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Surgery and involutions on 4-manifolds. (English) Zbl 1084.57017
The 4 dimensional surgery theorem is known to be equivalent to the existence of a family of canonical 4-manifolds with free fundamental group. This paper shows that by passing to a double cover these canonical problems can be solved. This reformulates the surgery conjecture in terms of existence of free involutions on a class of 4-manifolds.
##### MSC:
 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57M10 Covering spaces and low-dimensional topology 57M60 Group actions on manifolds and cell complexes in low dimensions
##### Keywords:
4-manifolds; surgery; involutions
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##### References:
 [1] S K Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983) 279 · Zbl 0507.57010 [2] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357 · Zbl 0528.57011 [3] M H Freedman, The disk theorem for four-dimensional manifolds, PWN (1984) 647 · Zbl 0577.57003 [4] M H Freedman, A geometric reformulation of 4-dimensional surgery, Topology Appl. 24 (1986) 133 · Zbl 0898.57005 [5] M H Freedman, X S Lin, On the $$(A,B)$$-slice problem, Topology 28 (1989) 91 · Zbl 0845.57016 [6] M H Freedman, F Quinn, Topology of 4-manifolds, Princeton Mathematical Series 39, Princeton University Press (1990) · Zbl 0705.57001 [7] M H Freedman, P Teichner, 4-manifold topology I: Subexponential groups, Invent. Math. 122 (1995) 509 · Zbl 0857.57017 [8] V S Krushkal, On the relative slice problem and four-dimensional topological surgery, Math. Ann. 315 (1999) 363 · Zbl 0935.57016 [9] V S Krushkal, R Lee, Surgery on closed 4-manifolds with free fundamental group, Math. Proc. Cambridge Philos. Soc. 133 (2002) 305 · Zbl 1012.57047 [10] V S Krushkal, F Quinn, Subexponential groups in 4-manifold topology, Geom. Topol. 4 (2000) 407 · Zbl 0954.57005 [11] F Quinn, Ends of maps III: Dimensions 4 and 5, J. Differential Geom. 17 (1982) 503 · Zbl 0533.57009 [12] J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170 · Zbl 0135.05201
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