In abelian category, if is splittable, , then clearly we have a short exact sequence, . But exactness dosen’t necessarily imply splitting, generalized from Hatcher’s example, we have a short exact sequence

,     (1)

where , , and are positive integers, , and is a prime. And we define the homomorphism

      (2)

by , and the homomorphism

by and . However, is not split as the direct sum of and .

The space of affine lines in a plane is a 2 dimensional manifold. This space carries a natural nondegenerate closed two-form , which gives  a symplectic structure on the space of affine lines.

There is an interesting general topological statement that a proper map which is local homeomorphism between compact simply-connected Hausdorff spaces is a global homeomorphism. The main idea of C.W. Ho’s proof in 1975 was that, a local homeomorphism between compact Hausdorff spaces is an open map, and since it is a proper map, therefore is a surjective map, and then is a covering map. Finally, by the uniqueness of universal covering, the proper map is a global homeomorphism.

This is a Youtube video from Adobe TED of introducing Multi-touch interface presented by Jefferson Y. Han. Multi-touch interface is a pretty cool technology, and it may be used in teaching in future, like showing geometric pictures, applying to electronic blackboard, making dynamic presentations, etc.

I uploaded some circle packing pictures drawn by using Stephenson’s Circlepack:

“Distorting Mirror” Packing

more to come…

I’d like to share a very funny geometry video clip by Stephen Sawin on Youtube, in which “one geometry” means the geometric uniformization of three dimensional manifolds.

In Riemannian geometry, there are some theorems describing the relationship between local geometry and global topology. Next Monday I’ll give a talk in our topology seminar, which will focus on the notable Synge Theorem for orientable manifolds with positive sectional curvature. We’ll use the second variation formula for variations on geodesics to prove the theorem in detail. In addition, I’ll talk about its consequence on non-orientable manifolds. Moreover, there are some recent developments on this topic due to X. C. Rong, who proved that, if has positive pinched sectional curvature, then has a normal finite cyclic subgroup, by using Ricci flow. This result also partially backed up Chern’s Conjecture that Abelian subgroups of are cyclic.

The Fundamental Theorem of Finitely Generated Abelian Group(Module) is an important theorem in group(module) theory because it is very useful, effective, and convenient for analyzing the structure of a finite Abelian group(module). Sometimes we cannot reduce a quotient group without using it, as we came up a lot of examples in computing homology groups of complexes or cell-complexes in algebraic topology. This theorem was also used to analyze the symmetry group of composite links .