Category Archives: algebraic-topology

Computing the Alexander polynomial: a guide

Given a knot $K$, which is an embedding $S^1\to \R^3$, we have see how to compute the fundamental group of $K$, defined as $\pi_1(\R^3 – K)$. For example, we have computed the fundamental group of the trefoil knot and the fundamental group of the cinquefoil knot. The fundamental group of the trefoil can be given […]


Ravenel’s “Nilpotence and periodicity in stable homotopy theory” Free Download

Doug Ravenel has made his book Nilpotence and periodicity in stable homotopy theory available for free download along with a list of errata, also available at the same page as the book. Here is the official description from Princeton University Press: Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic […]


A is Homotopy Equivalent to A^op via Functors

Let $ \mathcal{A}$ be a small category and $ \mathbf{B}\mathcal{A}$ its geometric realisation. It is evident that $ \mathbf{B}\mathcal{A}$ and $ \mathbf{B}\mathcal{A}^\circ$ are homotopy equivalent, and in fact homeomorphic. However, can we find functors that realise this equivalence? This post summarises some informal notes I have written on this following D. Quillen’s paper Higher Algebraic […]