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Register a new userThe ER(2)-cohomology of X^nCP^infty and BU(n)
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We continue the development of the computability of the second real Johnson-Wilson theory. As ER(2) is not complex orientable, this gives some difficulty even with basic spaces. In this paper we compute the second real Johnson-Wilson theory for products of infinite complex projective spaces and for the classifying spaces for the unitary groups.
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The $ER(2)$-cohomology of $Bmathbb{Z}/(2^q)$ and $mathbb{C}P^n$ are computed along with the Atiyah-Hirzebruch spectral sequence for $ER(2)^*(mathbb{C}P^infty)$. This, along with other papers in this series, gives us the $ER(2)$-cohomology of all Eilenberg-MacLane spaces. Since $ER(2)$ is $TMF_0(3)$ after a suitable completion, these computations also take care of that theory.
In this paper we complete a chain of explicit Quillen equivalences between the model category for $Theta_{n+1}$-spaces and the model category of small categories enriched in $Theta_n$-spaces. The Quillen equivalences given here connect Segal category objects in $Theta_n$-spaces, complete Segal objects in $Theta_n$-spaces, and $Theta_{n+1}$-spaces.
We give describe several models for $(infty,n)$-categories, with an emphasis on models given by diagrams of sets and simplicial sets. We look most closely at the cases when $n leq 2$, then summarize methods of generalizing for all $n$.
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.
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).
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