Kitada, Yasuhiko; Nagura, Maki The first Pontrjagin classes of homotopy complex projective spaces. (English) Zbl 1476.57050 Topology Appl. 287, Article ID 107447, 19 p. (2021). Authors’ abstract: Let \(M^{2n}\) be an oriented closed smooth manifold homotopy equivalent to the complex projective space \(\mathbb{C} P(n)\). The main purpose of this paper is to show that when \(n\) is even, the difference between the first Pontrjagin class of \(M^{2n}\) and that of \(\mathbb{C} P(n)\) is divisible by 16. As a geometric application of this result, we prove that the Kervaire sphere of dimension \(4k + 1\) does not admit any free circle group action if \(k + 1\) is not a power of 2. Reviewer: Yanghyun Byun (Seoul) MSC: 57R20 Characteristic classes and numbers in differential topology 57R55 Differentiable structures in differential topology 57R67 Surgery obstructions, Wall groups Keywords:Pontrjagin class; homotopy complex projective space; surgery PDFBibTeX XMLCite \textit{Y. Kitada} and \textit{M. Nagura}, Topology Appl. 287, Article ID 107447, 19 p. (2021; Zbl 1476.57050) Full Text: DOI arXiv References: [1] Atiyah, M. F., Thom complexes, Proc. Lond. Math. Soc. (3), 11, 291-310 (1961) · Zbl 0124.16301 [2] Browder, W., Surgery and the theory of differentiable transformation groups, (Proceedings of the Conference on Transformation Groups. Proceedings of the Conference on Transformation Groups, New Orleans, 1967 (1968), Springer-Verlag), 1-46 [3] Brumfiel, G., Homotopy equivalences of almost smooth manifolds, Comment. Math. Helv. Masuda, 46, 381-407 (1971) · Zbl 0222.57021 [4] Brumfiel, G.; Madsen, I.; Milgram, R. J., PL characteristic classes and cobordism, Ann. Math., 97, 82-159 (1973) · Zbl 0248.57006 [5] Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1979), Oxford Univ. Press · Zbl 0423.10001 [6] Hizebruch, F., Topological Methods in Algebraic Geometry (1966), Springer-Verlag [7] Kitada, Y., On the Kervaire classes of homotopy real projective spaces, J. Math. Soc. Jpn., 43, 2, 219-227 (1991) · Zbl 0867.57030 [8] Kitada, Y., Kervaire’s obstructions of free actions of finite cyclic groups on homotopy spheres, (Bak, A.; etal., Current Trends in Transformation Groups (2002), Kluwer Academic Publishers), 117-128 · Zbl 1025.57036 [9] Kitada, Y., On the first Pontrjagin class of homotopy complex projective spaces, Math. Slovaca, 62, 3, 551-566 (2012) · Zbl 1324.57003 [10] Masuda, M.; Tsai, Y-D., Tangential representations of cyclic group actions on homotopy complex projective spaces, Osaka J. Math., 22, 907-919 (1985) · Zbl 0575.57023 [11] Milgram, R. J., The mod 2 spherical characteristic classes, Ann. Math., 92, 238-261 (1970) · Zbl 0208.51601 [12] Montgomery, D.; Yang, C. T., Differentiable actions on homotopy seven spheres II, (Proceedings of the Conference on Transformation Groups. Proceedings of the Conference on Transformation Groups, New Orleans, 1967 (1968), Springer-Verlag), 125-134 [13] Montgomery, D.; Yang, C. T., Free differentiable actions on homotopy spheres, (Proceedings of the Conference on Transformation Groups. Proceedings of the Conference on Transformation Groups, New Orleans, 1967 (1968), Springer-Verlag), 175-192 [14] Obiedat, M., A note on localization of J-groups, Hiroshima Math. J., 29, 299-312 (1999) · Zbl 0932.55016 [15] Quillen, D., The Adams conjecture, Topology, 10, 67-80 (1971) · Zbl 0219.55013 [16] Sanderson, B. J., Immersions and embeddings of projective spaces, Proc. Lond. Math. Soc., 3, 137-153 (1964) · Zbl 0122.41703 [17] Steenrod, N. E., Cohomology Operations, Annals of Mathematics Studies, vol. 50 (1962), Princeton Univ. Press · Zbl 0102.38104 [18] Wall, C. T.C., Surgery on Compact Manifolds (1970), Academic Press · Zbl 0219.57024 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.