Available online at www.sciencedirect.com ScienceDirect Indagationes Mathematicae 29 (2018) 807–818 www.elsevier.com/locate/indag Involutions fixing Fn ∪ F3 Evelin M. Barbarescoa, Pedro L.Q. Pergherb,∗ a Departamento de Matemática, Universidade Estadual Paulista - Ibilce, São José do Rio Preto, SP 15054-000, Brazil b Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, São Carlos, SP 13565-905, Brazil Received 10 October 2017; received in revised form 12 January 2018; accepted 12 January 2018 Communicated by J. van Mill Abstract Let Mm be a closed smooth manifold equipped with a smooth involution having fixed point set of the form Fn ∪F3, where Fn and F3 are submanifolds with dimensions n and 3, respectively, where 3 < n < m and with the normal bundles over Fn and F3 being nonbounding. The authors of this paper, together with Patricia E. Desideri, previously showed that, when n is even, then m ≤ n + 4, which we call a small codimension phenomenon. Further, they showed that this small bound is best possible. In this paper we study this problem for n odd, which is much more complicated, requiring more sophisticated techniques involving characteristic numbers. We show in this case that m ≤ M(n − 3) + 6, where M(n) is the Stong– Pergher number (see the definition of M(n) in Section 1). Further, we show that this bound is almost best possible, in the sense that there exists an example with m = M(n − 3) + 5, which means that for n odd the small codimension phenomenon does not occur and the bound in question is meaningful. The existence of these bounds is guaranteed by the famous Five Halves Theorem of J. Boardman, which establishes that, under the above hypotheses, m ≤ 5 2 n. c⃝ 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. Keywords: Involution; Fixed-data; Whitney number; Wu formula; Steenrod operation; Stong–Pergher number ∗ Corresponding author. E-mail addresses: evelin@ibilce.unesp.br (E.M. Barbaresco), pergher@dm.ufscar.br (P.L.Q. Pergher). https://doi.org/10.1016/j.indag.2018.01.003 0019-3577/ c⃝ 2018 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved. http://crossmark.crossref.org/dialog/?doi=10.1016/j.indag.2018.01.003&domain=pdf http://www.elsevier.com/locate/indag https://doi.org/10.1016/j.indag.2018.01.003 http://www.elsevier.com/locate/indag mailto:evelin@ibilce.unesp.br mailto:pergher@dm.ufscar.br https://doi.org/10.1016/j.indag.2018.01.003 808 E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 1. Introduction Let F be a disjoint (finite) union of smooth and closed manifolds and M be a smooth and closed m-dimensional manifold equipped with a smooth involution T : M → M whose fixed point set is F . Write F = ∪ n j=0 F j , where F j denotes the union of those components of F having dimension j . Up to equivariant cobordism, each F j can be supposed connected; also, if the normal bundle over some F j bounds as a bundle, then after an equivariant surgery an appropriate tubular neighborhood over F j can be removed, and thus again, up to equivariant cobordism, we can suppose that F has not the j-dimensional component (see [4]). Then suppose that the normal bundle over each F j occurring does not bound. If n is the dimension of the component of F of maximal dimension, then m ≤ 5 2 n; this follows from the famous Five Halves Theorem of J. Boardman, announced