Inertia groups and smooth structures on quaternionic projective spaces
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Samik Basu
Abstract
This paper deals with certain results on the number of smooth structures on quaternionic projective spaces, obtained through the computation of inertia groups and their analogues, which in turn are computed using techniques from stable homotopy theory. We show that the concordance inertia group is trivial in dimension 20, but there are many examples in high dimensions where the concordance inertia group is non-trivial. We extend these to computations of concordance classes of smooth structures. These have applications to 3-sphere actions on homotopy spheres and tangential homotopy structures.
Acknowledgements
The research of the first author was partially supported by NBHM (project ref. number 2/48(11)/2015/NBHM(R.P.)/R&D II/3743).
References
[1] C. S. Aravinda and F. T. Farrell, Exotic structures and quaternionic hyperbolic manifolds, Algebraic Groups and Arithmetic, Tata Inst. Fund. Res., Mumbai (2004), 507–524. Suche in Google Scholar
[2] S. Basu and R. Kasilingam, Inertia groups of high-dimensional complex projective spaces, Algebr. Geom. Topol. 18 (2018), no. 1, 387–408. 10.2140/agt.2018.18.387Suche in Google Scholar
[3] A. Borel, Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963), 111–122. 10.1016/0040-9383(63)90026-0Suche in Google Scholar
[4] W. Browder, Surgery on Simply-Connected Manifolds, Ergeb. Math. Grenzgeb. (3) 65, Springer, New York, 1972. 10.1007/978-3-642-50020-6Suche in Google Scholar
[5]
G. Brumfiel,
Differentiable
[6] G. Brumfiel, Homotopy equivalences of almost smooth manifolds, Comment. Math. Helv. 46 (1971), 381–407. 10.1090/pspum/022/0319217Suche in Google Scholar
[7]
D. Burghelea,
Free differentiable
[8] A. M. Gleason, Spaces with a compact Lie group of transformations, Proc. Amer. Math. Soc. 1 (1950), 35–43. 10.1090/S0002-9939-1950-0033830-7Suche in Google Scholar
[9] D. N. Hertz, Ambient surgery and tangential homotopy quaternionic projective spaces, Trans. Amer. Math. Soc. 145 (1969), 517–545. 10.1090/S0002-9947-1969-0250320-6Suche in Google Scholar
[10]
W.-C. Hsiang,
A note on free differentiable actions of
[11] I. M. James, Relative Stiefel Manifolds, London Math. Soc. Lecture Note Ser. 24, Cambridge University, Cambridge, 1976. 10.1112/jlms/s2-13.2.331Suche in Google Scholar
[12]
I. H. Kaddoura,
De-suspension of free
[13] R. Kasilingam, Homotopy inertia groups and tangential structures, JP J. Geom. Topol. 20 (2017), 91–114. 10.17654/GT020020091Suche in Google Scholar
[14] K. Kawakubo, Inertia groups of low dimensional complex projective spaces and some free differentiable actions on spheres. I, Proc. Japan Acad. 44 (1968), 873–875. 10.3792/pja/1195520972Suche in Google Scholar
[15] K. Kawakubo, On the inertia groups of homology tori, J. Math. Soc. Japan 21 (1969), 37–47. 10.2969/jmsj/02110037Suche in Google Scholar
[16] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. 10.1142/9789812836878_0002Suche in Google Scholar
[17] R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann. of Math. Stud. 88, Princeton University, Princeton, 1977. 10.1515/9781400881505Suche in Google Scholar
[18] S. O. Kochman, Bordism, Stable Homotopy and Adams Spectral Sequences, Fields Inst. Monogr. 7, American Mathematical Society, Providence, 1996. 10.1090/fim/007Suche in Google Scholar
[19] L. Kramer and S. Stolz, A diffeomorphism classification of manifolds which are like projective planes, J. Differential Geom. 77 (2007), no. 2, 177–188. 10.4310/jdg/1191860392Suche in Google Scholar
[20]
H.-T. Ku,
A note on semifree actions of
[21]
H.-T. Ku and M.-C. Ku,
Free differentiable actions of
[22]
H.-T. Ku and M.-C. Ku,
Characteristic spheres of free differentiable actions of
[23] N. H. Kuiper and R. K. Lashof, Microbundles and bundles. I. Elementary theory, Invent. Math. 1 (1966), 1–17. 10.1007/BF01389695Suche in Google Scholar
[24] R. Lashof and M. Rothenberg, Microbundles and smoothing, Topology 3 (1965), 357–388. 10.1016/0040-9383(65)90003-0Suche in Google Scholar
[25] C.-N. Lee, Detection of some elements in the stable homotopy groups of spheres, Math. Z. 222 (1996), no. 2, 231–245. 10.1007/BF02621865Suche in Google Scholar
