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On the inertia group of a product of spheres


Author: Reinhard Schultz
Journal: Trans. Amer. Math. Soc. 156 (1971), 137-153
MSC: Primary 57.10
DOI: https://doi.org/10.1090/S0002-9947-1971-0275453-9
MathSciNet review: 0275453
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Abstract: In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples.


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  • [1] J. F. Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. (2) 72 (1960), 20-104. MR 25 #4530. MR 0141119 (25:4530)
  • [2] P. L. Antonelli, On the stable diffeomorphism of homotopy spheres in the stable range, $ n < 2p$, Bull. Amer. Math. Soc. 75 (1969), 343-346. MR 39 #2174. MR 0240829 (39:2174)
  • [3] M. A. Armstrong, Lectures on the Hauptvermutung according to Lashof and Rothenberg, Institute for Advanced Study, Princeton, N. J., 1968 (mimeographed).
  • [4] J. M. Boardman and R. M. Vogt, Homotopy-eoerything H-spaces, Bull. Amer. Math. Soc. 74 (1968), 1117-1122. MR 38 #5215. MR 0236922 (38:5215)
  • [5] E. H. Brown, Jr. and F. P. Peterson, Whitehead products and cohomology operations, Quart. J. Math. Oxford Ser. (2) 15 (1964), 116-120. MR 28 #4549. MR 0161341 (28:4549)
  • [6] R. DeSapio, Differential structures on a product of spheres. I, Comment. Math. Helv. 44 (1969), 61-69; II, Ann. of Math. (2) 89 (1969), 305-314. MR 39 #4857; MR 39 #7611. MR 0243536 (39:4857)
  • [7] M. W. Hirsch, On homotopy spheres of low dimension, Differential and Combinatorial Topology (A Symposium in Honor of M. Morse), Princeton Math. Series, no. 27, Princeton Univ. Press, Princeton, N. J., 1965, pp. 199-204. MR 31 #4042. MR 0179800 (31:4042)
  • [8] W.-C. Hsiang, J. Levine, and R. H. Szczarba, On the normal bundle of a homotopy sphere in Euclidean space, Topology 3 (1965), 173-181. MR 30 #5323. MR 0175138 (30:5323)
  • [9] W.-C. Hsiang and J. Shaneson, Fake tori, the annulus conjecture, and the conjectures of Kirby, Proc. Nat. Acad. Sci. U.S.A. 63 (1969), 687-691. MR 0270378 (42:5267)
  • [10] N. Jacobson, Lectures in abstract algebra. Vol. II. Linear algebra, Van Nostrand, Princeton, N. J., 1953. MR 14. 837. MR 0053905 (14:837e)
  • [11] M. Kato, A concordance classification of PL homeomorphisms of $ {S^p} \times {S^q}$, Topology 8 (1969), 371-384. MR 0256401 (41:1057)
  • [12] K. Kawakubo, Smooth structures on $ {S^p} \times {S^q}$, Proc. Japan Acad. 45 (1969), 215-218. MR 40 #2107. MR 0248857 (40:2107)
  • [13] M. A. Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161-169. MR 22 #4075. MR 0113237 (22:4075)
  • [14] A. Kosinski, On the inertia group of $ \pi $-manifolds, Amer. J. Math. 89 (1967), 227-248. MR 35 #4936. MR 0214085 (35:4936)
  • [15] R. Lashof and M. Rothenberg, Microbundles and smoothing, Topology 3 (1965), 357-388. MR 31 #752. MR 0176480 (31:752)
  • [16] J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15-50. MR 31 #5211. MR 0180981 (31:5211)
  • [17] -, Self-equivalences of $ {S^n} \times {S^k}$, Trans. Amer. Math. Soc. 143 (1969), 523-543. MR 40 #2098. MR 0248848 (40:2098)
  • [18] M. Mahowald, The metastable homotopy of $ {S^n}$, Mem. Amer. Math. Soc. No. 72 (1967). MR 38 #5216. MR 0236923 (38:5216)
  • [19] B. C. Mazur, Stable equivalence of differentiable manifolds, Bull. Amer. Math. Soc. 67 (1961), 377-384. MR 24 #A557. MR 0130697 (24:A557)
  • [20] J. Milnor, Two complexes which are homeomorphic but combinatorial distinct, Ann. of Math. (2) 74 (1961), 575-590. MR 24 #A2961 ; errata, MR 25, p. 1242. MR 0133127 (24:A2961)
  • [21] S. P. Novikov, Homotopically equivalent smooth manifolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 365-474; English transl., Amer. Math. Soc. Transl. (2) 48 (1965), 271-396. MR 28 #5445. MR 0162246 (28:5445)
  • [22] -, Differentiable sphere bundles, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 71-96; English transl., Amer. Math. Soc. Transl. (2) 63 (1967), 217-244. MR 30 #4266. MR 0174059 (30:4266)
  • [23] C. P. Rourke, The Hauptvermutung according to Sullivan, Institute for Advanced Study, Princeton, N. J., 1968 (mimeographed).
  • [24] H. Sato, Diffeomorphism groups and classification of manifolds, J. Math. Soc. Japan 21 (1969), 1-36. MR 39 #3525. MR 0242192 (39:3525)
  • [25] -, Diffeomorphism groups of $ {S^p} \times {S^q}$ and exotic spheres, Quart. J. Math. Oxford Ser. (2) 79 (1969), 255-276.
  • [26] R. Schultz, Smooth structures on $ {S^p} \times {S^q}$, Ann. of Math. (2) 90 (1969), 187-198. MR 40 #3560. MR 0250321 (40:3560)
  • [27] -, Smoothings of sphere bundles over spheres in the stable range, Invent. Math. 9 (1969), 81-88. MR 0256407 (41:1063)
  • [28] J. L. Shaneson, Wall's surgery obstruction groups for $ G \times Z$, Ann. of Math. (2) 90 (1969), 296-334. MR 39 #7614. MR 0246310 (39:7614)
  • [29] D. Sullivan, Triangulating and smoothing homotopy equivalences and homeomorphisms, Geometric Topology Seminar Notes, Princeton University, Princeton, N. J., 1967 (mimeographed).
  • [30] E. C. Turner, Diffeomorphisms of a product of spheres, Invent. Math. 8 (1969), 69-82. MR 40 #3562. MR 0250323 (40:3562)
  • [31] C. T. C. Wall, Killing the middle homotopy groups of odd-dimensional manifolds, Trans. Amer. Math. Soc. 103 (1962), 421-433. MR 25 #2621. MR 0139185 (25:2621)
  • [32] -, Classification problems in differential topology. II. Diffeomorphisms of handlebodies, Topology 2 (1963), 263-272. MR 27 #6278. MR 0156354 (27:6278)
  • [33] -, Surgery of compact manifolds (to appear).
  • [34] -, On homotopy tori and the annulus theorem, Bull. London Math. Soc. 1 (1969), 95-97. MR 39 #3498. MR 0242164 (39:3498)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0275453-9
Keywords: Inertia group, homotopy sphere, fiber homotopy equivalence, self-equivalence group, automorphisms of homology, surgery, smoothings of PL manifolds, homotopy smoothings, normal invariants, homotopy composition
Article copyright: © Copyright 1971 American Mathematical Society

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