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Transactions of the American Mathematical Society

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Adjacent connected sums and torus actions


Author: Dennis McGavran
Journal: Trans. Amer. Math. Soc. 251 (1979), 235-254
MSC: Primary 57S25; Secondary 57N15, 57Q15, 57R05
DOI: https://doi.org/10.1090/S0002-9947-1979-0531977-5
MathSciNet review: 531977
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Abstract: Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension $ n < m$ with embeddings of $ X\, \times \,{D^{m - n}}$ into M and N. Suppose the interior of $ X\, \times \,{D^{m - n}}$ is removed from M and N and the resulting manifolds are attached via a homeomorphism $ f:\,X \times \,{S^{m - n - 1}}\, \to \,X\, \times \,{S^{m - n - 1}}$. Let this homeomorphism be of the form $ f(x,\,t)\, = \,(x,\,F(x)(t))$ where $ F:\,X \to \,SO(m - n)$. The resulting manifold, written as $ M\,{\char93 _X}\,N$, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus $ {T^n}$ on closed, compact, connected, simply connected $ (n\, + \,2)$-manifolds, $ n \geqslant \,4$.


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  • [1] D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365-385. MR 0184241 (32:1714)
  • [2] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 0413144 (54:1265)
  • [3] R. Goldstein and L. Lininger, A classification of 6-manifolds with free $ {S^1}$-actions, Proc. of the Second Conf. on Compact Transformation Groups, Univ. of Mass., 1971, Part 1, Springer-Verlag, Berlin and New York, 1972, pp. 316-323. MR 0362378 (50:14820)
  • [4] A. Haefliger, Knotted $ (4k\, - \,1)$-spheres in 6k-space, Ann. of Math. (2) 75 (1962), 452-466. MR 0145539 (26:3070)
  • [5] J. F. P. Hudson, Piecewise linear topology, Benjamin, New York, 1969. MR 0248844 (40:2094)
  • [6] S. Kim, D. McGavran and J. Pak, Torus group actions on simply connected manifolds, Pacific J. Math. 53 (1974), 435-444. MR 0368051 (51:4293)
  • [7] S. Kim and J. Pak, Isotropy subgroups of torus $ {T^n}$ actions on $ (n\, + \,2)$-manifolds $ {M^{n + 2}}$, Michigan Math. J. 20 (1973), 353-359. MR 0343304 (49:8046)
  • [8] R. C. Kirby, Lectures on triangulations of manifolds (mimeographed), University of California at Los Angeles, 1969.
  • [9] Dennis McGavran, $ {T^3}$-actions on simply connected 6-manifolds. I, Trans. Amer. Math. Soc. 220 (1976), 59-85. MR 0415649 (54:3729)
  • [10] -, $ {T^3}$-actions on simply connected 6-manifolds. II, Indiana Univ. Math. J. 26 (1977), 125-136. MR 0440583 (55:13457)
  • [11] -, $ {T^n}$-actions on simply connected $ (n\, + \,2)$-manifolds, Pacific J. Math. 71 (1977), 487-497. MR 0461542 (57:1527)
  • [12] J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies, no. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR 0440554 (55:13428)
  • [13] P. Orlik and F. Raymond, Actions of $ SO(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups, New Orleans, 1967, Springer-Verlag, Berlin and New York, 1968, pp. 297-318. MR 0263112 (41:7717)
  • [14] -, Actions of the torus on 4-manifolds. I, Trans. Amer. Math. Soc. 152 (1970), 531-559. MR 0268911 (42:3808)
  • [15] J. Pak, Actions of the torus $ {T^n}$ on $ (n\, + \,1)$-manifolds $ {M^{n + 1}}$, Pacific J. Math. 44 (1973), 671-674. MR 0322892 (48:1253)
  • [16] F. Raymond, A classification of the actions of the circle on 3-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51-78. MR 0219086 (36:2169)
  • [17] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 0210112 (35:1007)
  • [18] C. T. C. Wall, Classification problems in differential topology. V: On certain 6-manifolds, Invent. Math. 1 (1966), 355-374.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0531977-5
Keywords: Adjacent connected sums, torus actions, simply connected manifolds, orbit space
Article copyright: © Copyright 1979 American Mathematical Society

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