On (Z2)k actions

Pergher, Pedro L. Q.

doi:10.1016/s0166-8641(00)00107-3

Cited by 4 publications

(5 citation statements)

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“…Specifically, we determine all possible equivariant cobordism classes of G-actions (M, Φ), Φ = (T 1 , T 2 ), for which F Φ is connected, dim(F Φ ) = n and dim(M ) = 4n−1 or 4n−2; here, F Φ is the fixed point set of Φ, that is, F Φ = {x ∈ M / T i (x) = x, i = 1, 2}. We remark that, for dim(M ) ≥ 4n, this type of classification was established in [3]. To state the results, we need to describe certain G-actions (M, Φ) with F Φ connected and n-dimensional, and with dim(M ) = 4n−1 and 4n−2.…”

Section: Introductionmentioning

confidence: 82%

$Z_2^2$-actions with $n$-dimensional fixed point set

Pergher¹

2007

Proc. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 82%

$Z_2^2$-actions with $n$-dimensional fixed point set

Pergher¹

2007

Proc. Amer. Math. Soc.

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“…We will apply this using certain special classes, which are polynomials in the above-displayed ones, and were initially introduced in [4] and also used in [5] and in [6]. The argument is identical with that of [6; part a) of Lemma of Section 3]; to ease the reading and mainly to establish some notations, we will rewrite it.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…Another way to generalize Theorem 2 would be an extension of the Pergher's result of [6] to F n ∪ F n−1 : if (M m , ) is a Z k 2 -action whose fixed set F has the form F = F n ∪ F n−1 , where F n and F n−1 are connected, and if m > 2 k n, then (M m , ) bounds equivariantly. Again we are not able to handle this situation, even in the case k = 2.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…Later, in [6], P. Pergher extended this result to Z k 2 -actions, under the restriction that the fixed set is connected: if (M m , ) is a Z k 2 -action whose fixed set F has dimension n and is connected, and if m > 2 k n, then (M m , ) bounds equivariantly. Connectedness is redundant for k = 1, since any involution is equivariantly cobordant to an involution with the property that the p-dimensional part of the fixed set is connected.…”

Section: Introductionmentioning

confidence: 96%

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Two Commuting Involutions Fixing F n ∪ F n-1

Pergher

Figueira

2006

Geom Dedicata

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Consider G = Z 2 2 as the group generated by two commuting involutions, and let : G × M → M be a smooth G-action on a smooth and closed manifold M. Suppose that the fixed point set of consists of two connected components, F n and F n−1 , with dimensions n and n − 1, respectively. In this paper we prove that, if in the fixed data of at least two eigenbundles over F n have dimension greater than n, and at least one eigenbundle over F n−1 has dimension greater than n − 1, then the action (M, ) bounds equivariantly. It is well known that, if T : M m → M m is a smooth involution on a smooth and closed m-dimensional manifold M m such that the fixed point set of T has constant dimension n, and if m > 2n, then (M m , T ) bounds equivariantly; this fact was proved by R. E. Stong and C. Kosniowski 27 years ago. As a consequence of our result, we will see that the same fact is true when, besides n-dimensional components, the fixed point set contains additional (n − 1)-dimensional components. Mathematics Subject Classifications (2000). Primary 57R85, Secondary 57R75.

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“…, T k , and the fixed data of Φ is η = ε → F, where F is the fixed point set of Φ and η = ε is the normal bundle of F in M m decomposed into eigenbundles ε with running through the 2 k − 1 nontrivial irreducible representations of Z k 2 . For example, see [12] (F = real projective space RP 2n and k = 1), [3] (F with constant dimension n, dim(η) ≥ n and k = 1), [2, Section 31] (F = a set of isolated points and k = 2), [10] (F = union of two real projective spaces and k = 1) and [5] (F = a connected n-dimensional manifold, dim(η) ≥ (2 k − 1)n and any k).…”

Section: Introductionmentioning

confidence: 99%

Commuting involutions whose fixed point set consists of two special components

Pergher

Oliveira

2008

Fund. Math.

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Abstract. Let F n be a connected, smooth and closed n-dimensional manifold. We call F n a manifold with property H when it has the following property: if N m is any smooth closed m-dimensional manifold with m > n and T : N m → N m is a smooth involution whose fixed point set is F n , then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces RP 2n , CP 2n and HP 2n , and the connected sum of RP 2n and any number of copies of S n × S n , where S n is the n-sphere and n is not a power of 2. In this paper we describe the equivariant cobordism classification of smooth actions (M m ; Φ) of the group Z k 2 on closed smooth m-dimensional manifolds M m for which the fixed point set of the action consists of two components K and L with property H, and where dim(K) < dim(L). The description is given in terms of the set of equivariant cobordism classes of involutions fixing K ∪ L.

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On (Z2)k actions

Cited by 4 publications

References 4 publications

$Z_2^2$-actions with $n$-dimensional fixed point set

$Z_2^2$-actions with $n$-dimensional fixed point set

Two Commuting Involutions Fixing F n ∪ F n-1

Commuting involutions whose fixed point set consists of two special components

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