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).
Stallings-Zeeman Theorem
See also
Poincaré ConjectureExplore with Wolfram|Alpha
References
Milnor, J. "The Poincaré Conjecture." http://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.Referenced on Wolfram|Alpha
Stallings-Zeeman TheoremCite this as:
Weisstein, Eric W. "Stallings-Zeeman Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Stallings-ZeemanTheorem.html