in [2], and its strengthened version of [12]. In fact, the Five Halves Theorem asserts that this is valid when M is not a boundary, and in [12] R.E. Stong and C. Kosniowski established the same conclusion under the weaker hypothesis that (M, T ) is a nonbounding involution. The assertion then follows from the fact that the equivariant cobordism class of (M, T ) is determined by the cobordism class of the normal bundle of F in M (see [4]). The generality of this result, which is valid for every n ≥ 1, allows the possibility that fixed components of all dimensions j, 0 ≤ j ≤ n, occur; in this way, it is natural to ask whether there exist better bounds for m when we omit some components of F and restrict the set of the involved maximal dimensions n. This class of problems was introduced by P. Pergher in [13], with its general formulation and with the following particular result: if F has the form F = Fn ∪ {point}, where n = 2p with p odd, then m ≤ 3p + 3 (which is better than 5 2 n = 5p). This case (F = Fn ∪ {point}) was completed by R. Stong and P. Pergher in [17], where they introduced the mysterious number M(n): writing n = 2pq, where p ≥ 0 and q is odd, then M(n) = 2n + p − q + 1 if p ≤ q and M(n) = 2n + 2p−q if p > q . They proved that, for every n ≥ 1, m ≤ M(n) and this bound is best possible. Remark. In [12], R. Stong and C. Kosniowski proved the following relevant result: if F = Fn has constant dimension n and m = 2n, then (M, T ) is equivariantly cobordant to the twist involution (Fn × Fn, S), S(x, y) = (y, x); further, if m > 2n, then (M, T ) bounds equivariantly. Thus, if the normal bundle over Fn does not bound, m ≤ 2n (which is better than m ≤ 5 2 n), and for each fixed n, with the exception of the dimensions n = 1 and n = 3, the maximal value m = 2n is achieved by taking the involution (Fn × Fn, twist), where Fn is any nonbounding n-dimensional manifold. This completely solves the Pergher problem when F has one compo- nent. Note that, if n is odd (that is, p = 0), then M(n) = n+1; this special small bound had been obtained by D.C. Royster in [18], when classifying, up to equivariant cobordism, involutions fixing two real projective spaces. With the cases F = Fn and F = Fn ∪ {point} completed, the next natural step is the case F = Fn ∪ F j , 0 < j < n. Concerning this more general case, one has the following results, which show the relevance of M(n): for j = 1, m ≤ M(n − 1) + 1 if n is odd and m ≤ M(n − 1) + 2 if n is even, the two bounds being best possible (see [10] and [11]). For j = 2, one has the best possible bound m ≤ M(n − 2) + 4 (see [7,6] and [9]). For j = n − 1, m ≤ 2n, which is best possible [8]. For every 2 ≤ j < n not of the form j = 2t − 1, if F j is indecomposable, then m ≤ M(n− j)+2 j +1, and there is an example with m = M(n− j)+2 j [16]. Also, in [5,15] and [14], we find related results. In this paper, we contribute to this problem, dealing with the case j = 3. As mentioned in the abstract, in [1] it was shown that in this case m ≤ n + 4 if n is even, and this bound is best possible. We will prove the following E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 809 Theorem 1.1. Let M be a closed smooth m-dimensional manifold equipped with a smooth involution having fixed point set of the form Fn ∪ F3, where 3 < n < m is odd and with the normal