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

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 with embeddings of into *M* and *N*. Suppose the interior of is removed from *M* and *N* and the resulting manifolds are attached via a homeomorphism . Let this homeomorphism be of the form where . The resulting manifold, written as , 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 on closed, compact, connected, simply connected -manifolds, .

**[1]**D. Barden,*Simply connected five-manifolds*, Ann. of Math. (2)**82**(1965), 365–385. MR**0184241****[2]**Glen E. Bredon,*Introduction to compact transformation groups*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR**0413144****[3]**Richard Z. Goldstein and Lloyd Lininger,*A classification of 6-manifolds with free 𝑆¹ actions*, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Springer, Berlin, 1972, pp. 316–323. Lecture Notes in Math., Vol. 298. MR**0362378****[4]**André Haefliger,*Knotted (4𝑘-1)-spheres in 6𝑘-space*, Ann. of Math. (2)**75**(1962), 452–466. MR**0145539****[5]**J. F. P. Hudson,*Piecewise linear topology*, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0248844****[6]**Soon Kyu Kim, Dennis McGavran, and Jingyal Pak,*Torus group actions on simply connected manifolds*, Pacific J. Math.**53**(1974), 435–444. MR**0368051****[7]**Soon Kyu Kim and Jingyal Pak,*Isotropy subgroups of torus 𝑇ⁿ-actions on (𝑛+2)-manifolds*, Michigan Math. J.**20**(1973), 353–359. MR**0343304****[8]**R. C. Kirby,*Lectures on triangulations of manifolds*(mimeographed), University of California at Los Angeles, 1969.**[9]**Dennis McGavran,*𝑇³-actions on simply connected 6-manifolds. I*, Trans. Amer. Math. Soc.**220**(1976), 59–85. MR**0415649**, 10.1090/S0002-9947-1976-0415649-0**[10]**Dennis McGavran,*𝑇³-actions on simply connected 6-manifolds. II*, Indiana Univ. Math. J.**26**(1977), no. 1, 125–136. MR**0440583****[11]**Dennis McGavran,*𝑇ⁿ-actions on simply connected (𝑛+2)-manifolds*, Pacific J. Math.**71**(1977), no. 2, 487–497. MR**0461542****[12]**John W. Milnor and James D. Stasheff,*Characteristic classes*, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR**0440554****[13]**Peter Orlik and Frank Raymond,*Actions of 𝑆𝑂(2) on 3-manifolds*, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR**0263112****[14]**Peter Orlik and Frank Raymond,*Actions of the torus on 4-manifolds. I*, Trans. Amer. Math. Soc.**152**(1970), 531–559. MR**0268911**, 10.1090/S0002-9947-1970-0268911-3**[15]**Jingyal Pak,*Actions of torus 𝑇ⁿ on (𝑛+1)-manifolds 𝑀ⁿ⁺¹*, Pacific J. Math.**44**(1973), 671–674. MR**0322892****[16]**Frank Raymond,*Classification of the actions of the circle on 3-manifolds*, Trans. Amer. Math. Soc.**131**(1968), 51–78. MR**0219086**, 10.1090/S0002-9947-1968-0219086-9**[17]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112****[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