𝑇³-actions on simply connected 6-manifolds. I

McGavran, Dennis

doi:10.1090/S0002-9947-1976-0415649-0

$T^{3}$-actions on simply connected $6$-manifolds. I
HTML articles powered by AMS MathViewer

by Dennis McGavran PDF

Trans. Amer. Math. Soc. 220 (1976), 59-85 Request permission

Abstract:

We are concerned with ${T^3}$-actions on simply connected 6-manifolds ${M^6}$. As in the codimension two case, there exists, under certain restrictions, a cross-section. Unlike the codimension two case, the orbit space need not be a disk and there can be finite stability groups. C. T. C. Wall has determined (Invent. Math. 1 (1966), 355-374) a complete set of invariants for simply connected 6-manifolds with ${H_\ast }({M^6})$ torsion-free and ${\omega _2}({M^6}) = 0$. We establish sufficient conditions for these two requirements to be met when M is a ${T^3}$-manifold. Using surgery and connected sums, we compute the invariants for manifolds satisfying these conditions. We then construct a ${T^3}$-manifold ${M^6}$ with invariants different than any well-known manifold. This involves comparing the trilinear forms (defined by Wall) for two different manifolds.

References

Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Marvin J. Greenberg, Lectures on algebraic topology, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0215295
André Haefliger, Differential embeddings of $S^{n}$ in $S^{n+q}$ for $q>2$, Ann. of Math. (2) 83 (1966), 402–436. MR 202151, DOI 10.2307/1970475
André Haefliger, Differentiable links, Topology 1 (1962), 241–244. MR 148076, DOI 10.1016/0040-9383(62)90106-4
André Haefliger, Knotted $(4k-1)$-spheres in $6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR 145539, DOI 10.2307/1970208
J. F. P. Hudson, Piecewise linear topology, W. A. Benjamin, Inc., New York-Amsterdam, 1969. University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees. MR 0248844
Sze-tsen Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, New York-London, 1959. MR 0106454
Soon Kyu Kim, Dennis McGavran, and Jingyal Pak, Torus group actions on simply connected manifolds, Pacific J. Math. 53 (1974), 435–444. MR 368051
Lloyd Lininger, $S^{1}$ actions on $6$-manifolds, Topology 11 (1972), 73–78. MR 281227, DOI 10.1016/0040-9383(70)90054-6
William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390

actions on simply connected

manifolds

John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554

The representation rings of some classical groups

Peter Orlik and Frank Raymond, Actions of $\textrm {SO}(2)$ on 3-manifolds, Proc. Conf. on Transformation Groups (New Orleans, La., 1967) Springer, New York, 1968, pp. 297–318. MR 0263112
Peter Orlik and Frank Raymond, Actions of the torus on $4$-manifolds. I, Trans. Amer. Math. Soc. 152 (1970), 531–559. MR 268911, DOI 10.1090/S0002-9947-1970-0268911-3
Jingyal Pak, Actions of torus $T^{n}$ on $(n+1)$-manifolds $M^{n+1}$, Pacific J. Math. 44 (1973), 671–674. MR 322892
Frank Raymond, Classification of the actions of the circle on $3$-manifolds, Trans. Amer. Math. Soc. 131 (1968), 51–78. MR 219086, DOI 10.1090/S0002-9947-1968-0219086-9
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
C. T. C. Wall, Classification problems in differential topology. V. On certain $6$-manifolds, Invent. Math. 1 (1966), 355–374; corrigendum, ibid. 2 (1966), 306. MR 215313, DOI 10.1007/BF01425407