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Register a new userTetrahedra on deformed spheres and integral group cohomology
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We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.
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We give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.
We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.
Let $G$ be a profinite group, $X$ a discrete $G$-spectrum with trivial action, and $X^{hG}$ the continuous homotopy fixed points. For any $N trianglelefteq_o G$ ($o$ for open), $X = X^N$ is a $G/N$-spectrum with trivial action. We construct a zigzag $text{colim},_N ,X^{hG/N} buildrelPhioverlongrightarrow text{colim},_N ,(X^{hN})^{hG/N} buildrelPsioverlongleftarrow X^{hG}$, where $Psi$ is a weak equivalence. When $Phi$ is a weak equivalence, this zigzag gives an interesting model for $X^{hG}$ (for example, its Spanier-Whitehead dual is $text{holim},_N ,F(X^{hG/N}, S^0)$). We prove that this happens in the following cases: (1) $|G| < infty$; (2) $X$ is bounded above; (3) there exists ${U}$ cofinal in ${N}$, such that for each $U$, $H^s_c(U, pi_ast(X)) = 0$, for $s > 0$. Given (3), for each $U$, there is a weak equivalence $X buildrelsimeqoverlongrightarrow X^{hU}$ and $X^{hG} simeq X^{hG/U}$. For case (3), we give a series of corollaries and examples. As one instance of a family of examples, if $p$ is a prime, $K(n_p,p)$ the $n_p$th Morava $K$-theory $K(n_p)$ at $p$ for some $n_p geq 1$, and $mathbb{Z}_p$ the $p$-adic integers, then for each $m geq 2$, (3) is satisfied when $G leqslant prod_{p leq m} mathbb{Z}_p$ is closed, $X = bigvee_{p > m} (Hmathbb{Q} vee K(n_p,p))$, and ${U} := {N_G mid N_G trianglelefteq_o G}$.
This survey paper describes two geometric representations of the permutation group using the tools of toric topology. These actions are extremely useful for computational problems in Schubert calculus. The (torus) equivariant cohomology of the flag variety is constructed using the combinatorial description of Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation representations on equivariant and ordinary cohomology are identified in terms of irreducible representations of the permutation group. We show how to use the permutation actions to construct divided difference operators and to give formulas for some localizations of certain equivariant classes. This paper includes several new results, in particular a new proof of the Chevalley-Monk formula and a proof that one of the natural permutation representations on the equivariant cohomology of the flag variety is the regular representation. Many examples, exercises, and open questions are provided.
Let an n-algebra mean an algebra over the chain complex of the little n-cubes operad. We give a proof of Kontsevichs conjecture, which states that for a suitable notion of Hochschild cohomology in the category of n-algebras, the Hochschild cohomology complex of an n-algebra is an (n+1)-algebra. This generalizes a conjecture by Deligne for n=1, now proven by several authors.
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