[26] I. Madsen and R. J. Milgram, The Classifying Spaces for Surgery and Cobordism of Manifolds, Ann. of Math. Stud. 92, Princeton University, Princeton, 1979. 10.1515/9781400881475Suche in Google Scholar
[27] I. Madsen, L. R. Taylor and B. Williams, Tangential homotopy equivalences, Comment. Math. Helv. 55 (1980), no. 3, 445–484. 10.1007/BF02566699Suche in Google Scholar
[28] I. Madsen, C. B. Thomas and C. T. C. Wall, The topological spherical space form problem. II. Existence of free actions, Topology 15 (1976), no. 4, 375–382. 10.1016/0040-9383(76)90031-8Suche in Google Scholar
[29] J. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. (2) 64 (1956), 399–405. 10.1142/9789812836878_0001Suche in Google Scholar
[30] R. E. Mosher, Some stable homotopy of complex projective space, Topology 7 (1968), 179–193. 10.1016/0040-9383(68)90026-8Suche in Google Scholar
[31] R. E. Mosher and M. C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper & Row, New York, 1968. Suche in Google Scholar
[32] B. Okun, Exotic smooth structures on nonpositively curved symmetric spaces, Algebr. Geom. Topol. 2 (2002), 381–389. 10.2140/agt.2002.2.381Suche in Google Scholar
[33] K. Ramesh, Farrell–Jones spheres and inertia groups of complex projective spaces, Forum Math. 27 (2015), no. 5, 3005–3015. 10.1515/forum-2013-0072Suche in Google Scholar
[34]
K. Ramesh,
Inertia groups and smooth structures of
[35] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, 2nd ed., AMS Chelsea, Providence, 2004. 10.1090/chel/347Suche in Google Scholar
[36] R. Schultz, On the inertia group of a product of spheres, Trans. Amer. Math. Soc. 156 (1971), 137–153. 10.1090/S0002-9947-1971-0275453-9Suche in Google Scholar
[37] R. Schultz, Homology spheres as stationary sets of circle actions, Michigan Math. J. 34 (1987), no. 2, 183–200. 10.1307/mmj/1029003551Suche in Google Scholar
[38] F. Sigrist and U. Suter, Cross-sections of symplectic Stiefel manifolds, Trans. Amer. Math. Soc. 184 (1973), 247–259. 10.1090/S0002-9947-1973-0326728-8Suche in Google Scholar
[39] D. P. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms, The Hauptvermutung Book, K-Monogr. Math. 1, Kluwer Academic, Dordrecht (1996), 69–103. 10.1007/978-94-017-3343-4_3Suche in Google Scholar
[40] I. Tamura, Sur les sommes connexes de certaines variétés différentiables, C. R. Math. Acad. Sci. Paris 255 (1962), 3104–3106. Suche in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- On the variance of the nodal volume of arithmetic random waves
- On length densities
- On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
- Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
- On cutting blocking sets and their codes
- Inertia groups and smooth structures on quaternionic projective spaces
- On the number of zeros of diagonal quartic forms over finite fields
- Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
- A dual version of Huppert's ρ-σ conjecture for character codegrees
- A topological correspondence between partial actions of groups and inverse semigroup actions
- On homotopy braids
- Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
- The Shintani double zeta functions
- Homological dimensions relative to preresolving subcategories II
- Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
- A converse theorem for quasimodular forms
Artikel in diesem Heft
- Frontmatter
- On the variance of the nodal volume of arithmetic random waves
- On length densities
- On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
- Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
- On cutting blocking sets and their codes
- Inertia groups and smooth structures on quaternionic projective spaces
- On the number of zeros of diagonal quartic forms over finite fields
- Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
- A dual version of Huppert's ρ-σ conjecture for character codegrees
- A topological correspondence between partial actions of groups and inverse semigroup actions
- On homotopy braids
- Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
- The Shintani double zeta functions
- Homological dimensions relative to preresolving subcategories II
- Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
- A converse theorem for quasimodular forms
