Assuming equal variance in populations is important if we want to use simple formula for the difference of the means based on pooled variance  instead of long formula of degree of freedom and diffrent variances.

http://demosophy.org/estimators-for-the-difference-of-the-means-of-two-populations/

to check the assumption of equality we consider:

H0:     $\sigma_1^{2}=&space;\sigma&space;_2^{2}$

H1:      $\sigma_1^{2}\neq&space;\sigma&space;_2^{2}$

according to Cochran’s theorem s2 follows a scaled chi-square distribution:

this means
${s}^{2}=\frac{{{\chi&space;}^{2}}_{n-1}*{\sigma}^{2}&space;}{n-1}$
therefore”
$\frac{{s_1}^{2}}{{s_2}^{2}}=\frac{{}\frac{{{\chi&space;}^{2}}_{m-1}*{\sigma_1}^{2}&space;}{m-1}}{\frac{{{\chi&space;}^{2}}_{n-1}*{\sigma_2}^{2}&space;}{n-1}}$
if null hypothesis is true
$\frac{{s_1}^{2}}{{s_2}^{2}}=\frac{{}\frac{{{\chi&space;}^{2}}_{m-1}&space;}{m-1}}{\frac{{{\chi&space;}^{2}}_{n-1}&space;}{n-1}}$
and  we know that  if and be independent variates distributed as chi-squared with and degrees of freedom.We Define a statistic as the ratio of the dispersions of the two distributions

this means that
$\frac{{s_1}^{2}}{{s_2}^{2}}$
has a $F_{m-1,n-1}$ distribution.