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Register a new userOn CW-complexes over groups with periodic cohomology
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Authors
John Nicholson
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If $G$ has $4$-periodic cohomology, then D2 complexes over $G$ are determined up to polarised homotopy by their Euler characteristic if and only if $G$ has at most two one-dimensional quaternionic representations. We use this to solve Walls D2 problem for several infinite families of non-abelian groups and, in these cases, also show that any finite Poincar{e} $3$-complex $X$ with $pi_1(X)=G$ admits a cell structure with a single $3$-cell. The proof involves cancellation theorems for $mathbb{Z} G$ modules where $G$ has periodic cohomology.
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We determine the mod $2$ cohomology over the Steenrod algebra of the classifying spaces of the free loop groups $LG$ for compact groups $G=Spin(7)$, $Spin(8)$, $Spin(9)$, and $F_4$. Then, we show that they are isomorphic as algebras over the Steenrod algebra to the mod $2$ cohomology of the corresponding Chevalley groups of type $G(q)$, where $q$ is an odd prime power. In a similar manner, we compute the cohomology of the free loop space over $BDI(4)$ and show that it is isomorphic to that of $BSol(q)$ as algebras over the Steenrod algebra.
We present a detailed description of a fundamental group algorithm based on Formans combinatorial version of Morse theory. We use this algorithm in a classification problem of prime knots up to 14 crossings.
Let $pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${sf cd}_G(pi)$, the equivariant geometric dimension ${sf gd}_G(pi)$, and the equivariant Lusternik-Schnirelmann category ${sf cat}_G(pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $pirtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg-Ganea and Stallings-Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $pi$ with ${sf cat}_G(pi)={sf cd}_G(pi)=2$ and ${sf gd}_G(pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings-Swan type result for families of subgroups which do not contain all finite subgroups.
In this paper we study the cohomology of (strict) Lie 2-groups. We obtain an explicit Bott-Shulman type map in the case of a Lie 2-group corresponding to the crossed module $Ato 1$. The cohomology of the Lie 2-groups corresponding to the universal crossed modules $Gto Aut(G)$ and $Gto Aut^+(G)$ is the abutment of a spectral sequence involving the cohomology of $GL(n,Z)$ and $SL(n,Z)$. When the dimension of the center of $G$ is less than 3, we compute explicitly these cohomology groups. We also compute the cohomology of the Lie 2-group corresponding to a crossed module $Gto H$ whose kernel is compact and cokernel is connected, simply connected and compact and apply the result to the string 2-group.
The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology.
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