We generally observe that when men work, and women do not, women do the majority of the housework and childcare. As women move into the workplace we see, statistically, men take over more and more of the housework. This does not, on average, ever surpass the amount of housework the woman does, but it almost gets there. At the point where the wife's income surpasses the husband's some studies find that this trend reverses, and the wife starts taking over more housework again.
One theory as to why this might be so, called gender deviance neutralization, is that both partners are invested in preserving traditional gender roles. Specifically, these roles are more important to them than is fairness in divvying up housework.
There is some evidence for this. Some studies have found that both partners will work to find roles for the man so that he can increase his role as a provider -- increasing his role in managing finances or undertaking DIY projects -- while downplaying the woman's role as primary breadwinner. On the other hand, other studies have not found a decrease in the husband's completion of housework at all, although even in these studies it does seem consistent that the woman's allotment of housework is very resistant to dropping below that of the man's.
It seems reasonable that gender deviance neutralization is an explanation of some people's behavior -- you can probably think of a couple whom this model fits -- but it is still a matter of some debate as to whether it can be applied to populations as a whole, and if so, which populations.
Brevity Quest 2016