Tag Archives: stupidity

Discrete groups and FPD actions

Joey’s question about discreteness and FPD actions caught me a little unprepared this morning, and I may not have answered as clearly as I should.  If \(G\) is a topological group, then the notion of action of \(G\) on \(X\) needs to be modified to require that \(g \mapsto g\cdot x\) should be a continuous function of \(g\), for each fixed \(x\).  (Note that the continuity in \(x\) of such an expression follows from the isometry condition.)  Now if a metrizable (or Hausdorff if you prefer) topological group \(G\) acts on a metric space via an FPD action, then the topology of \(G\) must be discrete. (If not, consider a sequence \( \{g_n\}\) of non-identity elements of \(G\) that converges to the identity, and ask what happens to \(g_n\cdot x\).)  Thus discreteness of \(G\) is actually an automatic consequence of the other conditions; for which reason, I didn’t mention it at all.

I see that the in-class exercise for this time got pushed to the next lecture.  So, whoever it was who was due to do today’s in-class assignment, try writing out and posting  the details of the above argument instead (maybe in a comment to this post).

John

 

Homework 2, q3 flakiness

Hi all

There are a couple of points in homework 2 question 3 which seem to have caused some confusion. These are

  • Topologies on function spaces
  • Base point issues

I should have been clearer that you should not worry too much about the first; and as far as the second goes, I made things unnecessarily complicated by the way I worded the question.  My apologies.  (I should add that these are both slightly sophisticated issues; the basic argument for this question can be understood without worrying about either of the points I will bring up below. If none of what I am about to say makes any sense, just ignore it.)

Function space topologies I asked you to prove various loop and path spaces are contractible, and I defined contractibility in terms of the function space \(X^X\). Since loop and path spaces are seldom compact, I have not told you what it means for a path of maps from such a space to itself to be continuous.  So, how to do the question?

The basic answer is that the contraction you produce is given by such a simple formula that it should be continuous under any “reasonable” definition (i.e., “don’t sweat it”).  But I know that this is not satisfactory for some of you.  The appropriate topology for function spaces in this context is usually the compactly generated modification of the compact-open topology.  My British colleague Neil Strickland  has some nice notes about this topology, why it is good, and why it does what you need for function spaces; you can find those here. (An earlier reference is Steenrod’s A Convenient Category of Topological Spaces.)

Base points I screwed up here by using base points in one part of the question but not in another part.  I should have gone “all or nothing”.   Since in the second and third parts of the question I use based loops and paths, the appropriate thing to do would have been to define contractibility in the first part of the question in terms of the path-connectedness of the space of basepoint-preserving maps (rather than all maps) from \(X\) to \(X\).  (Alternatively, I could have kept the first part as it was and worked in the second two parts with unbased loops and paths).

There are examples where “contractibility” and “based contractibility” are not the same thing, though that is impossible for ‘nice’ spaces such as CW-complexes.  See here for some more information about this.

John

The importance of stupidity in scientific research

I mentioned this paper in class today.  Here is a link to it.

It starts like this:

I recently saw an old friend for the first time in many years. We had been Ph.D. students at the same time, both studying science, although in different areas. She later dropped out of graduate school, went to Harvard Law School and is now a senior lawyer for a major environmental organization. At some point, the conversation turned to why she had left graduate school. To my utter astonishment, she said it was because it made her feel stupid. After a couple of years of feeling stupid every day, she was ready to do something else.

I had thought of her as one of the brightest people I knew and her subsequent career supports that view. What she said bothered me. I kept thinking about it; sometime the next day, it hit me. Science makes me feel stupid too. It’s just that I’ve gotten used to it. So used to it, in fact, that I actively seek out new opportunities to feel stupid. I wouldn’t know what to do without that feeling. I even think it’s supposed to be this way. Let me explain.