Smooth embeddings of homologically similar manifolds

Roseman, Dennis M.

doi:10.1090/S0002-9947-1972-0310910-9

Smooth embeddings of homologically similar manifolds
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by Dennis M. Roseman PDF

Trans. Amer. Math. Soc. 174 (1972), 107-126 Request permission

Abstract:

We consider the situation where we have two smooth n-manifolds $N \subseteq M$ with ${H_\ast }(M,N) = 0$ and show that given a smooth embedding of N into some manifold Q we may, under suitable conditions, extend this to embeddings of M into Q, $Q \times I$, or $Q \times {I^2}$ (where I is the unit interval). We can apply these results to obtain smooth embeddings of homologically k-connected manifolds into $(2n - k + 1)$-dimensional euclidian space.

References

Structure of manifolds

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 and Morris W. Hirsch, On the existence and classification of differentiable embeddings, Topology 2 (1963), 129–135. MR 149494, DOI 10.1016/0040-9383(63)90028-4
Morris W. Hirsch, Embeddings and compressions of polyhedra and smooth manifolds, Topology 4 (1966), 361–369. MR 189051, DOI 10.1016/0040-9383(66)90034-6
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
Michel A. Kervaire, Le théorème de Barden-Mazur-Stallings, Comment. Math. Helv. 40 (1965), 31–42 (French). MR 189048, DOI 10.1007/BF02564363
Michel A. Kervaire, On higher dimensional knots, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 105–119. MR 0178475
Michel A. Kervaire and John W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR 148075, DOI 10.1090/S0273-0979-2015-01504-1
Barry Mazur, A note on some contractible $4$-manifolds, Ann. of Math. (2) 73 (1961), 221–228. MR 125574, DOI 10.2307/1970288
Barry Mazur, Differential topology from the point of view of simple homotopy theory, Inst. Hautes Études Sci. Publ. Math. 15 (1963), 93. MR 161342
John Milnor, A duality theorem for Reidemeister torsion, Ann. of Math. (2) 76 (1962), 137–147. MR 141115, DOI 10.2307/1970268
John Milnor, Lectures on the $h$-cobordism theorem, Princeton University Press, Princeton, N.J., 1965. Notes by L. Siebenmann and J. Sondow. MR 0190942
John R. Stallings, On infinite processes leading to differentiability in the complement of a point, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 245–254. MR 0180983
Stephen Smale, Generalized Poincaré’s conjecture in dimensions greater than four, Ann. of Math. (2) 74 (1961), 391–406. MR 137124, DOI 10.2307/1970239
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258