A spectral sequence for Bredon cohomology. (English) Zbl 1282.55012
Let \(G\) be a finite group and \(H\) be a subgroup of \(G\). Denote by \(\text{len}_GH\) the maximal number \(k\) such that there is a sequence of proper inclusions of \(H=H_0 < H_1 < \cdots < H_k=G\). Let \(\mathcal{O}^{\text{op}}_G\) be the opposite orbit category of \(G\). Then this has a filtration \(\mathcal{F}_0 \subset \mathcal{F}_1 \subset \cdots \subset \mathcal{F}_N=\mathcal{O}^{\text{op}}_G\) consisting of the full subcategories \(\mathcal{F}_k\) of \(\mathcal{O}^{\text{op}}_G\) whose objects are those \(G/H\) of \(\mathcal{O}^{\text{op}}_G\) such that \(\text{len}_GH \leq k\) where \(N=\text{len}_G\{e\}\).
For a \(G\)-\(CW\) complex \({\mathbb X}\) we define an \(\mathcal{O}^{op}_G\)-diagram \(\underline{X}\) in \(\mathcal{T}\) as a functor \(\mathcal{O}^{op}_G \to \mathcal{T}\) defined by sending \(G/H\) to \({\mathbb X}^H\) where \(\mathcal{T}\) denotes the category of topological spaces. Also by \(\underline{Y}=\underline{K}(\underline{M}, n)\) we denote the \(\mathcal{O}^{op}_G\)-diagram of Eilenberg-Mac Lane spaces in \(\mathcal{T}\) corresponding to \(\underline{M}\) an \(\mathcal{O}^{op}_G\)-diagram in abelian groups. For these two diagrams \(\underline{X}\), \(\underline{Y}\) we obtain a tower of simplicial sets \[ \cdots \to \text{Map}(\underline{X}|_{\mathcal{F}_k}, \underline{Y}|_{\mathcal{F}_k})\overset{p_k}{\to} {\text{Map}}(\underline{X}|_{\mathcal{F}_{k-1}}, \underline{Y}|_{\mathcal{F}_{k-1}}{)} \cdots \] which is induced by the inclusions \(\cdots \subset \mathcal{F}_{k-1} \subset \mathcal{F}_k \subset \cdots\). The authors prove that this tower becomes a tower of fibrations, and hence we have a spectral sequence such that \[ E^{k+t, t}_1\cong \pi_t (F_k(\underline{X}, \underline{Y})) \Rightarrow \pi_t(\text{Map}(\underline{X}, \underline{Y})) \cong {\mathbf H}^{n-t}_G({\mathbb X}; \underline{M}) \] where \(F_k(\underline{X}, \underline{Y})\) denotes the fiber of \(p_k\). The authors also show that this \(E_1\)-term can be written as a direct sum of the reduced local cohomology groups \(\tilde{H}^{n-t}_{W_L}({\mathbf E}W_L\times_{W_L}{\mathbb X}^L_L; M_L)\) where \({\mathbb X}^L_L={\mathbb X}^L/\bigcup_{K>L}{\mathbb X}^K\). This spectral sequence is the main result of this paper. But additionally, the authors construct one more spectral sequence in a similar way, which allows us to compute \({\mathbf H}^{*}_{W_L}({\mathbb X}^L; \hat{M}_L)\) similarly from the reduced local cohomology groups.
For a \(G\)-\(CW\) complex \({\mathbb X}\) we define an \(\mathcal{O}^{op}_G\)-diagram \(\underline{X}\) in \(\mathcal{T}\) as a functor \(\mathcal{O}^{op}_G \to \mathcal{T}\) defined by sending \(G/H\) to \({\mathbb X}^H\) where \(\mathcal{T}\) denotes the category of topological spaces. Also by \(\underline{Y}=\underline{K}(\underline{M}, n)\) we denote the \(\mathcal{O}^{op}_G\)-diagram of Eilenberg-Mac Lane spaces in \(\mathcal{T}\) corresponding to \(\underline{M}\) an \(\mathcal{O}^{op}_G\)-diagram in abelian groups. For these two diagrams \(\underline{X}\), \(\underline{Y}\) we obtain a tower of simplicial sets \[ \cdots \to \text{Map}(\underline{X}|_{\mathcal{F}_k}, \underline{Y}|_{\mathcal{F}_k})\overset{p_k}{\to} {\text{Map}}(\underline{X}|_{\mathcal{F}_{k-1}}, \underline{Y}|_{\mathcal{F}_{k-1}}{)} \cdots \] which is induced by the inclusions \(\cdots \subset \mathcal{F}_{k-1} \subset \mathcal{F}_k \subset \cdots\). The authors prove that this tower becomes a tower of fibrations, and hence we have a spectral sequence such that \[ E^{k+t, t}_1\cong \pi_t (F_k(\underline{X}, \underline{Y})) \Rightarrow \pi_t(\text{Map}(\underline{X}, \underline{Y})) \cong {\mathbf H}^{n-t}_G({\mathbb X}; \underline{M}) \] where \(F_k(\underline{X}, \underline{Y})\) denotes the fiber of \(p_k\). The authors also show that this \(E_1\)-term can be written as a direct sum of the reduced local cohomology groups \(\tilde{H}^{n-t}_{W_L}({\mathbf E}W_L\times_{W_L}{\mathbb X}^L_L; M_L)\) where \({\mathbb X}^L_L={\mathbb X}^L/\bigcup_{K>L}{\mathbb X}^K\). This spectral sequence is the main result of this paper. But additionally, the authors construct one more spectral sequence in a similar way, which allows us to compute \({\mathbf H}^{*}_{W_L}({\mathbb X}^L; \hat{M}_L)\) similarly from the reduced local cohomology groups.
Reviewer: Haruo Minami (Nara)
MSC:
| 55N91 | Equivariant homology and cohomology in algebraic topology |
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 55N25 | Homology with local coefficients, equivariant cohomology |
| 55T99 | Spectral sequences in algebraic topology |
Keywords:
Bredon cohomology; cohomology with local coefficients; equivariant homotopy theory; spectral sequenceReferences:
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