bundles over Fn and F3 being nonbounding. Then m ≤ M(n − 3) + 6, and there is an example with m = M(n − 3) + 5. Remark. For example, take n = 2p + 3, where p ≥ 1. Then M(n − 3) + 6 = 2p+1 + 2p−1 + 6, which is equal to one less than the Boardman bound associated to n (= 2p+1 + 2p−1 + 7). To obtain the bound in question, we introduce some special cohomology classes associated to line bundles over closed smooth manifolds, by using the splitting principle, and mix them with some special polynomials in the characteristic classes of total spaces of projective space bundles, introduced by R. Stong and P. Pergher in [17]; the basic theoretical support is the equivariant cobordism theory of Conner and Floyd of [4]. As will be seen, this will require more sophisticated computations than the case n even. 2. Preliminaries If (M, T ) is an involution pair as discussed above, we call η → F the fixed-data of (M, T ) when F is the fixed point set of T and η → F is the normal bundle of F in M . Let η be a general k-dimensional vector bundle over a closed smooth n-dimensional manifold F . Write W (η) = 1 + w1(η) + w2(η) + · · · + wk(η) ∈ H∗(F,Z2) for the Stiefel–Whitney class of η, and W (F) = 1 + w1(F) + w2(F) + · · · + wn(F) for the Stiefel–Whitney class of the tangent bundle of F . From [4], one has an algebraic scheme to determine the cobordism class of η, given by the set of Whitney numbers (or characteristic numbers) of η; such modulo 2 numbers are obtained by evaluating n-dimensional Z2-cohomology classes of the form wi1 (F)wi2 (F)...wir (F)w j1 (η)w j2 (η)...w js (η) ∈ H n(F,Z2) (that is, with i1 + i2 + · · · + ir + j1 + j2 + · · · + js = n) on the fundamental homology class [F] ∈ Hn(F,Z2). For example, suppose F three-dimensional. In this case, we can denote W (F) = 1+w1 +w2 +w3 and W (η) = 1+v1 +v2 +v3. Then the set of Whitney numbers comes from the ten three-dimensional cohomology classes w3, w1w2, w3 1 , v3, v1v2, v3 1 , w1v 2 1 , w1v2, w2 1v1 and w2v1. However, this number can be reduced. Indeed, from [1] one has the following Lemma 2.1. Let η a vector bundle over a three-dimensional manifold as above. Then w3 = w1w2 = w3 1 = v2 1w1 = 0, w2v1 = w2 1v1 and v2w1 = v1v2 + v3. So any cobordism class of a bundle over a three-dimensional manifold is determined by the numbers coming from the three-dimensional classes v3 1 , v3, v2w1 and v1w 2 1 . Further, any nonempty subset of this set of cohomology classes is realized by a stable cobordism class, in the sense that there is a bundle over a three-dimensional manifold whose set of nonzero Whitney numbers comes from the subset in question (which means that one has fifteen nonzero stable cobordism classes of bundles over closed three-dimensional manifolds). 3. The bound m ≤ M(n − 3) + 6 This section will be devoted to the proof of the part “m ≤ M(n − 3) + 6” of Theorem 1.1. One then has an involution pair (M, T ) with fixed set of the form Fn ∪ F3, where 3 < n < m is odd and with the normal bundles over Fn and F3 being nonbounding, and wants to prove the bound in question, where m = dim(M). Let (µ ↦→ Fn) ∪ (η ↦→ F3) be the fixed-data of (M, T ) and, as in Lemma 2.1, write W (F3) = 1 + w1 + w2 + w3 and W (η) = 1 + v1 + v2 + v3. The following lemma is crucial: 810 E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 Lemma 3.1. If m > M(n − 3) + 