Connected sum at infinity and 4–manifolds

Calcut, Jack S¹ ; Haggerty, Patrick V²

¹Department of Mathematics, Oberlin College, Oberlin, OH 44074, USA
²Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3281-3303

Résumé

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m ≥ 3, we construct an infinite family of pairs of m–manifolds on which the connected sum at infinity operation yields distinct manifolds for certain ray choices. We use cohomology algebras at infinity to distinguish these manifolds.

Détail

DOI : 10.2140/agt.2014.14.3281

Classification : 57R19, 55P57
Keywords: connected sum at infinity, end sum, ladder manifold, cohomology algebra at infinity, proper homotopy, direct limit, stringer sum, lens space

Bibliographie

[1] R H Bing, A surface is tame if its complement is $1$–ULC, Trans. Amer. Math. Soc. 101 (1961) 294

[2] J S Calcut, H C King, L C Siebenmann, Connected sum at infinity and Cantrell–Stallings hyperplane unknotting, Rocky Mountain J. Math. 42 (2012) 1803

[3] J Cerf, Sur les difféomorphismes de la sphère de dimension trois $(\Gamma_{4}=0)$, Lecture Notes in Mathematics 53, Springer (1968)

[4] W T Eaton, The sum of solid spheres, Michigan Math. J. 19 (1972) 193

[5] R H Fox, E Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. 49 (1948) 979

[6] R E Gompf, Minimal genera of open $4$–manifolds

[7] R E Gompf, An infinite set of exotic $\mathbb{R}^4$'s, J. Differential Geom. 21 (1985) 283

[8] R E Gompf, A I Stipsicz, 4–manifolds and Kirby calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc. (1999)

[9] V Guillemin, A Pollack, Differential topology, Prentice-Hall (1974)

[10] O G Harrold Jr., E E Moise, Almost locally polyhedral spheres, Ann. of Math. 57 (1953) 575

[11] A Hatcher, Notes on basic $3$–manifold topology (2000)

[12] A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)

[13] M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1994)

[14] B Hughes, A Ranicki, Ends of complexes, Cambridge Tracts in Mathematics 123, Cambridge Univ. Press (1996)

[15] J W Milnor, Topology from the differentiable viewpoint, Princeton Univ. Press (1997)

[16] J Munkres, Differentiable isotopies on the $2$–sphere, Michigan Math. J. 7 (1960) 193

[17] R Myers, End sums of irreducible open $3$–manifolds, Quart. J. Math. Oxford Ser. 50 (1999) 49

[18] S Schröer, Baer's result: The infinite product of the integers has no basis, Amer. Math. Monthly 115 (2008) 660

[19] C D Sikkema, A duality between certain spheres and arcs in $S^{3}$, Trans. Amer. Math. Soc. 122 (1966) 399

[20] S Smale, Diffeomorphisms of the $2$–sphere, Proc. Amer. Math. Soc. 10 (1959) 621

[21] W P Thurston, Three-dimensional geometry and topology, Vol. $1$ (editor S Levy), Princeton Mathematical Series 35, Princeton Univ. Press (1997)

[22] F C Tinsley, D G Wright, Some contractible open manifolds and coverings of manifolds in dimension three, Topology Appl. 77 (1997) 291