## On proper homotopy theory for noncompact $3$-manifolds

HTML articles powered by AMS MathViewer

- by E. M. Brown and T. W. Tucker PDF
- Trans. Amer. Math. Soc.
**188**(1974), 105-126 Request permission

## Abstract:

Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincaré conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product $F \times [0,1]$ or a half-open product $F \times [0,1)$ where*F*is a 2-manifold is actually homeomorphic to $F \times [0,1]$ or $F \times [0,1)$, respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a well-known result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case.

## References

- R. H. Bing,
*Locally tame sets are tame*, Ann. of Math. (2)**59**(1954), 145–158. MR**61377**, DOI 10.2307/1969836 - W. Browder, J. Levine, and G. R. Livesay,
*Finding a boundary for an open manifold*, Amer. J. Math.**87**(1965), 1017–1028. MR**189046**, DOI 10.2307/2373259 - Edward M. Brown,
*The Hauptvermutung for $3$-complexes*, Trans. Amer. Math. Soc.**144**(1969), 173–196. MR**251729**, DOI 10.1090/S0002-9947-1969-0251729-7
—, - 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 - C. H. Edwards Jr.,
*Open $3$-manifolds which are simply connected at infinity*, Proc. Amer. Math. Soc.**14**(1963), 391–395. MR**150745**, DOI 10.1090/S0002-9939-1963-0150745-3 - Hans Freudenthal,
*Über die Enden topologischer Räume und Gruppen*, Math. Z.**33**(1931), no. 1, 692–713 (German). MR**1545233**, DOI 10.1007/BF01174375 - L. S. Husch and T. M. Price,
*Finding a boundary for a $3$-manifold*, Ann. of Math. (2)**91**(1970), 223–235. MR**264678**, DOI 10.2307/1970605 - Wolfgang Heil,
*On $P^{2}$-irreducible $3$-manifolds*, Bull. Amer. Math. Soc.**75**(1969), 772–775. MR**251731**, DOI 10.1090/S0002-9904-1969-12283-4 - D. R. McMillan Jr.,
*Some contractible open $3$-manifolds*, Trans. Amer. Math. Soc.**102**(1962), 373–382. MR**137105**, DOI 10.1090/S0002-9947-1962-0137105-X - Edwin E. Moise,
*Affine structures in $3$-manifolds. VIII. Invariance of the knot-types; local tame imbedding*, Ann. of Math. (2)**59**(1954), 159–170. MR**61822**, DOI 10.2307/1969837 - L. C. Siebenmann,
*Infinite simple homotopy types*, Nederl. Akad. Wetensch. Proc. Ser. A 73 = Indag. Math.**32**(1970), 479–495. MR**0287542**, DOI 10.1016/S1385-7258(70)80052-X - L. C. Siebenmann,
*On detecting open collars*, Trans. Amer. Math. Soc.**142**(1969), 201–227. MR**246301**, DOI 10.1090/S0002-9947-1969-0246301-9
—, - Friedhelm Waldhausen,
*On irreducible $3$-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**224099**, DOI 10.2307/1970594
J. H. C. Whitehead, - E. M. Brown and R. H. Crowell,
*Deformation retractions of 3-manifolds into their boundaries*, Ann. of Math. (2)**82**(1965), 445–458. MR**182012**, DOI 10.2307/1970707 - Marianne Brown,
*Constructing isotopies in noncompact $3$-manifolds*, Bull. Amer. Math. Soc.**78**(1972), 461–464. MR**298667**, DOI 10.1090/S0002-9904-1972-12945-8

*On proper homotopy type*, Lecture Notes in Math., vol. 375, Springer-Verlag, Berlin and New York.

*The obstruction to finding a boundary for an open manifold of dimension greater than*5, Thesis, Princeton University, Princeton, N. J., 1965.

*Simplicial spaces, nuclei, and m-groups*, Proc. London Math. Soc.

**45**(1939), 243-327. E. C. Zeeman,

*Seminar on combinatorial topology*, Inst. Hautes Etudes Sci., Paris, 1963. (mimeographed)

## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**188**(1974), 105-126 - MSC: Primary 57A65
- DOI: https://doi.org/10.1090/S0002-9947-1974-0334225-X
- MathSciNet review: 0334225