ON PETRIE’S THEOREM FOR TORUS MANIFOLDS
Suyoung Choi
Department of Mathematics, Osaka City University
Abstract. Petrie [9] has shown that all homotopy equivalence between homotopy projective spaces admitting effective smooth half-dimensional compact torus
actions should preserve their Pontrjagin classes. In this article, we propose several
problems about the invariance of Pontrjagin classes of torus manifolds, which canbe regarded as a generalization of Petrie’s theorem. In addition, we introduce
known results [1] and their applications.

1. Petrie’s Theorem
A torus manifold, introduced by Hattori and Masuda [5], is a closed smoothmanifold of dimension 2n admitting an effective smooth T n -action with non-empty
fixed point set. Let M be a torus manifold homotopy equivalent to the projective
space CP n of the same dimension. In other words, M is a homotopy projective spaceof dimension 2n admitting an effective smooth T n -action because its fixed point setnn
M T is always non-empty; χ(M T ) = χ(M ) = n + 1 6= 0, where χ(X) is the euler
characteristic number of X.
Theorem 1.1 (Petrie [9]). Let M1 and M2 be torus manifolds homotopy equivalent
to CP n . Then, any homotopy equivalence between M1 and M2 preserves their Pontrjagin classes, namely, if f : M1 → M2 is a homotopy equivalence, then f ∗ (p(M2 )) =p(M1 ), where p(X) denotes the total Pontrjagin class of X and f ∗ : H ∗ (M2 ) →H ∗ (M1 ) is the induced map of f .
More precisely, he has shown that the total Pontrjagin class of the torus homotopy projective space M of dimension 2n is (1 + x2 )n+1 ∈ H ∗ (M ) = Z[x]/xn+1 ,where deg x = 2. Since any cohomology ring isomorphism between two homotopy
projective spaces sends a generator to a generator up to sign, we can conclude thatit should preserve their Pontrjagin classes. Surprisingly, using the theory of Masuda[6] without hard difficulties, one can show that Theorem 1.1 holds even if we replace the condition which Mi is homotopy equivalent to CP n to the condition whichH ∗ (Mi ) is isomorphic to H ∗ (CP n ) as a graded ring for i = 1, 2.
Theorem 1.2. Let M be a torus manifold whose cohomology ring is isomorphic toH ∗ (CP n ) as a graded ring. Then, its total Pontrjagin class is(1 + x2 )n+1 ∈ H ∗ (M ) = Z[x]/xn+1 ,where deg x = 2.
Date: January 18, 2011.
The author was supported by the Japanese Society for the Promotion of Sciences (JSPS grantno. P09023).
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Motivated by this, we may ask whether any cohomology ring isomorphism preserves the Pontrjagin classes of torus manifolds or not. The purpose of this article isto propose several problems related on this question and to introduce partial answers
and their applications based on [1].
2. Torus manifolds
In this section, we briefly review torus manifolds following [7]. A torus manifold
is a 2n-dimensional closed connected manifold M with an effective smooth action of
an n-dimensional torus T = (S 1 )n such that the fixed point set M T is non-empty.
Since dim M = 2 dim T and M is compact, M T is a finite set of isolated points.
A codimension-two connected component of the set fixed pointwisely by a circle
subgroup of T is called a characteristic submanifold of M . Since M is compact,
there are only finitely many characteristic submanifolds, and we denote them by
Mi , i = 1, . . . , m.
Example 2.1. Let n be a positive integer with n ≥ 2. Let S 2n be the 2n-dimensional
sphere identified with the subset {(z1 , . . . , zn , y) ∈ Cn ×R | |z1 |2 +· · ·+|zn |2 +y 2 = 1}
of Cn × R, and define an action of T n := (S 1 )n on S 2n by(t1 , . . . , tn ) · (z1 , . . . , zn , y) = (t1 z1 , . . . , tn zn , y),where S 1 is the unit circle in C1 . Then this action is effective and smooth, and thepoints (0, . . . , 0, ±1) are fixed by T n -action. Hence, S 2n is a torus manifold. A map(z1 , . . . , zn , y) 7→ (|z1 |, . . . , |zn |, y)induces a homeomorphism from the orbit space S 2n /T n onto the manifold withcorners{(x1 , . . . , xn , y) ∈ Rn+1 | x21 + · · · + x2n + y 2 = 1, x1 ≥ 0, . . . , xn ≥ 0}.
Note that every face of the orbit space S 2n /T n is contractible. The facets are imagesof characteristic submanifolds {zi = 0} of S 2n (i = 1, . . . , n) under the quotient map
above and the intersection of the n codimension-one faces, called facets, consists of
two points (0, . . . , 0, ±1).
