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Every closed odd-dimensional manifold has Euler characteristic zero. For an orientable manifold this follows by pairing complementary Betti numbers using Poincare duality; the general case follows from the orientable double cover and Euler characteristic under a finite covering.

A closed manifold of odd dimension has Euler characteristic zero by the Euler characteristic of an odd-dimensional closed manifold. Decomposing $M$ into $P$ and the removed closed ball along $S^{2k}$ givesso $\chi (P)=1$.

If the antipodal boundary map extended to a fixed-point-free involution of $P$, the quotient map $P\to P/⟨\tau ⟩$ would be a double covering. The formula for Euler characteristic under a finite covering would implywhich is impossible. Hence no such extension exists.