6, then v3 1 = v1w 2 1 , v3 = 0 and v2w1 = v1w 2 1 . Taking into account the nonempty subsets of {v3 1, v3, v2w1, v1w 2 1} mentioned in Lemma 2.1, we note that the unique nonzero stable cobordism class over a three-manifold which satisfies the relations of Lemma 3.1 is the one whose nonzero Whitney numbers come from v1w 2 1 , v3 1 and v2w1; call this class β. Thus this lemma will reduce our task to the following: Theorem 3.1. In the statement of Theorem 1.1, suppose that η ↦→ F3 represents β. Then m ≤ M(n − 3) + 6. The following basic fact, which follows from the Conner and Floyd exact sequence of [4], will be necessary for the proof of Lemma 3.1 (and Theorem 3.1): if Eµ and Eη denote the total spaces of the projective space bundles RP(µ) and RP(η), respectively, and λµ ↦→ Eµ and λη ↦→ Eη denote the line bundles of the double covers S(µ) → Eµ and S(η) → Eη, S( ) meaning sphere bundles, then λµ ↦→ Eµ and λη ↦→ Eη are cobordant as elements of the cobordism group Nm−1(BO(1)), that is, the cobordism group of 1-dimensional real vector bundles over (m − 1)-dimensional closed smooth manifolds. Therefore any cohomology class of dimension m − 1, given by a product of the classes wi (Eµ) and w1(λµ), evaluated on the fundamental homology class [Eµ], gives the same characteristic number as the one obtained by the corresponding product of the classes wi (Eη) and w1(λη), evaluated on [Eη]. With this tool in hand, our strategy will be: first, we will use a very special class, denoted by X , introduced by Pergher and Stong in [17]. X is associated to Eµ and, as above required, is a product of the classes wi (Eµ) and w1(λµ); further, X has dimension M(n − 3). Second, by using the splitting principle and the partitions of 3, ω1, ω2 and ω3, we introduce three special cohomology classes of dimension 6 associated to line bundles λ over closed smooth s-dimensional manifolds Bs , denoted by fω1 (λ), fω2 (λ) and fω3 (λ), and which are special polynomials in the characteristic classes of λ and Bs . Write Y for the cohomology class of Eη which corresponds to X . Then, if m > M(n − 3) + 6, m − 1 ≥ M(n − 3) + 6 and we can form a modulo 2 system of equations⎧⎪⎨⎪⎩ X. fω1 (λµ).w1(λµ)m−(M(n−3)+6)[Eµ] = Y. fω1 (λη).w1(λη)m−(M(n−3)+6)[Eη] X. fω2 (λµ).w1(λµ)m−(M(n−3)+6)[Eµ] = Y. fω2 (λη).w1(λη)m−(M(n−3)+6)[Eη] X. fω3 (λµ).w1(λµ)m−(M(n−3)+6)[Eµ] = Y. fω3 (λη).w1(λη)m−(M(n−3)+6)[Eη]. The solution of this system will be given by the relations of Lemma 3.1, thus providing the proof. Next, we detail the technical steps regarding this strategy, and the first thing to do is to describe the class X of Stong and Pergher of [17]. Set k = m−n, and write W (Fn) = 1+θ1+θ2+· · ·+θn , W (µ) = 1 + u1 + u2 + · · · + uk and W (λµ) = 1 + w1(λµ) = 1 + c for the Stiefel–Whitney classes of Fn, µ and λµ, respectively. One has (see [3]): W (Eµ) = (1 + θ1 + θ2 + · · · + θn){(1 + c)k + (1 + c)k−1u1 + · · · + uk}, where here we are suppressing bundle maps. First, for any integer r , Stong and Pergher introduced the following variant of W (Eµ): W [r ] = W (Eµ) (1 + c)k−r , noting that each class W [r ] j is still a polynomial in the classes wi (Eµ) and c. Next, for n ≥ 5, write n − 3 = 2pq , where p ≥ 1 (n − 3 is even) and q is odd; X is built in terms of p and q . Specifically, suppose first that p < q + 1. Then, in this case, X is