Topological invariants to prove non topological equivalence for a ball and the quotient of two balls

Question

Consider a ball $B_1$ and the quotient of two balls $B_2$ and $B_3$ at some common point $p \in B_2,B_3$ that we denote as $B_2 \#_p B_3$.

What topological invariants can be used in order to prove that $B_1$ and $B_2 \#_p B_3$ are not topologically equivalent?

Thanks in advance.

Answer 1

The operation that you consider is usually called a wedge-product.

The easiest way to see that these two spaces are not homeomorphic is to consider the wedge product deprived from point $p$: the result is a disconnected space. On the other hand, there is no point in the ball the removal of which would result in a disconnected space.