On certain submodules of Weyl modules for SO(2n+1,F) with char(F) = 2

For $k = 1, 2,...,n-1$ let $V_k = V(\lambda_k)$ be the Weyl module for the special orthogonal group G = \mathrm{SO}(2n+1,\F) with respect to the $k$-th fundamental dominant weight $\lambda_k$ of the root system of type $B_n$ and put $V_n = V(2\lambda_n)$. It is well known that all of these modules are irreducible when \mathrm{char}(\F) \neq 2 while when \mathrm{char}(\F) = 2 they admit many proper submodules. In this paper, assuming that \mathrm{char}(\F) = 2, we prove that $V_k$ admits a chain of submodules $V_k = M_k \supset M_{k-1}\supset ... \supset M_1\supset M_0 \supset M_{-1} = 0$ where $M_i \cong V_i$ for $1,..., k-1$ and $M_0$ is the trivial 1-dimensional module. We also show that for $i = 1, 2,..., k$ the quotient $M_i/M_{i-2}$ is isomorphic to the so called $i$-th Grassmann module for $G$. Resting on this fact we can give a geometric description of $M_{i-1}/M_{i-2}$ as a submodule of the $i$-th Grassmann module. When \F is perfect G\cong \mathrm{Sp}(2n,\F) and $M_i/M_{i-1}$ is isomorphic to the Weyl module for \mathrm{Sp}(2n,\F) relative to the $i$-th fundamental dominant weight of the root system of type $C_n$. All irreducible sections of the latter modules are known. Thus, when \F is perfect, all irreducible sections of $V_k$ are known as well

The de Rham cohomology of the Suzuki curves

For a natural number $m$, let $\mathcal{S}_m/\mathbb{F}_2$ be the $m$th Suzuki curve. We study the mod $2$ Dieudonn\'{e} module of $\mathcal{S}_m$, which gives the equivalent information as the Ekedahl-Oort type or the structure of the $2$-torsion group scheme of its Jacobian. We accomplish this by studying the de Rham cohomology of $\mathcal{S}_m$. For all $m$, we determine the structure of the de Rham cohomology as a $2$-modular representation of the $m$th Suzuki group and the structure of a submodule of the mod $2$ Dieudonn\'{e} module. For $m=1$ and $2$, we determine the complete structure of the mod $2$ Dieudonn\'{e} module

The Fundamental Crossed Module of the Complement of a Knotted Surface

We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.Comment: A perfected version will appear in Transactions of the American Mathematical Societ