Splitting isomorphisms of mapping tori

Lawson, Terry C.

doi:10.1090/S0002-9947-1975-0358821-X

Splitting isomorphisms of mapping tori
HTML articles powered by AMS MathViewer

by Terry C. Lawson PDF

Trans. Amer. Math. Soc. 205 (1975), 285-294 Request permission

Abstract:

Necessary and sufficient conditions involving invertible cobordisms are given for two mapping tori to be isomorphic. These are used to give conditions under which a given isomorphism ${M_f} \to {N_g}$ is pseudoisotopic to an isomorphism which sends $M$ to $N$. An exact sequence for the group of pseudoisotopy classes of automorphisms of $M \times {S^1}$ is derived. The principal tools are an imbedding technique due to C. T. C. Wall as well as arguments involving invertible cobordisms. Applications and examples are given, particularly for manifolds of higher dimension where the $s$-cobordism theorem is applied.

References

Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
William Browder, Diffeomorphisms of $1$-connected manifolds, Trans. Amer. Math. Soc. 128 (1967), 155–163. MR 212816, DOI 10.1090/S0002-9947-1967-0212816-0
E. M. Brown, Unknotting in $M^{2}\times I$, Trans. Amer. Math. Soc. 123 (1966), 480–505. MR 198482, DOI 10.1090/S0002-9947-1966-0198482-0

Arrondissement des arêtes

28

Michel A. Kervaire, Smooth homology spheres and their fundamental groups, Trans. Amer. Math. Soc. 144 (1969), 67–72. MR 253347, DOI 10.1090/S0002-9947-1969-0253347-3
R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI 10.1090/S0002-9904-1969-12271-8

Classification theorem for sphere bundles and mapping tori

Terry C. Lawson, Inertial $h$-cobordisms with finite cyclic fundamental group, Proc. Amer. Math. Soc. 44 (1974), 492–496. MR 358820, DOI 10.1090/S0002-9939-1974-0358820-2
Terry C. Lawson, Remarks on the four and five dimensional $s$-cobordism conjectures, Duke Math. J. 41 (1974), 639–644. MR 362361
J. Levine, Inertia groups of manifolds and diffeomorphisms of spheres, Amer. J. Math. 92 (1970), 243–258. MR 266243, DOI 10.2307/2373505
J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. MR 196736, DOI 10.1090/S0002-9904-1966-11484-2
L. C. Siebenmann, Pseudo-annuli and invertible cobordisms, Arch. Math. (Basel) 19 (1968), 528–535. MR 239611, DOI 10.1007/BF01898777
L. C. Siebenmann, Disruption of low-dimensional handlebody theory by Rohlin’s theorem. , Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 57–76. MR 0271955
G. Ananda Swarup, Pseudo-isotopies of $S^{3}\times S^{1}$, Math. Z. 121 (1971), 201–205. MR 293640, DOI 10.1007/BF01111592
Edward C. Turner, Diffeomorphisms of a product of spheres, Invent. Math. 8 (1969), 69–82. MR 250323, DOI 10.1007/BF01418871
C. T. C. Wall, Unknotting tori in codimension one and spheres in codimension two, Proc. Cambridge Philos. Soc. 61 (1965), 659–664. MR 184249, DOI 10.1017/s0305004100039001