# A∞-algebras and group cohomology

In A-infinity, Algebra, English, Homology and Homotopy, Mathematics, PhD, Research.

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

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
μ_{1}=d and μ_{2}=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:

_{i,i}=a and D

_{i+1,i}=b.

Recall that [tex]H^*(G,k)=H^*(\Hom_{kG}(P,P))[/tex]. Thus [tex]\Hom_\Lambda(P,P)[/tex] is a dg-algebra, whose homology is precisely the group cohomology.

Now, by the minimality theorem (proven by Kadeishvili first, and
reproven by a veritable host of mathematicians), there is a
quasi-isomorphism of A_{∞}-algebras [tex]H^*A \to A[/tex] that
lifts the identity in homology. This we can use to figure out the
A_{∞}-structure for our cohomology ring: we know that the
[tex]\Hom_\Lambda(P,P)[/tex] is an honest-to-glod dg-algebra, and
thus has an A_{∞}-structure with all higher multiplications (by
which I mean 3-ary and higher) vanishing. We can also pick
representatives for our cohomology ring elements as representatives of
homotopy classes of chain maps [tex]P_{.}\to P_{.}[/tex]. This gives
us a quasi-isomorphism of dg-algebras [tex]H^*A\to A[/tex], lifting
the identity, and which we can augment to an
A_{∞}-quasi-isomorphism.

Which is what we'll want to do now.

We'll (at this stage) use the fact that we know what [tex]H^*(C_2\times C_2,\mathbb F_2)[/tex] looks like: it's the algebra [tex]\mathbb F_2[x,y][/tex]. There are two 1-coclasses, both represented by a morphism [tex]\Lambda^2\to\mathbb F_2[/tex], namely one composing the projection onto the first factor with the augmentation map, and one composing the projection onto the second factor. We'll name the first of these x, and the second y, and note that they lift to chain maps [tex]P_{.}\to P_{.}[/tex] that shave off the first and last summand of [tex]\Lambda^i[/tex] respectively in each degree.

_{∞}-maps states that

_{1}=0 and so this reduces to

_{2}is a map such that its differential is equal to the "commutator" of f

_{1}and multiplication.

Now, pick some coclasses u,v. These will map to shaving maps as
described above, and their product will map to a shaving map that does
just the same as the composition of the individual shaving maps; so if u
is x^{i}y^{j}, and v is x^{k}y^{l}, then
f_{1}(u) shaves off i components in the front and j components
in the back, and f_{1}(v) shaves off k in the front and l in the
back. So, the composition of these two maps is the map that drops i+k
components from the front, and j+l components from the back. On the
other hand f_{1}(uv)=f:sub:1(x:sup:i+ky^{j+l}) is
the map that drops i+k components from the front, and j+l components
from the back.

So they are the same. And thus we don't need to bother with any
homotopies, or any higher order operations or higher order maps. We set
f^{2} to be the zero map, and consider ourself finished and happy.
The cohomology is a dg-algebra in its own right, and this is all there
is to it in A_{∞}-terms. And we're done.

This result implies, by the way, via a proposition from Keller, that the elementary abelian 2-groups have Koszul group algebras. An argument using restrictions of non-nilpotent coclasses to cyclic subgroups will tell you that these are the only finite groups that have Koszul group algebras.

With this example.

## Cyclic 4-group

This is the one canonical example known beforehand in group cohomology. It was calculated by Dag Madsen in his PhD-thesis, and cited ever since. I will perform the same calculation, but in a blinding detail you won't find in a thesis or a paper on the subject.

_{i}are all known, and m

_{1}=0, and m

_{2}is the multiplication in Γ.

Once we've choosen representatives for the coclasses in
[tex]\Hom_\Lambda(P,P)[/tex], we can lift this choice to an
A_{∞}-quasi-isomorphism. In this process, we'll find and define
the relevant higher multiplications for Γ, thus finding the
A_{∞}-structure for that algebra.

_{1}sends

_{1}(η)=(1 1 1 1)[2]

_{1}(ξ)=(1 x

^{2}1 x

^{2})[1]

_{1}(ξ)f:sub:1(ξ) = (x:sup:2 x

^{2}x

^{2}x

^{2})[2]

Composing the lowest degree component of the map (x:sup:2 x^{2}
x^{2} x^{2})[2] with the augmentation map, we see that in
cohomology, it corresponds to the 0 element. So it is actually homotopic
to the image of [tex]\xi^2[/tex] under f_{1}, and this particular
homotopy is what we'll want [tex]f_2(\xi,\xi)[/tex] to be.

^{2}x

^{2}x

^{2})[2]

^{3}h(o)

^{3}e) = x

^{3}h(e)

^{3}all involve h applied on elements of odd degree, and so we'll set those to vanish, and fill in the needed values by letting h(e)=x. Thus we get the chain map

Thus [tex]f_2(\xi,\xi)=(x\; 0\; x\; 0)[1][/tex], and f_{2}
on odd and odd elements are all translates of this, and all other
parameters to f_{2} give us a zero map. This brings us to a point
where we can start investigating m_{3}.

We can extract f_{1}m_{3} and m_{1}f_{3}
from the 3rd A_{∞}-morphism axiom, and put the rest into a map of
its own. This will end up to be something supposed to be homotopic to
the image of m_{3}, and so we can define m_{3} and the
homotopy once we have them.

If only one element is of odd degree, then every f_{2} occurring
will have at least one even argument, and so will vanish.

_{2}for higher odd coclasses into account: these are just translates of the behaviour defined in f

_{2}(ξ,ξ), and so should vanish, since we defined f

_{2}on pairs of odd classes to just be translates of the value on ξ and ξ.

Remains the case with all three elements of odd degree. Again, higher odd elements behave by translating the behaviour of ξ, and so it is enough to study the behaviour on ξ, ξ, ξ.

^{3}x

^{3}x

^{3})[2]

^{3}vanishes), so we put m

_{3}=0, to correspond to what this is homotopic to. And f

_{3}(ξ,ξ,ξ) needs to be mapped precisely to this homotopy.

^{3}x

^{3}x

^{3})[2] is a map h with

^{3}x

^{3}x

^{3})[2]

^{3}h(o)

^{3}e) = x

^{3}h(e)

^{3}to occur, we can pick h to be 0 on even degree components and the identity on odd degree components, giving us

_{3}(ξ,ξ,ξ).

_{1}f

_{4}and f

_{1}m

_{4}out of the mix, we'll get

For only one odd argument, all the f_{2} and f_{3} will
vanish.

For two odd arguments, the same thing will happen.

_{1}f

_{3}and

_{3}f

_{1}and

_{2}f

_{2}

_{2}f

_{2}vanishes, and that the complete expression for [tex]\Phi_4(\xi,\xi,\xi,\xi)[/tex] is the one we'd write down as

And this concludes the calculation of the A_{∞}-structure on
[tex]H^*(C_4,\mathbb F_2)[/tex], and also gives a rather clear hint
as to how to do it for [tex]H^*(C_{2^i},\mathbb F_2)[/tex] in
general.