E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 811 X = W [2p − 1]q+1−p 2p+1−1 · W [r1]2r1 · W [r2]2r2 · · · W [rp]2r p , where ri = 2p − 2p−i for 1 ≤ i ≤ p. If p ≥ q + 1, X is X = W [r1]2r1 · W [r2]2r2 · · · W [rq+1]2rq+1 , where ri = 2p − 2p−i for 1 ≤ i ≤ q + 1. Stong and Pergher proved that X has the following two crucial properties: (i) dimension(X ) = M(2pq) = M(n − 3); (ii) X has the form X = At · cM(2pq)−t + terms with smaller c powers, where At is a cohomology class of dimension t ≥ 2pq + 1 and comes from the cohomology of Fn . In our case, X = At · cM(n−3)−t + terms with smaller c powers, where At is a cohomology class of dimension t ≥ n − 2 coming from the cohomology of Fn . The next technical step is, as above announced, to introduce the three special 6-dimensional cohomology classes fωi (λ), i = 1, 2, 3, where λ is a line bundle over a smooth closed s-dimensional manifold Bs . Using the splitting principle, write W (Bs) = (1+x1)·(1+x2) · · · (1+ xs) and W (λ) = 1 + c. We then consider the following symmetric polynomials in the variables x1, x2, . . . , xs, c, of degree 6 and related to the partitions of 3, ω1 = (1, 1, 1), ω2 = (2, 1) and ω3 = (3): fω1 = ∑ i< j M(n − 3) + 6, then it is possible to find special polynomials in the characteristic classes so that the corresponding characteristics numbers are zero on Fn and nonzero on F3, thus giving the contradiction. Taking into account our previous knowledge on the behavior of the class X of Stong and Pergher on Fn , the additional key point will be a subtle (but routine) calculation based on the structure of the cohomology ring of Eη, described as follows: let α ∈ H 1(RP1,Z2) and β ∈ H 1(RP2,Z2) be the respective generators. Then H∗(Eη,Z2) is the free H∗(F3,Z2)-module on 1, d, d2, . . . , dm−4, subject to the relation dm−3 = dm−4(α + β) + dm−5αβ. From this relation, we obtain dm−1 = dm−2(α + β) + dm−3αβ, 814 E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 dm−2α = dm−3αβ, dm−2β = dm−3αβ+dm−3β2 +dm−4αβ2, dm−3β2 = dm−4αβ2 and dm−3αβ = dm−4αβ2. Combining these relations, we obtain dm−1 = dm−2β = dm−3αβ = dm−4αβ2, which is the (top-dimensional) generator of H m−1(Eη,Z2). Write n − 3 = 2pq , where p > 0 and q is odd, and first suppose p > 1. On Fn one takes the same class X considered before; that is, X ∈ HM(n−3)(Eµ,Z2) and each term of X has a factor of dimension at least n − 2 from the cohomology of Fn . Note that, on Fn , W [0]2 = u1c+u2+θ1u1+θ2. Hence every term of W [0]3 2 = (u2 1c2 +u2 2+θ2 1 u2 1+θ2 2 )(u1c+u2+θ1u1+θ2) has a factor of dimension at least 3 from Fn . If m > M(n−3)+6, one then has the zero characteristic number X · W [0]3 2 · cm−1−(M(n−3)+6)[Eµ]. Our next task will be to show that, on F3, the corresponding characteristic number Y · W [n − 3]3 2 · dm−1−(M(n−3)+6)[Eη] is nonzero. One has W (Eη) = (1 + β + β2) ( (1 + d)n+k−3 + (1 + d)n+k−4(α + β) + (1 + d)n+k−5(αβ) ) . Then W [n − 3]2 = ( n − 3 2 ) d2 + d(α + β). Since n − 3 = 2pq with p > 1 and q odd, one has that 2 does not belong to the 2-adic expansion of n − 3, and thus W [n − 3]2 = d(α + β). Hence, W [n − 3]3 2 = d3αβ2. Concerning the class Y , which is obtained from X by replacing each W [r ]i by W [n + r − 3]i , we are exactly in the same situation of Lemma 3.1. In fact, W [n −3]3 2 · A = d3αβ2 · A = 0 for each A ∈ I, the ideal of H∗(Eη,Z2) generated