Pedro L. Q. Pergher

(Z2)k-actions with fixed point set of constant codimension☆

Abstract In this work we start the study of the ideals I r ∗, k of cobordism classes in N ∗ containing a representative M n admitting a ( Z 2 ) k -action whose fixed point set is ( n − r )-dimensional.

BOUNDS ON THE DIMENSION OF MANIFOLDS WITH INVOLUTION FIXING F n F 2

Let M be a closed smooth manifold with an involution having fixed point set of the form F∪F , where F and F 2 are submanifolds with dimensions n and 2, respectively, and where n ≥ 4 is even (n < m). Suppose that the normal bundle of F 2 in M, μ→ F , does not bound, and denote by β the stable cobordism class of μ→ F . In this paper we determine the upper bound for m in terms of the pair (n, β) for many such pairs. The similar question for n odd (n ≥ 3) was completely solved in a previous paper of the authors. The existence of these upper bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5/2n.

G-coincidences for maps of homotopy spheres into CW-complexes

Let G be a finite group acting freely in a CW-complex Σ m which is a homotopy m-dimensional sphere and let f: Σ m → Y be a map of Σ m to a finite k-dimensional CW-complex Y. We show that if m ≥ |G|k, then f has an (H, G)-coincidence for some nontrivial subgroup H of G.

Involutions fixing an arbitrary product of spheres and a point

SummaryIn this paper we identify up to cobordism all involutions whose fixed point set is the disjoint union of an arbitrary product of spheres and a point.

The union of a connected manifold and a point as fixed set of commuting involutions

Abstract Let A be the family of all equivariant bordism classes of involutions containing a representative ( M , T ) with M connected and with the fixed point set of T being the disjoint union of a fixed connected n -dimensional manifold V n and a point. In this paper we analyse the influence of A in the determination of the equivariant bordism classes of ( Z 2 ) κ -actions which contain a representative fixing V n ∪ point. The results obtained are used to determine, up to bordism, all possible ( Z 2 ) κ -actions fixing RP (n) ∪ point with n odd.

Involutions Fixing F n ?{Indecomposable}

Let Mm be an m-dimensional, closed and smooth manifold, equipped with a smooth involution T : Mm → Mm whose fixed point set has the form Fn ∪ F j , where Fn and F j are submanifolds with dimensions n and j, F j is indecomposable and n > j. Write n − j = 2pq, where q ≥ 1 is odd and p ≥ 0, and set m(n− j) = 2n + p− q + 1 if p ≤ q + 1 and m(n− j) = 2n + 2p−q if p ≥ q. In this paper we show that m ≤ m(n − j) + 2 j + 1. Further, we show that this bound is almost best possible, by exhibiting examples (Mm(n− j)+2 j , T) where the fixed point set of T has the form Fn ∪ F j described above, for every 2 ≤ j < n and j not of the form 2t − 1 (for j = 0 and 2, it has been previously shown that m(n − j) + 2 j is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses m ≤ 5 2 n. Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, São Carlos, SP 13565905, Brazil e-mail: pergher@dm.ufscar.br Received by the editors January 15, 2009; revised May 19, 2009. Published electronically March 24, 2011. The author was partially supported by CNPq and FAPESP AMS subject classification: 57R85.

Bordism of two commuting involutions

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, ε1 ⊕ ε2 ⊕ ε3 → F , to be the fixed data of a (Z2)-action; these conditions yield the fact that if (ε1 ⊕ R) ⊕ ε2 ⊕ ε3 → F is the fixed data of a (Z2)-action, where R → F is the trivial one dimensional bundle, then the same is true for ε1 ⊕ ε2 ⊕ ε3 → F . The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible (Z2)-actions fixing the disjoint union of an even projective space and a point.

Limiting Cases of Boardman’s Five Halves Theorem

The famous Five Halves Theorem of J. Boardman says that, if T : M m ! M m is a smooth involution defined on a nonbounding closed smooth m-dimensional manifold M m (m > 1) and if F = Sn j=0 F j (n 6 m) is the fixed-point set of T, where F j denotes the union of those components of F having dimension j, then 2m 6 5n. If the dimension m is written as m = 5k c, where k > 1 and 0 6 c 1, the equivariant cobordism classification of involutions (M m ,T) for which the fixed submanifold F attains

A coincidence theorem for commuting involutions

Let M be an m-dimensional, closed and smooth manifold, and S, T : M → M two smooth and commuting diffeomorphisms of period 2. Suppose that S = T on each component of M. Denote by FS and FT the respective sets of fixed points. In this paper we prove the following coincidence theorem: if FT is empty and the number of points of FS is of the form 2p, with p odd, then Coinc(S, T ) = {x ∈ M m | S(x) = T (x)} has at least some component of dimension m − 1. This generalizes the classic example given by M = S, the m-dimensional sphere, S(x0, x1, ..., xm) = (−x0,−x1, ...,−xm−1, xm) and T the antipodal map.