# FP on simplicial groups

It’s a good exercise to create fun problems (FPs). Here’s one: Show that every simplicial group is a Kan complex. In case you aren’t familiar with the terminology, let me explain. A simplicial group is a functor where is the simplex category. A Kan complex is a simplial set (i.e., a functor such that for any […]

# A solution to the fun question in the last post

Recall the question: Show that and are irreducible over , but reducible in . First we try the standard methods; the rational root test shows that there aren’t any rational roots. Also, you can’t use Eisenstein here. So we’ll resort to Galois theory :o) The polynomial is the polynomial in the case , , and […]

# A fun question on polynomials

Here’s a somewhat basic but fun question on polynomials that I prepared for the Putnam seminar. I’ll post the solution by the end of this week. Show that and are irreducible over , but reducible in .

# Hopf invariant one, III: K-theory

Let be a vector bundle over , and let be a continuous map. There’s an induced bundle over , given by (what’s the projection map?). Suppose is a vector bundle; the following result states that this is classified by a map from to some space. Lemma: There’s a universal bundle, denoted , over a space , which […]

# Hopf invariant one, part II: cohomology operations and vector fields on spheres

Let’s begin by talking about cohomology operations. Cohomology Operations I will state a few definitions and results now, and explain their significance in a bit. Definition: Let and be cohomology theories. A cohomology operation of “type and degree ” is a natural transformation . A stable cohomology operation of degree is a collection of homomorphisms […]

# Hopf invariant one, part I

Computing for is easy. By definition, , and by cellular approximation, we note that every map can be homotoped to a cellular map. It is obvious that all such cellular maps are trivial, so that if . The homotopy groups are more interesting, and can be computed via the Hurewicz theorem: Theorem (Hurewicz): There is […]

# The (co)homology of the loop space of a n-sphere

Computing (co)homology can be rather hard without the appropriate tools. The Serre spectral sequence is one such tool. We compute the important example of , and also provide a different proof of Proposition 3.22 in Hatcher’s Algebraic Topology, namely the computation of , where is the reduced James product (see section 4.J, for James). (Note […]

# Elliptic curves and cryptography

I taught a class on elliptic curves and cryptography to high school students. Here are some things that I wrote. Elliptic Curves Definition: An elliptic curve over a field is a curve defined by an equation like where and , along with a “point at infinity” denoted . This “point at infinity” is obtained by considering […]

# An Injective Tango! (Schroeder-Bernstein Theorem)

Now there’s a simple theorem in set theory whose proof has always appeared a bit cloudy to me, since I’ve never been able to find it written in a straightforward manner. This theorem is the Schroeder-Bernstein Theorem, whose statement is utterly intuitive: Schroeder-Bernstein Theorem: Let and be sets. If there exists an injection , and an […]

# What is homotopy coherence?

Today I want to talk about homotopy coherence, which is a very important notion in motivating a lot of the constructions in homotopy theory. Consider complex K-theory, . This is represented by the space , i.e., there is a bijection induced by pullback (see my post for an explanation). Because is valued in rings, the space is a commutative […]