Assume one is given a commutative square of spaces

 $A \quad \to \quad  C$

$
\downarrow \qquad  \qquad     \downarrow$


 $B\quad \to \quad  X$

which is a pushout and in which each map is a cofibration.
If $A \to B$ is $r$-connected and $A\to C$ is $s$-connected, 
then the Blakers-Massey theorem says that the square is 
$(r+s-1)$-cartesian (this means that the map from $A$ into the 
homotopy pullback of the remaining terms is $(r+s-1)$-connected).

The only proofs of this statement at this level of generality that I know of make use of transversality. However, if all spaces are <i> simply connected,</i> there are
proofs which avoid transversality (for example, when $B$ is a point, one can 
deduce it using the Serre exact sequence).

**Question**: <i> Is transversality intrinsic to a proof of the theorem in the general
 case? </i>