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Register a new userUn 3-polyGEM de cohomologie modulo 2 nilpotente
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Jiang Dong Hua
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In 1983, C. McGibbon and J. Neisendorfer have given a proof for one conjecture in J.-P. Serres famous paper (1953). In 1985, another proof was given by J. Lannes and L. Schwartz. Since then, one considers a more general conjecture: if the reduced mod 2 cohomology of any 1-connected polyGEM is of finite type and is not trivial, then it contains at least one element of infinite height, i.e., non nilpotent. This conjecture has been verified in several special situations, more precisely, by Y. Felix, S. Halperin, J.-M. Lemaire and J.-C. Thomas in 1987, by J. Lannes and L. Schwartz in 1988, and by J. Grodal in 1996. In this note, we construct an example, for which this conjecture fails.
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Let $V$ be an elementary abelian $2$-group and $X$ be a finite $V$-CW-complex. In this memoir we study two cochain complexes of modules over the mod2 Steenrod algebra $mathrm{A}$, equipped with an action of $mathrm{H}^{*}V$, the mod2 cohomology of $V$, both associated with $X$. The first, which we call the topological complex, is defined using the orbit filtration of $X$. The second, which we call the algebraic complex, is defined just in terms of the unstable $mathrm{A}$-module $mathrm{H}^*_V X$, the mod2 equivariant cohomology of $X$. Our study makes intensive use of the theory of unstable $mathrm{H}^{*}V$-$mathrm{A}$-modules which is a by-product of the researches on Sullivan conjecture. There is a noteworthy overlap between the topological part of our memoir and the paper Syzygies in equivariant cohomology in positive characteristic, by Allday, Franz and Puppe, which has just appeared; however our techniques are quite different from theirs (the name Steenrod does not show up in their article).
The principal result of this work is the freeness in the $ overline{mathbb Z}_l$-cohomology of the Lubin-Tate tower. The strategy is of global nature and relies on studying the filtration of stratification of the perverse sheaf of vanishing cycles of some Shimura varieties of Kottwitz-Harris-Taylor types, whose graduates can be explicited as some intermediate extension of some local system constructed in the book of Harris andTaylor. The crucial point relies on the study of the difference between such extension for the two classical $t$-structures $p$ and $p+$. The main ingredients use the theory of derivative for representations of the mirabolic group.
Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. For example, for all integers $ngeq0$ and $alphageq1$, we prove that begin{align*} p_{3,3}left(3^{alpha}n+dfrac{3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+1}} end{align*} and begin{align*} p_{3,3}left(3^{alpha+1}n+dfrac{5times3^{alpha}+1}{2}right) &equiv0pmod{3^{alpha+4}}. end{align*}
Let $A$ be the quotient of a graded polynomial ring $mathbb{Z}[x_1,cdots,x_m]otimesLambda[y_1,cdots,y_n]$ by an ideal generated by monomials with leading coefficients 1. Then we constructed a space~$X_A$ such that $A$ is isomorphic to $H^*(X_A)$ modulo torsion elements.
Consider the ring of holomorphic function germs in $C^n$ and denote by $M$ the maximal ideal of this ring. For any a holomorphic function germ $f$ with an isolated critical point, the finite determinacy theorem (Mather-Tougeron) asserts that there exists some $k$, such that $f+g$ can be brought back to $f$, via a holomorphic change of variables, for any $g in M^k$. In this paper, a generalisation of this theorem for functions defined in a neighbourhood of a Stein compact subset and for an arbitrary ideal is given.
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