dim(S) implies
2 codSF(σ0|S) + 2 codSF(σσ0|S)≤dim(S),
which implies P holds with P =S by the periodicity theorem.
This leaves us with the case dkSF(σ|S) ≤ 1. Let T00 denote a codimension 1
subtorus of T0 that acts almost effectively on F(σ|S). By minimality of dim(S),
we have dim(T00)<2 log2(dimF(σ|S)) + c2 −δ. Since dim(T00)≥dim(T0)−1 and
dim(T0)≥2 log2(dimS) +2c−δ, this implies 2 log2(dimF(σ|S))>2 log2(dimS)−1,
or
codF(σ|S) = dimS−dimF(σ|S)<
1−√1 2 dimS < 1 3dimS. Taking N =S and i= 0, we see that I holds with dim(N)≤dim(S).
We now show the second part of the proof, which is roughly that I implies P. Table 5.1 displays a summary of the notation used in the proof of Lemma 5.3. All submanifolds shown are inC, that is, each is theF-component of the fixed-point set of some subgroup of T, and each has dimension both divisible by four and at least c, and the inclusion of each inM isc-connected. The codimensionk of F(σ),→N satisfies 0 < k < dim3·2jS.
M x T where dim(T)≥2 log2(dimM) + c2 −δ
|
S x T0 where dim(T0)≥2 log2(dimS) + c2 −δ
|
N x T00 where dim(T00)≥dim(T0)−j
| |
F(σ) F(τ)
| |
F(hσ, τi)
Table 5.1: Summary of notation in the proof of Lemma 5.3.
Lemma 5.3. Assume dim(S) is minimal such that S holds and that dim(N) is minimal such that I holds. Then P holds.
As we already established, Lemmas 5.2 and 5.3 imply Theorem P. We spend the rest of the chapter proving Lemma 5.3.
Let T00 denote a subtorus of dimension dim(T00) = dim(T0)−dkS(N) acting
almost effectively on N. After dividingT00 by the kernel of this action if necessary, we assume that T00 acts effectively. Observe that the image of σ (which we also denote byσ) in the quotient still acts effectively. Without loss of generality, we may assume F(σ|N) has minimal codimension among those with codimension divisible
The assumption in S that dim(T0)≥2 log2(dimS) +2c−δ and the assumption in I that 4≤codNF (σ|N)<dim(S)/(2dkS(N)+1.5) imply
dim(T00) = dim(T0)−dkS(N)>log2(dimN) +
c
2+ 2.5.
Hence Proposition 4.1 implies the existence of τ ∈ T0 with F(τ|N) 6⊆ F(σ|N),
codNF(τ|N)≡4 0, codN(F(σ|N)∩F(τ|N))≡4 0, and
0<codNF(τ|N)≤
1
2(dim(N)−c−1). Choose such a τ with minimal codNF(τ|N).
Claim. F(τ|N), F (στ|N), and F (σ|N)∩F(τ|N) are in C.
Proof. First, our choice of τ implies all three codimensions are congruent to zero modulo four. Second, all three dimensions are at least csince F (σ|N)∩F(τ|N) has
dimension at least
dim(N)−codNF(σ|N)−codNF(σ|N)≥dimN −2
dimN −c−1 2
≥c. Third, this same estimate shows by the connectedness theorem that
F (σ|N)∩F(τ|N),→F(τ|N),→N
is c-connected. Since N is c-connected in M, the inclusions of F (σ|N)∩F(τ|N)
F (σ|N)∩F(τ|N),→F(στ|N) is t-connected with
t ≥ dimF (στ|N)−2 codF(στ|N)(F(σ|N)∩F(τ|N)) + 1
= dimN −codNF(σ|N)−codNF(τ) + 1
≥ c+ 1.
Hence F(στ|N),→M isc-connected by seeing the map on homotopy as that which
goes via F(σ|N)∩F(τ|N), F(τ|N), N, and finally M.
As in the previous lemma, we use the the minimality of codNF(τ|N) to prove
an upper bound for dkNF(τ|N): Claim. dkNF(τ|N)≤3.
Proof. Indeed, suppose a 4-torus T4 acting effectively on N fixes F(τ|N). Choose
a Z2
2 ⊆ T4 with the property that every τ
0 ∈
Z22 satisfies codNF(τ0|N) ≡4 0 and
codN(F(σ|N)∩F(τ0|N))≡4 0, then choose τ0 ∈Z22\ hτi.
It follows that F(τ0|N) lies strictly between F(τ|N) and N. Since F(τ|N) 6⊆
F(σ|N), we have F(τ0|N) 6⊆ F(σ|N) as well. This contradicts to the minimality of
codNF(τ|N).
The remainder of the proof now follows a sequence of claims. In each, we claim that we may assume something. The idea in each case is to use the periodicity theorem and the minimality of dim(S) to show that P holds if the claim does not. At the end of the sequence of claims, we conclude the proof by showing that, if
P does not hold, then the combination of assumptions from the claims implies a contradiction to the minimality of dim(N).
The first claim improves upon the previous one:
Claim. We may assume dkNF(τ|N)≤2.
Proof. Indeed, suppose dkNF(τ|N) = 3. We show that this impliesP.
