One may speak about the pseudo-homotopy groups of a 1-connected, finitely ... Assuming A = Q, that is, Ïâ. 1(A) = 0, set now: (1.5). kA := max{k | Ï2kâ1. 1 ... properties (i) and (ii) from Definition 1.2 were proved by Shiga and Tezuka in [17]. ... Further refinements led to various answers to the general fundamental question.
arXiv:math/0311340v1 [math.DG] 20 Nov 2003
ISOMETRY-INVARIANT GEODESICS AND NONPOSITIVE DERIVATIONS OF THE COHOMOLOGY † ˘ S¸TEFAN PAPADIMA∗† AND LAURENT ¸ IU PAUNESCU
Abstract. We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of Q-cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M with H ∗ (M, Q) belonging to this class, every isometry has a non-trivial invariant geodesic, for any metric on M . We use Qsurgery to construct large classes of new examples for which the above result may be applied.
1. Introduction 1.1. Complexes which look like homogeneous spaces. Let A be a weighted, zero–dimensional (that is, artinian), complete intersection (W ACI), i.e., a commutative graded Q-algebra of the form (1.1)
A = Q[x1 , . . . , xn ]/I ,
where the variables xi have positive even weights, wi :=| xi |, and the ideal I is generated by a regular sequence, (1.2)
I = (f1 , . . . , fn ) ,
of weighted-homogeneous polynomials, fi . We are going to introduce a special class of W ACI’s, with an eye for applications in Riemannian geometry. One may speak about the pseudo-homotopy groups of a 1-connected, finitely generated, graded-commutative Q-algebra, A: M j (1.3) π∗∗ (A) = πi (A) . i≥0,j>1
2000 Mathematics Subject Classification. Primary 53C22, 13C40; Secondary 57T15, 55P62, 57R65. Key words and phrases. isometry-invariant geodesic, artinian complete intersection, equal rank homogeneous space, Q-surgery. ∗ Partially supported by grant CERES 152/2003 of the Romanian Ministry of Education and Research. † Partially supported by grant U4249 Sesqui R&D/2003 of the University of Sydney. 1
˘ S ¸ TEFAN PAPADIMA AND LAURENT ¸ IU PAUNESCU
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These algebraic invariants of A are finite-dimensional graded Q-vector spaces, πi (A) := ⊕j πij (A), for all i ≥ 0. See [9]. When A is a W ACI, as above, one knows that π>1 (A) = 0. Moreover, for any 1-connected CW -complex, S, such that H ∗ (S, Q) = A∗ , as graded algebras, one has the following topological interpretation (see Sullivan [18]): ( π0∗ (A) = HomQ (π∗=even (S) ⊗ Q, Q), and (1.4) π1∗ (A) = HomQ (π∗=odd (S) ⊗ Q, Q) . Assuming A 6= Q, that is, π1∗ (A) 6= 0, set now: (1.5)
kA := max {k | π12k−1 (A) 6= 0} .
Denote by Der∗ (A) the graded Lie algebra of homogeneous derivations of A∗ , with Lie bracket given by graded commutator, and note that Der∗ (Q) = 0. Definition 1.2. We shall say that the W ACI A is simple if: (i) Der