by the classes coming from F3 and with positive dimension. Then, as in Lemma 3.1, in the computation of Y , one needs to consider only that W (Eη) ≡ (1 + d)n+k−3 mod I and, for each integer l, W [l] ≡ (1 + d)l mod I. In this way, similarly we conclude that Y ≡ dM(n−3) mod I. It follows that Y · W [n − 3]3 2 · dm−1−(M(n−3)+6)[Eη] = dm−4 · αβ2[Eη] = αβ2[F3] = 1, which proves Theorem 3.1 when p > 1. Now suppose p = 1. On RP(µ) we have W [1] = (1 + θ1 + · · · + θn) ( (1 + c) + u1 + u2 (1 + c) + · · · + uk (1 + c)k−1 ) . Then W [1]3 = u2c + u3 + θ1u2 + θ2c + θ2u1 + θ3. Hence every term of W [1]2 3 has a factor of dimension at least 4 from H∗(Fn,Z2). Since, as before, each term of X has a factor of dimension at least n − 2 from the cohomology of Fn , if m > M(n − 3) + 6, we have the zero characteristic number X · W [1]2 3 · cm−1−(M(n−3)+6)[RP(µ)]. E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 815 So, the next and final task will be to show that, on F3, the corresponding characteristic number Y · W [n − 2]2 3 · dm−1−(M(n−3)+6)[Eη] is nonzero. One has W [n − 2] = (1 + β + β2) ( (1 + d)n−2 + (1 + d)n−3(α + β) + (1 + d)n−4αβ ) . Thus W [n − 2]3 = ( n − 2 3 ) d3 + ( n − 3 2 ) d2(α + β) + ( n − 4 1 ) dαβ + ( n − 2 2 ) d2β + ( n − 3 1 ) d(αβ + β2) + ( n − 4 0 ) αβ2 + ( n − 2 1 ) dβ2 + ( n − 3 0 ) αβ2. An easy inspection of 2-adic expansions shows that 1 and 2 belong to the 2-adic expansion of n − 2 = 2q + 1, and 2 belongs to the 2-adic expansion of n − 3 = 2q. We conclude that W [n −2]3 = d3 +d2(α +β)+dαβ +d2β +dβ2 = d3 +d2α +dαβ +dβ2 and W [n −2]2 3 = d6. Now, since p = 1, the class X (on Eµ) is W [1]2.(W [1]3)q . Therefore Y = W [n −2]2.(W [n − 2]3)q . One has W [n − 2]2 = ( n − 2 2 ) d2 + ( n − 3 1 ) d(α + β) + ( n − 4 0 ) αβ + ( n − 2 1 ) dβ + ( n − 3 0 ) (αβ + β2) + ( n − 2 0 ) β2 = ( n − 2 2 ) d2 + dβ = d2 + dβ, and, as seen above, W [n − 2]3 = d3 + d2α + dαβ + dβ2. Thus, Y = (d3 + d2α + d(αβ + β2))q (d2 + dβ) = ( q∑ i=0 ( q i ) (d3 + d2α)q−i(d(αβ + β2) )i ) (d2 + dβ) = ( (d3 + d2α)q + ( (d3 + d2α)q−1 )( c(αβ + β2) )) (d2 + dβ), since ( d(αβ + β2) ) j = 0 if j ≥ 2. But (d3 + d2α)q = q∑ i=0 ( q i ) (d3)q−i + (d2α)i = d3q + d3q−1α 816 E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 and (d3 + d2α)q−1 = q−1∑ i=0 ( q − 1 i ) (d3)q−1−i + (d2α)i = d3q−1 because q is odd and α j = 0 if j > 1. Therefore, Y = ( d3q + d3q−1α + d3q−2(αβ + β2) ) (d2 + dβ) = d3q+2 + d3q+1(α + β) + d3qβ2 + d3q−1αβ2 = d t + d t−1(α + β) + d t−2β2 + d t−3αβ2, where t = 3q + 2 = M(n − 3). Thus Y · W [n − 2]2 3 · dm−1−(t+6) = ( d t + d t−1(α + β) + d t−2β2 + d t−3αβ2 ) · d6 · dm−1−(t+6) = dm−1 + dm−2(α + β) + dm−3β2 + dm−4αβ2. From the relation dm−3 = dm−4(α+β)+dm−5αβ one obtains dm−1 = dm−2α+dm−2β+dm−3αβ; replacing in the above expression, we get Y · W [n − 2]2 3 · dm−1−(t+6) = dm−3αβ + dm−3β2 + dm−4αβ2. Again, from dm−3 = dm−4(α+β)+dm−5αβ, one obtains dm−2β = dm−3αβ+dm−3β2 +dm−4αβ2. But, as seen before, dm−2β, dm−3αβ and dm−4αβ2 are the top-dimensional generator, which means that dm−3β2 also is. It follows that Y · W [n − 2]2 3 · dm−1−(t+6) = dm−3αβ + dm−3β2 + dm−4αβ2 = dm−4αβ2, which ends the proof. 