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1. CJM 2007 (vol 59 pp. 845)
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions |
Representations of the Fundamental Group of an $L$-Punctured Sphere Generated by Products of Lagrangian Involutions In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixed-point set of an
anti-symplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixed-point set is
non-empty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasi-Hamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasi-Hamiltonian $U$-space
$(M, \w, \mu\from M \to U)$ for it to induce an anti-symplectic involution on
the reduced space $M/\!/U := \mu^{-1}(\{1\})/U$.
Keywords:momentum maps, moduli spaces, Lagrangian submanifolds, anti-symplectic involutions, quasi-Hamiltonian Categories:53D20, 53D30 |
2. CJM 2003 (vol 55 pp. 42)
$*$-Subvarieties of the Variety Generated by $\bigl( M_2(\mathbb{K}),t \bigr)$ Let $\mathbb{K}$ be a field of characteristic zero, and $*=t$ the
transpose involution for the matrix algebra $M_2 (\mathbb{K})$. Let
$\mathfrak{U}$ be a proper subvariety of the variety of algebras with
involution generated by $\bigl( M_2 (\mathbb{K}),* \bigr)$. We define
two sequences of algebras with involution $\mathcal{R}_p$,
$\mathcal{S}_q$, where $p,q \in \mathbb{N}$. Then we show that
$T_* (\mathfrak{U})$ and $T_* (\mathcal{R}_p \oplus \mathcal{S}_q)$
are $*$-asymptotically equivalent for suitable $p,q$.
Keywords:algebras with involution, asymptotic equivalence Categories:16R10, 16W10, 16R50 |
3. CJM 2001 (vol 53 pp. 212)
Group Actions and Codes A $\mathbb{Z}_2$-action with ``maximal number of isolated fixed
points'' ({\it i.e.}, with only isolated fixed points such that
$\dim_k (\oplus_i H^i(M;k)) =|M^{\mathbb{Z}_2}|, k = \mathbb{F}_2)$
on a $3$-dimensional, closed manifold determines a binary self-dual
code of length $=|M^{\mathbb{Z}_2}|$. In turn this code determines
the cohomology algebra $H^*(M;k)$ and the equivariant cohomology
$H^*_{\mathbb{Z}_2}(M;k)$. Hence, from results on binary self-dual
codes one gets information about the cohomology type of $3$-manifolds
which admit involutions with maximal number of isolated fixed points.
In particular, ``most'' cohomology types of closed $3$-manifolds do
not admit such involutions. Generalizations of the above result are
possible in several directions, {\it e.g.}, one gets that ``most''
cohomology types (over $\mathbb{F}_2)$ of closed $3$-manifolds do
not admit a non-trivial involution.
Keywords:Involutions, $3$-manifolds, codes Categories:55M35, 57M60, 94B05, 05E20 |