Wednesday, 18 January 2012

gt.geometric topology - Two solid N_3 glued by its boundary

It is a general fact that a closed manifold of odd Euler characteristic cannot bound a compact manifold. This can be deduced pretty easily from the fact that a closed manifold of odd dimension has Euler characteristic zero (a consequence of Poincaré duality) as follows. Suppose N is the boundary of a compact manifold P. Let M be the double of P, the union of two copies of P glued along N. Then the Euler characteristics of M, N, and P are related by:



chi(M)=chi(P)+chi(P)chi(N)



Thus chi(M) and chi(N) are congruent mod 2. If the dimension of N is even, then M is a closed manifold of odd dimension so chi(M)=0, hence chi(N) is even. And if the dimension of N is odd then chi(N)=0 anyhow.



I should have put this in my book!

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