There exists τ0 ∈ T00 such that F(τ0|N) 6⊆ F(σ|N), codNF(τ|N) ≡4 0, and
F(τ|N) ⊆ F(τ0|N) ⊆ N with all inclusions strict. Choose such a τ0 with minimal
codNF(τ0|N). Choose a basis of the isotropy representationT0 ,→SO(TpN) so that
the actions of σ,τ, and τ0 are given by
σ = diag(−I,−I,−I, I, I, I), τ = diag(−I,−I, I,−I,−I, I), τ0 = diag(−I, I, I,−I, I, I),
where the blocks are of sizey,x−y, k−x,z,l−x−z, and dim(F(σ|N)∩F(τ|N)),
where k = codNF(σ|N) and l = codN F(τ|N). Note that we can replace τ0 by τ τ0
if necessary to ensure that 2y ≤ x. In addition, note that maximality of F(τ|N)
implies 2x≤k, y >0, andz >0. Observe thatF τ0|F(στ|N) andF τ τ0|F(στ|N) intersect transversely inF (στ|N)
and have (positive) codimensions x and x−y, respectively. If 2y+ 2(x−y)≤dimF(στ|N),
then the periodicity theorem impliesP withP =F(στ|N). We claim the other case
leads to a contradiction. Indeed, if
2x >dimF(στ|N) = dimF(τ|N)−k+ 2x,
then the assumption in I implies dimF(τ|N)< dimS 2dkSN+1.5 ≤ dimS 2(dkSN+3)/2 = dimS 2(dkSF(τ|N))/2.
Using the assumption in S on the size of dim(T0), we conclude from this that S
also holds for F(τ|N), a contradiction to the minimality of dimS.
We will soon show that, in fact, we may assume dkN F(τ|N)≤1, but we must
first show the following:
Claim. We may assume dimF(τ|N)≥2k.
Proof. We assume dimF(τ|N) < 2k and show that P holds. Suppose first that
dkSN ≥1 or dkNF(τ|N)≤1. This together with the previous claim implies
dkNF(τ|N)≤dkSN + 1.
Combining this with the equality
dkSF(τ|N) = dkSN + dkNF(τ|N),
we have
Using this together with the assumption in I, we have dimF(τ|N)≤2k ≤ dimS 2dkSN+0.5 ≤ dimS 2(dkSF(τ|N))/2,
which by the assumption in S implies that S holds with S replaced by F(τ|N), a
contradiction to the minimality of dimS. Assume therefore that dkSN = 0 and
dkNF(τ|N) = 2.
By minimality of dimS, we have N =S, which implies
dkSF(τ|N) = dkNF(τ|N) = 2.
Using minimality of dimS again, it follows that dimF(τ|N) > 12dim(N). Now
dkNF(τ|N) = 2 implies that an involutionτ0 exists so thatF(τ|N)⊆F(τ0|N)⊆N
with all inclusions strict. Since dimF(τ|N)> 12dimN, it follows that
2 codNF(τ0|N) + 2 codNF(τ τ0|N) = 2 codNF(τ|N)≤n,
so we conclude from the periodicity theorem that P holds.
As mentioned before the previous claim, our goal is to show the following:
Claim. We may assume dkNF (τ|N)≤1.
Proof. As in the previous proofs, our goal is to show thatP holds if the statement in the claim does not. Suppose therefore that dkNF (τ|N) = 2.
We can chooseτ0 ∈T00 satisfyingF(σ|N)6⊆F(τ0|N) andF(τ|N)⊆F(τ0|N)⊆N
under the isotropy representation as above. Observe that, by replacing τ0 byτ τ0 if necessary, we may assume y≤x−y.
Using the assumption that dimF(τ|N)≥2k, we apply the periodicity theorem
in each of three cases to conclude P. The relevant transverse intersections in the three cases are the following:
• If y > 0, we consider the intersection ofF(τ0|F(στ|N)) and F(τ τ 0|
F(στ)) inside
F(στ|N), and we conclude P with P =F(στ|N).
• If y = 0 and x > 0, we consider the intersection of F(σ|F(στ τ0|
N)) and
F(τ0|F(στ τ0|
N)) inside F(στ τ 0|
N), and we conclude P for P = F(στ|N) =
F(τ0|F(στ τ0| N)).
• If y = 0 and x = 0, we first replace τ0 by τ τ0 if necessary to suppose that z ≤l−z. We then consider the intersection ofF(σ|N) andF(τ0|N) inside N,
and we conclude P for P =N.
To summarize, we now have dkNF(τ|N) ≤ 1 and dimF(τ|N) ≥ 2k. We must
prove three more claims in the same manner before concluding the theorem.
Claim. We may assume that dkN(F(σ|N)∩F(τ|N))≤2.
Proof. If this is not the case, then dkF(τ|N)(F(σ|N)∩F(τ|N)) ≥ 2. Hence there
F(τ|N) =F(σ|F(τ|N)) and F(τ|N). Because F(σ 0| F(τ|N)) and F(σσ 0| F(τ|N) intersect transversely in F(τ|N) with 2 codF(τ|N)F(σ 0| F(τ|N)) + 2 codF(τ|N)F(σσ 0| F(τ|N))≤2k≤dimF(τ|N),
the periodicity theorem implies P with P =F(τ|N).
Claim. We may assume that F (σ|N)∩F(τ|N) is not transverse.
Proof. Suppose instead that F(σ|N)∩F(τ|N) is transverse. If dkS(N) = 0, then
the minimality of dim(S) implies S = N and dimF(σ|N)∩F(τ|N) ≥ 12dim(N).
Hence
2 codF(σ|N) + 2 codF(τ|N) = 2 codNF(σ|N)∩F(τ|N)≤dim(N),