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Register a new userThe Omega spectrum for mod 2 KO-theory
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W Stephen Wilson
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The 8-periodic theory that comes from the KO-theory of the mod 2 Moore space is the same as the real first Morava K-theory obtained from the homotopy fixed points of the Z/(2) action on the first Morava K-theory. The first Morava K-theory, K(1), is just mod 2 KU-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum. There are a lot of maps into and out of these spaces and the spaces for KO- theory, KU-theory and the first Morava K-theory. For every one of these 98 maps (counting suspensions) there is a spectral sequence. We describe all 98 maps and spectral sequences. 48 of these maps involve our new spaces and 56 of the spectral sequences do. In addition, the maps on homotopy are all written down.
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Let $Gamma$ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic $Gamma$-CW complex on which $Gamma$ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel constructon of Xs with respect to the action of $Gamma$. This cohomology coincides with the mod-2 cohomology of $Gamma$ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the strucure of the cohomology ring H * ($Gamma$; Z/2).
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 study the mod-p cohomology of the group Out(F_n) of outer automorphisms of the free group F_n in the case n=2(p-1) which is the smallest n for which the p-rank of this group is 2. For p=3 we give a complete computation, at least above the virtual cohomological dimension of Out(F_4) (which is 5). More precisley, we calculate the equivariant cohomology of the p-singular part of outer space for p=3. For a general prime p>3 we give a recursive description in terms of the mod-p cohomology of Aut(F_k) for k less or equal to p-1. In this case we use the Out(F_{2(p-1)})-equivariant cohomology of the poset of elementary abelian p-subgroups of Out(F_n).
The equivariant cohomology of the classical configuration space $F(mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnold (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojevic, Luck & Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $ell$-skew embeddings. Using our new proof for Hungs main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.
In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrods student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrods original cochain definition of the Square operations.
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