Stallings-Zeeman Theorem

If  is a finite simplicial complex of dimension  that has the homotopy type of the sphere  and is locally piecewise linearly homeomorphic to the Euclidean space , then  is homeomorphic to  under a homeomorphism which is piecewise linear except at a single point. In other words, the complement  is piecewise linearly homeomorphic to  (Milnor).

REFERENCES:

Milnor, J. "The Poincaré Conjecture." https://www.claymath.org/millennium/Poincare_Conjecture/Official_Problem_Description.pdf.

Stallings, J. "The Piecewise-Linear Structure of Euclidean Space." Proc. Cambridge Philos. Soc. 58, 481-488, 1962.

Zeeman, E. C. "The Generalised Poincaré Conjecture." Bull. Amer. Math. Soc. 67, 270, 1961.

Zeeman, E. C. "The Poincaré Conjecture for ." In Topology of 3-Manifolds and Related Topics, Proceedings of the University of Georgia Institute, 1961. Englewood Cliffs, NJ: Prentice-Hall, pp. 198-204, 1961.