Abstract
Erratum to: Morse relations and Fredholm deformations of v-convex contact forms
Highlights
The arguments of original article for the conclusions (iii) and (iv) of Theorem 3 are not complete.Our arguments in the original article, p. 181 assert that c2k−1 is a non-zero cycle in H2k−1(Cβ, L+ ∪LJ∞−−(∪)J∪∞−∂(∞)c∪2k−∂∞1)c∪2kW−1u)(.hT2hk−is1,d∞o)e)s, not necessarily imply that c2k−1 is which is the conclusion that would non-zero in be requiredH2k−1(Cβ, for (iii) and L+ ∪ (iv)
Under the assumption of (i) and (ii), this does not happen since the Fadell–Rabinowitz index [3] of γF R(L+) and γF R(L−) are at most (k − 2), so that the second factor maps at most into PCk−2 × [−1, 1] ∪ PCk−1 × {0}
We briefly study the Fadell–Rabinowitz index of “sections” to Wu(h2k−1,∞) in order to decide whether this space has a classifying map, given its intersections with L± and its boundary trace ∂∞c2k−1, valued into PCk−1 × [−1, 1] or in a lower dimensional complex projective space: Starting from Wu(h2k−1,∞), we seek a section to the flow-lines of this subset that enter eg L+ (L−)
Summary
The arguments of original article for the conclusions (iii) and (iv) of Theorem 3 are not complete.Our arguments in the original article, p. 181 assert that c2k−1 is a non-zero cycle in H2k−1(Cβ , L+ ∪LJ∞−−(∪)J∪∞−∂(∞)c∪2k−∂∞1)c∪2kW−1u)(.hT2hk−is1,d∞o)e)s, not necessarily imply that c2k−1 is which is the conclusion that would non-zero in be requiredH2k−1(Cβ , for (iii) and L+ ∪ (iv). We observe that, starting from y2k, see original article, section 2.5, p. 125—which dominates all the h2k−1,∞-the ±v-jumps that remain zero all along a flow-line separating two critical points with a difference of Morse indexes equal to 1, cannot disappear; they survive as zero-H01 ± v-jumps in the closure of the flow-line, The online version of the original article can be found under doi:10.1007/s40065-014-0098-1.
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