My last few posts have been about zero sections of vector bundles, and how the homology class of a zero section is governed by a cohomology class associated with the vector bundle, known as the Euler class. I worked through some examples where we can compute the Euler class of a vector bundle using another mysterious cohomology class...

On an orientable nnn-dimensional manifold MMM, the following four spaces are all isomorphic: Hi(M;R),Hn−i(M;R),Hi(M;R),Hn−i(M;R).H_i(M;\RR), H_{n-i}(M;\RR), H^i(M;\RR), H^{n-i}(M;\RR).Hi​(M;R),Hn−i​(M;R),Hi(M;R),Hn−i(M;R). This fact really boils down to two distinct isomorphisms: Hi(M;R)≃Hi(M;R)H_i(M;\RR) \simeq H^i(M;\RR)Hi​(M;R)≃Hi(M;R) and Hi(M;R)≃Hn−i(M;R)H_i(M;\RR) \simeq H^{n-i}(M;\RR)Hi​(M;R)≃Hn−i(M;R). The first (known as the universal coefficients theorem), behaves surprisingly differently from the second (known as Poincaré duality) . This can be especially confusing on...

My last post discussed zero sections of vector bundles, and showed some examples where the zero sections are determined by a particular cohomology class called the Euler class. In this post, I'll go through two constructions of the Euler class and work through calculations of Euler classes using both constructions. The first construction—using Chern classes—is geometric and a...

This post walks through some interesting results on real and complex line bundles, leading up to a discussion of the Euler class for complex line bundles, which characterizes their zero sections. Briefly, a complex line bundle is a topological object which generalizes the space of complex-valued functions defined on some domain. The "sections" of the complex line bundle...

Suppose we have a 2−d-d−d quadratic form Q:R2→RQ : \mathbb{R}^2 \to \mathbb{R}Q:R2→R. How can we compute its determinant? This problem is a little under-specified. In fact, we can always pick a basis for R2\RR^2R2 so that our quadratic form is diagonal with diagonal entries 000 or ±1\pm 1±1. So there is always some basis where our quadratic form...

Today, I want to write about a pretty simple problem: finding the line where two planes intersect. At the end of the day, it boils down to a simple formula that's easy to find elsewhere online. However, I find the derivation interesting, and it serves as a nice introduction to some powerful techniques that can be used on...

Suppose we have two conic sections and we want to find their intersections. If the conics are circles, then this is easy, and can be done in closed form. However, if we have, say, a pair of hyperbolas then things are harder. For instance, a pair of hyperbolas can intersect in four points, unlike a pair of circles...