4. An example with m = M(n − 3) + 5 In this section we construct the example announced in the abstract, with m = M(n − 3) + 5. Consider the vector bundle τ ⊗ ξ1 ↦→ RP2 × RP1, where τ is the tangent bundle over RP2 (again we are omitting pullback notations). Maintaining the notations of the previous section, one has W (τ ⊗ ξ1) = (1 + α)2 + (1 + α)β + β2, which gives w1(τ ⊗ ξ1) = β and w2(τ ⊗ ξ1) = αβ + β2. Then w1(RP2 × RP1).w2(τ ⊗ ξ1) = α.β2 ̸= 0, which means that τ ⊗ ξ1 does not bound. Let E be the total space of the projective space bundle RP(τ ⊗ ξ1) and λ ↦→ E the usual line bundle. From the Conner–Floyd exact sequence of [4], τ ⊗ ξ1 is the fixed-data of some involution (V 5, S) if, and only if, λ ↦→ E bounds. Set W (λ) = 1 + c. One has W (E) = (1 + β + β2)((1 + c)2 + (1 + c)β + αβ + β2), subject to the relation c2 = cβ + αβ + β2. An easy calculation then shows that W (E) = 1. Thus the only relevant characteristic number of λ comes from c4. From the relation c2 = cβ + αβ + β2, we get c4 = c3β + c2αβ + c2β2, c2β2 = 0, c3β = c2β2 + cαβ2 and E.M. Barbaresco, P.L.Q. Pergher / Indagationes Mathematicae 29 (2018) 807–818 817 c2αβ = cαβ2. Since cαβ2 ∈ H 4(E, Z2) is the (top-dimensional) generator, c3β and c2αβ also are. Then c4 = 0 and λ ↦→ E bounds, which then gives an involution (V 5, S) with nonbounding fixed-data τ ⊗ ξ1 ↦→ RP2 × RP1. For n ≥ 5, now take the maximal involution (MM(n−3), T ) of Stong and Pergher, with fixed point set of the form Fn−3 ∪ {point}. The product involution (N = MM(n−3) × V 5, T × S) has dim(N ) = M(n − 3) + 5 and fixes the disjoint union (Fn−3 × RP2 × RP1) ∪ (RP2 × RP1). The normal bundle of RP2 × RP1 in N is (τ ⊗ ξ1) ⊕ M(n − 3)R, which does not bound. We assert that the normal bundle of Fn−3 × RP2 × RP1 in N does not bound. Otherwise, after an equivariant surgery, it can be removed to give an involution (W, L), where dim(W ) = M(n − 3) + 5 and the fixed-data is (τ ⊗ ξ1) ⊕ M(n − 3)R ↦→ RP2 × RP1. For n ≥ 5 odd, the lesser value of M(n − 3) + 5 is 10, corresponding to n = 5. Then (W, L) has fixed set RP2 ×RP1 of constant dimension 3 and dim(W ) ≥ 10 > 6 = 2dim(RP2 ×RP1). From the Kosniowski–Stong theorem cited in the first remark of Section 1, one then has that (W, L) bounds equivariantly, contradicting the fact that its fixed-data does not bound. Therefore (N , T × S) is the required example. Remark. Theorem 1.1 leaves open the question of either to construct a maximal example, that is, with m = M(n − 3) + 6, or to improve the bound m ≤ M(n − 3) + 6 to m ≤ M(n − 3) + 5. Regarding to the first alternative, since the desired dimension M(n − 3) + 6 involves the Stong–Pergher number, it is difficult to try anything other than the procedure used above to get our almost maximal example, that is, an example of the form (MM(n−3) × V 6, T × S), where S is a nonbounding involution defined on a 6-dimensional manifold V 6 fixing a three- dimensional manifold P3; for example, in the case F = Fn ∪ F2, the best possible bound is m ≤ M(n − 2) + 4 and a (simpler) maximal example is (MM(n−2), T ) × (RP2 × RP2, twist) (see [7]). However, any involution (V 6, S) fixing some P3 bounds equivariantly: again, this follows from the Kosniowski–Stong theorem cited in the first remark of Section 1. In fact, in this case (V 6, S) is equivariantly cobordant to (P3 × P3, twist), whose fixed-data is the tangent bundle τ 3 → P3. Since any three-dimensional manifold bounds, τ 3 → P3 bounds as a bundle, and thus (V 6, S) bounds. 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