\titleofpresentation{The Extension Class of a Subset Complex}

\presenter{Asl\i \ G\"{u}\c cl\"{u}kan}  %  Name just one speaker, please; collaborators
            %  can be credited in the body of the abstract.

%  [Body of abstract goes here.]
This is a joint work with E.~Yal\c c\i n. Let $G$ be a finite
group and $X$ be a finite $G$-set. The subset complex $\Delta(X)$
of $X$ is defined as the simplicial complex whose simplices are
subsets of $X$. The oriented complex of $\Delta(X)$ in mod 2
coefficients gives the following $\mathbb{F}_2G$-module extension
$$ 0 \rightarrow
\mathbb{F}_2 \rightarrow C_{n-1}(\Delta(X))\rightarrow \cdots
\rightarrow C_{0}(\Delta(X)) \rightarrow \mathbb{F}_2 \rightarrow
0.
$$
Let $\bar{\zeta}_{X} \in \mathrm{Ext}_{\mathbb{F}_2
G}^{n}(\mathbb{F}_2,\mathbb{F}_2)$ denote the extension class for
this extension. In the case $X= G/1$, the class $\zeta_{G/1}$ is
an essential class which makes it interesting for group
cohomology. This class is a mod 2 reduction of a class $\zeta_X$
which is introduced by Reiner and Webb \cite{Asli1}. They raised
the question that for which finite groups $\zeta_{G/1} \neq 0$? We
answer their question under mod 2 reduction:
\begin{theorem}Let $G$ be a finite group. Then $\bar{\zeta}_{G/1}$ is non-zero
if and only if $G$ is an abelian $2$-group isomorphic to
$(\mathbb{Z}/2)^n \times (\mathbb{Z}/4)^m$ for some $n\geq 0$ and
$m=0,1$.
\end{theorem}

\begin{ourbiblio}{99}  %  Delete this environment if you
               %  aren't giving a bibliography.

\bibitem{Asli1}
V. ~Reiner and P. ~Webb, {\it The combinatorics of the bar
resolution in group cohomology,} J. Pure Appl. Algebra 190 (2004),
no. 1-3, 291-327.
\end{ourbiblio}

\presenteraddress{Bilkent University}  % The name of a university is fine.

\email{guclukan@fen.bilkent.edu.tr}

