# Connectedness of the join of two spaces

Let $$X$$ be an $$n$$-connected space and $$Y$$ be an $$m$$-connected space. How can I prove that the join $$X*Y$$ is $$(n+m+1)$$-connected?

I thought that homotopy excision would do the trick, but it does not seem so.

• If $X$, $Y$ are CW, then there is an obvious CW structure on $X\ast Y$, and oce you have determined this you can use cellular methods to decide why the connectivity statement is true. – Tyrone Nov 15 '18 at 11:26
• I'm afraid I don't quite get your point. Wouldn't the CW structure on the join contain $X$ and $Y$ as subcomplexes? What results are you refering to? – user09127 Nov 20 '18 at 13:22
• The join of two CW complexes $X,Y$ is a quotient of $X\times I\times Y$ by a certain relation. Take the product CW structure on $X\times I\times Y$ and then figure out which cells you need to quotient out. You can also use the fact that $X\ast Y$ is the pushout of the inclusions $X\times CY\leftarrow X\times Y\rightarrow CX\times Y$ to get a CW structure. The end result is that the cells of $X\ast Y$ are the joins of the cells of $X$ and $Y$. I'll leave you to figure out what the joins $D^n\ast D^m$ and $S^{n-1}\ast S^{m-1}$ are. – Tyrone Nov 20 '18 at 13:47
• If $X$ is $n$ connected, and $Y$ is $m$ connected, then you'll see that the first cell of $X\ast Y$ above dimension $0$ that you need to worry about is $e^n\ast e^m$, so you can figure out the connectivity of $X\ast Y$ from, say, cellular homology, depending on what you are confident with. – Tyrone Nov 20 '18 at 13:49
• To move to the general case use CW approximation and the functorality of the join construction. – Tyrone Nov 20 '18 at 13:50