On not staying in your lane

This week I’ve been thinking about some problems outside my usual zone of expertise — namely, questions about the mapping class group and the Johnson kernel.  This has been my week:

• Three days of trying to prove a cohomology class is nonzero;
• Then over Thanksgiving I worked out an argument that it was zero and was confused about that for a couple of days because I feel quite deeply that it shouldn’t be zero;
• This morning I was able to get myself kind of philosophical peace with the class being zero and was working out which refined version of the class might not be zero;
• This afternoon I was able to find the mistake in my argument that the class was zero so now I hope it’s not zero again.
• But I still don’t know.

There’s a certain frustration, knowing that I’ve spend a week trying to compute something which some decently large number of mathematicians could probably sit down and just do, because they know their way around this landscape.  But on the other hand, I would never want to give up the part of math research that involves learning new things as if I were a grad student.  It is not the most efficient way, in the short term, to figure out whether this class is zero or not, but I think it probably helps me do math better in a global sense that I spend some of my weeks stumbling around unfamiliar rooms in the dark.  Of course I might just be rationalizing something I enjoy doing.  Even if it’s frustrating.  Man, I hope that class isn’t zero.