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 | References | Similar Articles | Additional Information

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 (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*-*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*-*spheres in*6*k-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**actions on*-*manifolds*, 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, -*actions on simply connected*6-*manifolds*. I, Trans. Amer. Math. Soc.**220**(1976), 59-85. MR**0415649 (54:3729)****[10]**-, -*actions on simply connected*6-*manifolds*. II, Indiana Univ. Math. J.**26**(1977), 125-136. MR**0440583 (55:13457)****[11]**-, -*actions on simply connected*-*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**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**on*-*manifolds*, 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