Groups of embedded manifolds

Agoston, Max K.

doi:10.1090/S0002-9947-1971-0273628-6

Groups of embedded manifolds
HTML articles powered by AMS MathViewer

by Max K. Agoston PDF

Trans. Amer. Math. Soc. 154 (1971), 365-375 Request permission

Abstract:

This paper defines a group $\theta ({M^n},{\nu _\varphi })$ which generalizes the group of framed homotopy n-spheres in ${S^{n + k}}$. Let ${M^n}$ be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let ${\nu _\varphi }$ be the normal bundle of some imbedding $\varphi :{M^n} \to {S^{n + k}}$, where $2k \geqq n + 3$. Then $\theta ({M^n},{\nu _\varphi })$ is the set of h-cobordism classes of triples $(F,{V^n},f)$, where $F:{S^{n + k}} \to T({\nu _\varphi })$ is a map which is transverse regular on M, ${V^n} = {F^{ - 1}}({M^n})$, and $f = F|{V^n}$ is a homotopy equivalence. ($T({\nu _\varphi })$ is the Thom complex of ${\nu _\varphi }$.) There is a natural group structure on $\theta ({M^n},{\nu _\varphi })$, and $\theta ({M^n},{\nu _\varphi })$ fits into an exact sequence similar to that for the framed homotopy n-spheres.

References

Max K. Agoston, $S$-tangential maps and imbeddings in the conditionally stable range, Topology 9 (1970), 13–23. MR 254863, DOI 10.1016/0040-9383(70)90045-5
Max K. Agoston, Browder-Novikov theory for maps of degree $d>1$. I, Topology 9 (1970), 251–265. MR 264677, DOI 10.1016/0040-9383(70)90015-7
William Browder, The Kervaire invariant of framed manifolds and its generalization, Ann. of Math. (2) 90 (1969), 157–186. MR 251736, DOI 10.2307/1970686
André Haefliger, Knotted $(4k-1)$-spheres in $6k$-space, Ann. of Math. (2) 75 (1962), 452–466. MR 145539, DOI 10.2307/1970208
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
Michel A. Kervaire, An interpretation of G. Whitehead’s generalization of H. Hopf’s invariant, Ann. of Math. (2) 69 (1959), 345–365. MR 102809, DOI 10.2307/1970187
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

Some theorems of Browder and Novikov on homotopy equivalent manifolds with an application

J. Levine, On differentiable imbeddings of simply-connected manifolds, Bull. Amer. Math. Soc. 69 (1963), 806–809. MR 155336, DOI 10.1090/S0002-9904-1963-11041-1
J. Levine, A classification of differentiable knots, Ann. of Math. (2) 82 (1965), 15–50. MR 180981, DOI 10.2307/1970561
S. P. Novikov, Homotopically equivalent smooth manifolds. I, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 365–474 (Russian). MR 0162246

Smoothing homotopy equivalences