Show that if \(X\) and \(Y\) are CW complexes with \(X\) \(m\)-connected and \(Y\) \(n\)-connected, then \((X \times Y, X \vee Y)\) is \((m+n+1)\)-connected, as is the smash product \(X \wedge Y\).
Solution
By the CW version of the relative Hurewicz/cellular approximation argument, an \(m\)-connected CW complex is homotopy equivalent to a CW complex with one \(0\)-cell and no cells in dimensions \(1,\dots,m\). Likewise, replace \(Y\) by a CW model with no positive-dimensional cells below dimension \(n+1\). This replacement does not change the connectivity assertion.
Give \(X\times Y\) the product CW structure. Relative to \[ X\vee Y=(X\times\{y_0\})\cup(\{x_0\}\times Y), \] the cells are precisely products \[ e^p\times e^q \] with \(p,q>0\). By the chosen CW structures, \[ p\ge m+1, \qquad q\ge n+1, \] so every relative cell has dimension at least \[ (m+1)+(n+1)=m+n+2. \] Hence the relative CW pair \[ (X\times Y,X\vee Y) \] has no relative cells in dimensions at most \(m+n+1\). Cellular approximation therefore gives \[ \pi_i(X\times Y,X\vee Y)=0 \qquad(i\le m+n+1). \] Thus the pair is \((m+n+1)\)-connected.
Since \(X\vee Y\hookrightarrow X\times Y\) is a cofibration, the quotient map identifies relative homotopy with the homotopy of the quotient in this range, and \[ (X\times Y)/(X\vee Y)=X\wedge Y. \] Equivalently, the quotient CW structure on \(X\wedge Y\) has no positive-dimensional cells below dimension \(m+n+2\). Therefore \[ \pi_i(X\wedge Y)=0 \qquad(i\le m+n+1), \] so \[ \boxed{(X\times Y,X\vee Y)\text{ and }X\wedge Y\text{ are }(m+n+1)\text{-connected}.} \]