Last Geometric Theorem
Can a quadratic Lefschetz theorem prove the last theorem of Poincare? At the moment, I think so. The usual Lefschetz theorem does not work as the Lefschetz number of any orientation preserving map on the annulus has Lefschetz number 0 as the trace on both zero and one dimensional cohomolgies are 1. As for quadratic cohomology, there are again two sectors as b=(0,0,0,1,1,0) (By the way, we had used this cohomology to distinguish the annulus from the Moebius strip, where b=(0,0,0,0,0,0), illustrating that the quadratic cohomology is not a homotopy invariant and can distinguish spaces which usual (linear) simplicial cohomology can not. It appears as if area preservation assures that the map induced on H^3 is 1 while the twist condition implies that the map induced on H^4 (dealing with intersecting triangles and edges) induces a map reversing orientation. Having a Lefschetz number of 2 would then imply two pairs of simplices which are invariant. In a continuum limit, this would then produce two fixed points. The pictures at the beginning were taken close, where George Birkhoff lived 100 years ago. The George D. Birkhoff house on Craigie street is a national historic Landmark. BIrkhoff lived in the house from 1920 to 1928. Birkhoff started to work at Harvard in 1912, the year of Birkhoff's death. At that time the Harvard College obervatory was already there. At the end, there are some footage taken above Observator hill. The department of astronomy came only into existence in 1931. Birkhoff lived until 1944. I could not find where the Birkhoffs lived from 1928 until 1944 nor where they were living from 1912 to 1920 while already working at Harvard.
Can a quadratic Lefschetz theorem prove the last theorem of Poincare? At the moment, I think so. The usual Lefschetz theorem does not work as the Lefschetz number of any orientation preserving map on the annulus has Lefschetz number 0 as the trace on both zero and one dimensional cohomolgies are 1. As for quadratic cohomology, there are again two sectors as b=(0,0,0,1,1,0) (By the way, we had used this cohomology to distinguish the annulus from the Moebius strip, where b=(0,0,0,0,0,0), illustrating that the quadratic cohomology is not a homotopy invariant and can distinguish spaces which usual (linear) simplicial cohomology can not. It appears as if area preservation assures that the map induced on H^3 is 1 while the twist condition implies that the map induced on H^4 (dealing with intersecting triangles and edges) induces a map reversing orientation. Having a Lefschetz number of 2 would then imply two pairs of simplices which are invariant. In a continuum limit, this would then produce two fixed points. The pictures at the beginning were taken close, where George Birkhoff lived 100 years ago. The George D. Birkhoff house on Craigie street is a national historic Landmark. BIrkhoff lived in the house from 1920 to 1928. Birkhoff started to work at Harvard in 1912, the year of Birkhoff's death. At that time the Harvard College obervatory was already there. At the end, there are some footage taken above Observator hill. The department of astronomy came only into existence in 1931. Birkhoff lived until 1944. I could not find where the Birkhoffs lived from 1928 until 1944 nor where they were living from 1912 to 1920 while already working at Harvard.