Thoughts on Fractional Cohomology

Before I get into this post, allow me to give a bit of back story.

A close friend of mine, Aaron Slipper, mentioned the notion of a set $S$ with $i$ elements, that is, $S \times S \cup \{*\} = \emptyset$, ($\sqrt{-1} \times \sqrt{-1} + 1 = 0$). The cardinality of the automorphism group of a set with n elements is n!, but n doesn’t have to be a natural number, we can also evaluate the factorial when n is a complex number, such as $i!$, using the Gamma function.

I responded that we could similarly construct a “manifold” M which squared to the n-sphere (and call it a radical manifold).

We then extended the category of manifolds by appending radical manifolds, then asking for “Grothendieck completion wrt cartesian product and disjoint union,” in this case, formally constructing the -n-dimensional manifold, $M^{-n}$.

  • $M \coprod -M = \emptyset$ (gives us a group, with disjoint union as the operation, $-M$ is the same dimension as M)
  • $M^n \times M^{-n} = *$ (gives us negative dimensional spaces)
  • take the algebraic closure, the $n$th root of an m-dimensional manifold should be $m/n$ dimensional (gives us a field)

Then, looking at polynomials with coefficients in this strange field.

There are a few immediate and natural questions:

  • What are ideals of this ring?
  • What is the cell decomposition of a “manifold” with a $\sqrt{2}$ dimensional cell? How do we compute the $\sqrt{2}$ cohomology group?

The following thoughts resulted from a dinner-time discussion (between Aaron Slipper, Alex Mennen and I) on potentially computing homology in fractional dimensions.

Sometimes we have a graded module $H^*$ indexed by the natural numbers, equipped with a filtration:

$$F^0H^* \supset F^1H^* \supset … \supset F^nH^* \supset F^{n+1}H^* \supset … \{0\}$$

such that  $F^kH^* \supset F^{k + 1}H^* \supset … \supset \{0\} = \bigoplus_{k \geq n} H^n$

Why restrict ourselves to indexing by the naturals? What if we index by the reals? What is the continuous version of a direct sum, a “direct integral”?

Continue reading Thoughts on Fractional Cohomology

Dirichlet Series and Homology Theories

Euler was a swashbuckler. He considered the following series, nevermind that it sums to infinity, let’s see what we can do with it!

I highly recommend that you follow along with a pen and paper to convince yourself that everything cancels as I say it will, if you’re a bit too lazy for that, or already familiar with the L-function, feel free to scroll ahead.

$x = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} +  … $

Let’s multiply by 1/2 and see what happens:

$\frac{1}{2}x = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{10} +  …$

Then subtract this from the original for funsies.

$x -\frac{1}{2}x$

$=\frac{1}{1}$ + 1/2 $ + \frac{1}{3}$ + 1/4 $+ \frac{1}{5} +  …$ Continue reading Dirichlet Series and Homology Theories

A Second Glimpse of Spectra

Spectra arise in the wild when studying a collection of functors (indexed by $n$) $$F_n:  \{\text{CW-complexes}\} \to \text{AbGrp}$$

Sometimes, there is an object $E_n$ in the source category (in this case, CW-complexes) of $F_n$ such that $F_n(X) \simeq [X, E_n]$. When $F_n(-)$ is a cohomology functor, we call these $E_n$ “spectra.”

Spectra also fill the need for negative dimensional spheres — the need for a category where suspension has an inverse and not just an adjoint!

If you want a fantastic introduction to the stable category of spectra, and the context of various topological theorems calling for a definition of negative dimensional spheres, this might help.


Continue reading A Second Glimpse of Spectra

What does the sphere spectrum have to do with formal group laws?

This post assumes that you’re familiar with the definition of a prime ideal, a local ring, $R_{(p)}$, the sphere spectrum, $\mathbb{S}$, and the Lazard ring, $L$.

During a talk Jacob Lurie gave at Harvard this April, he labeled the moduli space of (1-d commutative) formal group laws as $\text{Spec }\mathbb{S}$.

Eric Peterson kindly explained why $\text{Spec } \mathbb{S} \simeq \text{Spec } L$ and I found his answer so lovely that I wish to share (all mistakes are due to me).

Why is Spec L iso to Spec $\mathbb{S}$?

This is part of the story of geometers working with higher algebra asking “what is an ideal of a ring spectrum?” Continue reading What does the sphere spectrum have to do with formal group laws?