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An A_∞-structure on the cohomology of D₈

November 30th, 2006

As a first unknown (kinda, sorta, it still falls under the category of path algebra quotients treated by Keller) A_∞-calculation, I shall find the A_∞-structure of $H^*(D_8,\mathbb F_2)$ .

To do this, I fix the group algebra
$\Lambda=\mathbb F_2[a,b]/(a^2,b^2,abab+baba)$
and the cohomology ring
$\Gamma=\mathbb F_2[x,y,z]/(xy)$
with $x,y\in\Gamma_1$ , $z\in\Gamma_2$

Furthermore, we pick a canonical nice resolution P, continuing the one I calculated previously. This has the i:th component Λⁱ⁺¹, and the differentials looking like
$\begin{pmatrix} a&0&bab&0&0&0\\ 0&b&0&aba&0&0\\ 0&0&a&0&bab&0\\ 0&0&0&b&0&aba\\ 0&0&0&0&a&b \end{pmatrix}$
for differentials starting in odd degree, and
$\begin{pmatrix} a&0&bab&0&0\\ 0&b&0&aba&0\\ 0&0&a&0&bab\\ 0&0&0&b&aba \end{pmatrix}$
for differentials starting in even degree. The first few you can see on the previous calculation, or if you don’t want to bother, they are
$\partial_0=\begin{pmatrix}a&b\end{pmatrix}$
$\partial_1=\begin{pmatrix}a&0&bab\\0&b&aba\end{pmatrix}$
$\partial_2=\begin{pmatrix}a&0&bab&0\\0&b&0&aba\\0&0&a&b\end{pmatrix}$

Now, armed with this, we can get cracking. By lifting, we get canonical representating chain maps for x,y,z described, loosly, by the following:

x takes an element in $\Lambda^{2k}$ , keeps the first, third, et.c. elements and throws out the even ordered elements; so $x\cdot(a_1,a_2,a_3,a_4,a_5,a_6)=(a_1,0,a_3,0,a_5)$
For an element in $\Lambda^{2k+1}$ , the last element gets extra treatment, so
$x\cdot(a_1,a_2,a_3,a_4,a_5,a_6,a_7)=(a_1,0,a_3,0,a_5,ab\cdot a_7)$
For the lowest degrees, we also have
$x_0 = \begin{pmatrix}1&0\end{pmatrix}$
$x_1 = \begin{pmatrix}1&0&0\\0&0&ab\end{pmatrix}$
$x_2 = \begin{pmatrix}1&0&0&0\\0&0&1&0\end{pmatrix}$

This is a preview of An A_∞-structure on the cohomology of D₈. Read the full post (815 words, 61 images)

A_∞-algebras and group cohomology

November 23rd, 2006

In which the author, after a long session sweating blood with his advisor, manages to calculate the A_∞-structures on the cohomology algebras $H^*(C_4,\mathbb F_2)$ and $H^*(C_2\times C_2,\mathbb F_2)$ .

We will find the A_∞-structures on the group cohomology ring by establishing an A_∞-quasi-isomorphism to the endomorphism dg-algebra of a resolution of the base field. We’ll write m_i for operations on the group cohomology, and μ_i for operations on the endomorphism dg-algebra. The endomorphism dg-algebra has μ₁=d and μ₂=composition of maps, and all higher operations vanishing, in all our cases.

Elementary abelian 2-group

Let’s start with the easy case. Following to a certain the notation used in Dag Madsen’s PhD thesis appendix (the Canonical Source of the A_∞-structures of cyclic group cohomology algebras), and the recipe given in A-infinity algebras in representation theory, we may start by stating what we know as we start:

This is a preview of A_∞-algebras and group cohomology. Read the full post (1833 words, 70 images)

A-infinity, Algebra, English, Homology and Homotopy, Mathematics, PhD, Research
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Why Blog?

November 16th, 2006

The Community College Dean has written about why he blogs, and asks any and all readers to tack on to his effort.

My blog is not very anonymous. It is occasionally personal, occasionally political and throughout a venting location for thoughts, and a place where I formulate myself in higher detail – so to speak a scratchpad, but public enough for me to allow others to read it.

I write it to formulate my own thoughts further, find possible errors, start discussions, or just jot down the viewpoints that illuminated some point of some argument for me. I do it in public because I thouroughly enjoy the conversations it sparks.

This is a preview of Why Blog?. Read the full post (194 words)

A_∞ for the layman

November 7th, 2006

I recently had reason to describe my PhD work in complete laymans terms while writing letters to my grandparents. This being a good thing to do in order to digest your ideas properly, I thought I might try and write it up here as well.

It will, however, push through some 100-odd years of mathematical development rather swiftly. Try to keep up – I will keep it as light as I can while not losing what I want to say.

Algebra

In the 19th century, a number of different mathematical efforts ended up using more or less precisely the same structures, though not really recognizing that they were the same. This recognition came with Cayley, who first brought the first abstract definition of a group.

A group is a set G of elements, with a binary operation *, such that the following relations hold:

This is a preview of A_∞ for the layman. Read the full post (1250 words, 1 image)

A-infinity, Algebra, English, Homology and Homotopy, Mathematics, PhD, Topology
(1) Comment

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about: Michi is a recent PhD working in homological algebra and applied algebraic topology. This blog is his outlet for texts with some manner of thought put into them. Over at his LiveJournal intimate details and streams of consciousness might be found.
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