Is it true that the wedge of two manifolds is not a manifold?

Question

The wedge sum of two circles is not a manifold since it contains a cross point. Can we generalize this property? In other words, is it true that the wedge sum of two $n$-manifolds, $n \geq 1$, is not a $n$-manifold?

Answer 1

Yes, that's true. For a $k$- and $p$-manifold, a neighborhood of the wedge point will always have the topology of $A = \Bbb R^k \times 0 \cup 0 \times \Bbb R^p \subset \Bbb R^k \times \Bbb R^p$.

How do I know that this is not homeomorphic to $\Bbb R^{k+p}$? Consider deleting the origin. Then both $\pi_{k-1}(A)$ and $\pi_{p-1}(A)$ will be nonzero, while these groups are both zero (at least for $k, p > 1$) for $\Bbb R^{k+p} - \{0\}$.

Answer 2

Any point should have a punctured neighborhood that is connected. But any punctured neighborhood of the wedge point is disconnected.