A torus manifold M is said to be locally standard if every point in M has aninvariant neighborhood U weakly equivariantly diffeomorphic to an open subset W ⊂Cn invariant under the standard T -action on Cn , namely, there is an automorphism
ψ : T → T and a diffeomorphism f : U → V such that f (ty) = ψ(t)f (y) for all t ∈ Tand y ∈ U . Let M be a locally standard torus manifold. Let Q := M/T denote theorbit space of M and π : M → Q the quotient projection. Then, Q can be regarded
as a manifold with corners, and faces of Q can be defined in a natural way. Wenote that the projection π : M → Q maps every k-dimensional orbit to a point in
the interior of a codimension-k face of Q for all k = 0, . . . , n. We set Qi := π(Mi ).
Then, Qi is a codimension-one face of Q, called a facet of Q. Since M is locallystandard, any point in Q has a neighborhood diffeomorphic to an open subset in
the positive cone Rn≥0 := {(x1 , . . . , xn ) ∈ Rn | xi ≥ 0, i = 1, . . . , n}. Such manifolds
are called nice manifolds with corners.
2Let Q be a compact nice manifold with corners. Faces of Q are defined naturally.
A nice manifold Q with corners is called a homology cell if all faces of Q, including
Q itself, are acyclic, namely, their reduced cohomology rings vanish. We also saythat Q is a homology polytope if it is a homology cell and any multiple intersectionof faces is acyclic whenever it is non-empty. A simple polytope provides a typicalexample of homology polytope and the orbit space of S 2n /T n in Example 2.1 is ahomology cell but not a homology polytope.
It is shown in [7] that if H odd (M ) = 0 for a torus manifold M , then M is locallystandard. In addition, they have shown the following theorem.
Theorem 2.2 (Masuda-Panov [7]). Let M be a torus manifold and Q the orbit
space of M . Then
(1) Q is a homology cell if and only if H odd (M ) = 0, and(2) Q is a homology polytope if and only if H ∗ (M ) is generated by H 2 (M ) as aring.
3. Invariance of Pontrjagin classes of torus manifolds
We recall a torus manifold homotopy equivalent to the projective space in Theorem 1.1. We note that the orbit space of the torus homotopy projective spaces isa homology polytope. It is natural to ask whether Theorem 1.1 still holds when wereplace a torus homotopy projective space to a torus manifold whose orbit space isa homology polytope (or a homology cell), namely, we have the following problems.
Problem 3.1. Let M1 and M2 be torus manifolds whose orbit spaces are homology
polytopes or homology cells. Let f : M1 → M2 be a homotopy equivalence. Then isit true thatf ∗ (p(M2 )) = p(M1 )?
As we discussed in Section 1, we can also ask the following stronger question.
Problem 3.2. Let M1 and M2 be torus manifolds whose orbit spaces are homology
polytopes or homology cells. Let ϕ : H ∗ (M2 ) → H ∗ (M1 ) be a ring isomorphism.
Then is it true thatϕ(p(M2 )) = p(M1 )?
Unfortunately, the answer of Problem 3.2 is negative in general. We have thefollowing counter example.
Example 3.3. Let
M1 = S 7 ×S 1 S(C12 ⊕ R9 ) and M2 = S 7 ×S 1 S(C42 ⊕ R1 ),where S(C` ⊕ Rm ) ⊂ C` ⊕ Rm stands the unit sphere, and Cρ is C with S 1 -action
by t · z = tρ z. The author and Kuroki [2] have shown that H ∗ (M1 ) ∼= H ∗ (M2 ) =Z[x, y]/hx4 , z(z + 2x)4 i with deg x = 2, deg z = 8, and p1 (M1 ) = 4x2 and p1 (M2 ) =16x2 . One can easily check that both M1 and M2 are torus manifolds, and since thegenerators have even degree, their orbit spaces are homology cells (but not homology
polytopes). Because of the degree of generators, any cohomology ring isomorphism
ϕ : H ∗ (M2 ) → H ∗ (M1 ) sends a degree-two generator to a degree-two generator up
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to sign. Hence, ϕ(p1 (M2 )) 6= p1 (M1 ). Hence, it gives us the negative answer toProblem 3.2. But we still do not know whether Problem 3.2 is negative on the casewhere orbit spaces are homology polytopes.
We also have some
Q partial answers to the problems. We note that the product ofprojective spaces hi=1 CP ni is also torus manifold whose orbit space is a product ofQQ
simplices hi=1 ∆ni . It is easy to show that H ∗ ( hi=1 CP ni ) is generated by degree
two elements.
Theorem 3.4 (Choi [1]). Let M1 and M2 be torus manifolds whose cohomology
Qrings are isomorphic to H ∗ ( hi=1 CP ni ). Then, any cohomology ring isomorphism
between them preserves their Pontrjagin classes.
We remark that Theorem 3.4 generalizes Theorem 1.1 strictly.
