Inhalt

•  
   Introduction
•  
  Preliminaries on (poset) topology
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   Posets and their realisations
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  Tools from poset topology
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   Fibre theorems
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   Homotopic poset maps and monotonicity
•  
   Alexander duality for posets
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   The nerve of a covering
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   Spherical complexes
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   The Cohen–Macaulay property
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  Coset complexes
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  Definitions and basic properties
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   Coset complex and coset poset
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   Higher generation
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   Group actions and detecting coset complexes
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   Higher generation and Cohen–Macaulay complexes
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   Parabolic subgroups and buildings
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   Levi subgroups and the opposition complex
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   Characterisation of CM coset complexes
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   Coset complexes and short exact sequences
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   Coset posets
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   Coset complexes
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   Summary
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   Homotopy type of the complex of free factors
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  Preliminaries: Free products and their automorphisms
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   Free factors and free splittings
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   Relative automorphism groups
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   Relative Outer space
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   The case A=Fn: Culler–Vogtmann Outer space
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  Complexes of free factors
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   The free factor complex as a coset complex
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   Posets of graphs
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   The poset of core subgraphs
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   The poset of connected core subgraphs
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  Contractibility of relative free splitting complexes
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   Outline of the proof
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   Subgraphs of groups
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   Blow-up construction
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   Proof of lem blow-up construction
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   Contractibility of X(A0, …, Al : B0, …, Bm)
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   Proof of contractibility of free splitting complexes
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   Factor complexes at infinity
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  Higher connectivity of factor complexes
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   Projection to the second factor
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   Projection to the first factor
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   Homotopy type of F
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   Higher connectivity of FF
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  Boundary structures of Outer space
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   Reduced Outer space and Jewel space
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   Three boundary structures
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   Dualising module of Out(Fn)
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  A Cohen–Macaulay complex for Out(RAAGs)
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   Relative automorphism groups of RAAGs
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   RAAGs and their automorphism groups
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   Generators of relative automorphism groups
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   Restriction and projection homomorphisms
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   Restrictions to conical subgroups
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  A spherical complex for Out(A)
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   Rank and maximal parabolic subgroups
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   The parabolic sieve
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   Proof of Theorem A
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  Summary of the inductive procedure and examples
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   Consequences for the induction of Day–Wade
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   Examples
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  Cohen–Macaulayness, higher generation and rank
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   Cohen–Macaulayness
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   Parabolic subgroups of lower rank
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   Interpretation of rank in terms of Coxeter groups
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   Closing comments and open questions
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   Appendix: Graph posets