3

I am having problem computing the de Rham cohomology with compact support of an open mobius strip,it's aquestion from Bott's book, and Bott said its cohomology is identically zero which can be obtained from MV sequence, I want some suggestions on how to choose the open coverings U and V

gordon
  • 31

1 Answers1

1

Start the ordinary construction of the mobius band: gluing a rectangle $R$ with left side of $R$ identified to right side of $R$ (with a flip).

Next, cut the rectangle into pieces, lets say $R_1,R_2,R_3$.

Next, reconstruct the Mobius band by doing all the identifications at one time: the right side of $R_1$ is glued to the left side of $R_2$ (with no flip); the right side of $R_2$ is glued to the left side of $R_3$ (with no flip); and the right side of $R_3$ is glued to the left side of $R_1$ (with a flip).

Next, fatten the rectangles up a bit, replacing $R_i$ by $S_i$ such that a strip incident to the right edge of $S_1$ is glued to a strip incident to the left edge of $S_{i+1}$ (for $i=1,2,3$, with appropriate flipping).

What you now have is an open covering by three rectangles ($S_1,S_2,S_3$) together with overlap maps (the gluing maps) between their intersections (one intersection being the strip of $S_1$ identified a strip of $S_2$; the second intersection being the strip of $S_2$ identified with a strip of $S_3$; the third being the strip of $S_3$ identified with a strip of $S_1$.) A perfect setup for a Mayer-Vietoris computation.

Lee Mosher
  • 120,280
  • Thanks for giving me a wonderful idea for the open coverings of a mobius brand, but I found a similar question whose poster made a mistake similar to mine,and that question is perfectly resolved. So I think it's also feasible to cover the mobius brand by two open sets with either diffeomorphic to R2 and the intersections of which are diffeomorphic to two copies of R2 .We may also do the Mayer-Vietoris computation under that setup . – gordon Apr 25 '15 at 15:37
  • @gordon: Of course that will work too. I used three sets so as to have connected intersections. – Lee Mosher Apr 25 '15 at 15:58