For a complex vector bundle E, we denote the total space of its projectivization
by P (E). A generalized Bott tower of height h is a sequence of projective bundles(3.1)ππh−1ππ21hB1 −→B0 = {a point},
Bh −→Bh−1 −→ · · · −→where each πi : Bi = P (C⊕ξi ) → Bi−1 and ξi is the Whitney sum of ni (≥ 1) complex
line bundles over Bi−1 for i = 1, . . . , h. We call Bh an h-stage generalized Bottmanifold. Obviously, a complex projective space CP n is an one-stage generalized
Bott manifold. When all fibers in (3.1) are CP 1 , namely, ni = 1 for all i, Bh is calleda Bott manifold. A closed smooth manifold is called a cohomology Bott manifold if
its cohomology ring is isomorphic to that of some Bott manifolds, and is called atorus cohomology Bott manifold if it is both a cohomology Bott manifold and a torus
manifold. We note that a Bott manifold itself is a torus cohomology Bott manifold.
In addition, all manifolds homotopy equivalent to Bott manifolds are cohomology
Bott manifolds.
Theorem 3.5 (Choi [1]). Let M1 and M2 be torus cohomology Bott manifolds. Thenany ring isomorphism ϕ : H ∗ (M1 ) → H ∗ (M2 ) preserves their Pontrjagin classes,
namely, ϕ(p(M1 )) = p(M2 ).
We remark that both Theorems 3.4 and 3.5 provide affirmative evidences to Problems 3.1 and 3.2.
4. Applications
One of the most interesting problems in Toric topology is the topological classification of toric manifolds. A toric manifold is a non-singular compact complex
algebraic variety with an algebraic torus action having a dense orbit. Clearly, a toric
manifold is a torus manifold. Interestingly, many recent research provide evidences
for toric manifolds to be classified by their cohomology rings. In general, the cohomology ring as an invariant is too weak to determine the topological type. However,in the category of toric manifolds, we do not know any examples of two distinct
toric manifold having the same cohomology rings because of their tori-symmetries.
Hence, it raises the following problem, called the cohomological rigidity problem fortoric manifolds.
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Problem 4.1 (Cohomological rigidity problem for toric manifolds). Let M1 and M2
be two toric manifolds such that H ∗ (M1 ) ∼= H ∗ (M2 ) as graded rings. Then are theydiffeomorphic (or homeomorphic)?
See [8] for more details. By the classical theory on the low dimensional manifolds
such as [4], the cohomological rigidity holds for all toric manifolds up to 4 dimension
since toric manifolds are simply connected. In high dimensional case, this problem
is still open.
We note that both a product of projective spaces and a Bott manifold are not
only torus manifolds but also toric manifolds. Hence, by combining Theorems 3.4and 3.5 with the result of Sullivan [10], we can say the finiteness of such manifolds
having the isomorphic cohomology rings;
Corollary 4.2. There are at most a finite number of torus manifolds homotopy
equivalent to the given Bott manifold or the given product of projective spaces.
So the corollary also provides affirmative evidences of cohomological rigidity oftoric manifolds.
We remark that any diffeomorphism between two closed smooth manifolds preserves their Pontrjagin classes. Hence, we can ask the following problem, too.
Problem 4.3 (Strong cohomological rigidity problem for Bott manifolds). Let Bnand Bn0 be two Bott manifolds, and φ : H ∗ (Bn ) → H ∗ (Bn0 ) an isomorphism as agraded ring. Then, there is a diffeomorphism f : Bn0 → Bn such that f ∗ = φ.
On the other hand, it is well-known that the invariance of Stifel-Whitney classes
for a closed manifold whose cohomology ring is generated by the same degree elements.
Theorem 4.4 (Choi-Masuda-Suh [3]). Suppose that H ∗ (M ) is generated by H r (M )for some r as a ring and let M 0 be another connected closed manifold of the same
dimension such that H ∗ (M 0 ) is isomorphic to H ∗ (M ) as a ring. Then ϕ(w(M 0 )) =w(M ) for any ring isomorphism ϕ : H ∗ (M 0 ) → H ∗ (M ), where w(X) denotes for the
total Stifel-Whitney class of X.
Since any diffeomorphism preserves the total Stiefel-Whintey class, the above
theorem also supports to Problem 4.3 affirmatively.
References
1. S. Choi, Torus actions on cohomology generalized Bott manifolds, preprint.
2. S. Choi and S. Kuroki, Topological classification of torus manifolds which have codimension
one extended actions, preprint, arXiv:0906.1335.
3. S. Choi, M. Masuda and D. Y. Suh, Topological classification of generalized Bott towers, Trans.
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7. M. Masuda and T. E. Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006),
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139–146. MR 0322893
10. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math.
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Department of Mathematics, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: choi@sci.osaka-cu.ac.jp

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