Hairy Graphs and the Homology of Out ( F n ) Jim Conant Univ. of - PowerPoint PPT Presentation

Hairy Graphs and the Homology of Out ( F n ) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann July 11, 2012 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 1 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 2 Stable computations Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 2 Stable computations ∼ = H k ( Aut ( F n ); Q ) − → H k ( Aut ( F n +1 ); Q ) if n >> k . 1 (Hatcher-Vogtmann) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 2 Stable computations ∼ = H k ( Aut ( F n ); Q ) − → H k ( Aut ( F n +1 ); Q ) if n >> k . 1 (Hatcher-Vogtmann) ∼ = H k ( Aut ( F n ); Q ) − → H k ( Out ( F n ); Q ) if n >> k . 2 (Hatcher-Wahl-Vogtmann) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 2 Stable computations ∼ = H k ( Aut ( F n ); Q ) − → H k ( Aut ( F n +1 ); Q ) if n >> k . 1 (Hatcher-Vogtmann) ∼ = H k ( Aut ( F n ); Q ) − → H k ( Out ( F n ); Q ) if n >> k . 2 (Hatcher-Wahl-Vogtmann) H k ( Aut ( F n ); Q ) = H k ( Out ( F n ); Q ) = 0 if n >> k . (Galatius) 3 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Virtual cohomological dimension vcd ( Out ( F n )) = 2 n − 3 (Culler-Vogtmann) 1 vcd ( Aut ( F n )) = 2 n − 2 2 2 Stable computations ∼ = H k ( Aut ( F n ); Q ) − → H k ( Aut ( F n +1 ); Q ) if n >> k . 1 (Hatcher-Vogtmann) ∼ = H k ( Aut ( F n ); Q ) − → H k ( Out ( F n ); Q ) if n >> k . 2 (Hatcher-Wahl-Vogtmann) H k ( Aut ( F n ); Q ) = H k ( Out ( F n ); Q ) = 0 if n >> k . (Galatius) 3 cf. Mod ( S )! 4 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 2 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations H k ( Out ( F n ); Q ) = 0 = H k ( Aut ( F n ) for k ≤ 7 except 1 H 4 ( Aut ( F 4 ); Q ) = H 4 ( Out ( F 4 ); Q ) = Q and H 7 ( Aut ( F 5 ); Q ) = Q . (Hatcher-Vogtmann, Gerlits) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations H k ( Out ( F n ); Q ) = 0 = H k ( Aut ( F n ) for k ≤ 7 except 1 H 4 ( Aut ( F 4 ); Q ) = H 4 ( Out ( F 4 ); Q ) = Q and H 7 ( Aut ( F 5 ); Q ) = Q . (Hatcher-Vogtmann, Gerlits) H 8 ( Aut ( F 6 ); Q ) � = 0, H 8 ( Aut ( F 6 ); Q ) = Q , 2 H 12 ( Aut ( F 8 ); Q ) � = 0 � = H 12 ( Out ( F 8 ); Q ), H 11 ( Aut ( F 7 ); Q ) � = 0 (Ohashi, C.-Kassabov-Vogtmann, Gray) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations H k ( Out ( F n ); Q ) = 0 = H k ( Aut ( F n ) for k ≤ 7 except 1 H 4 ( Aut ( F 4 ); Q ) = H 4 ( Out ( F 4 ); Q ) = Q and H 7 ( Aut ( F 5 ); Q ) = Q . (Hatcher-Vogtmann, Gerlits) H 8 ( Aut ( F 6 ); Q ) � = 0, H 8 ( Aut ( F 6 ); Q ) = Q , 2 H 12 ( Aut ( F 8 ); Q ) � = 0 � = H 12 ( Out ( F 8 ); Q ), H 11 ( Aut ( F 7 ); Q ) � = 0 (Ohashi, C.-Kassabov-Vogtmann, Gray) 2 Orbifold Euler characteristic (If G 0 < G is of index n and is of finite cohomological dimension, then χ orb ( G ) := 1 n χ ( BG 0 )) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations H k ( Out ( F n ); Q ) = 0 = H k ( Aut ( F n ) for k ≤ 7 except 1 H 4 ( Aut ( F 4 ); Q ) = H 4 ( Out ( F 4 ); Q ) = Q and H 7 ( Aut ( F 5 ); Q ) = Q . (Hatcher-Vogtmann, Gerlits) H 8 ( Aut ( F 6 ); Q ) � = 0, H 8 ( Aut ( F 6 ); Q ) = Q , 2 H 12 ( Aut ( F 8 ); Q ) � = 0 � = H 12 ( Out ( F 8 ); Q ), H 11 ( Aut ( F 7 ); Q ) � = 0 (Ohashi, C.-Kassabov-Vogtmann, Gray) 2 Orbifold Euler characteristic (If G 0 < G is of index n and is of finite cohomological dimension, then χ orb ( G ) := 1 n χ ( BG 0 )) 1 n 2 3 4 5 6 − 1 − 1 − 161 − 367 − 120257 χ orb ( Out ( F n )) 24 48 5760 5760 580608 Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

Our knowledge of H ∗ ( Out ( F n ); Q ) and H ∗ ( Aut ( F n ); Q ) 1 Unstable computations H k ( Out ( F n ); Q ) = 0 = H k ( Aut ( F n ) for k ≤ 7 except 1 H 4 ( Aut ( F 4 ); Q ) = H 4 ( Out ( F 4 ); Q ) = Q and H 7 ( Aut ( F 5 ); Q ) = Q . (Hatcher-Vogtmann, Gerlits) H 8 ( Aut ( F 6 ); Q ) � = 0, H 8 ( Aut ( F 6 ); Q ) = Q , 2 H 12 ( Aut ( F 8 ); Q ) � = 0 � = H 12 ( Out ( F 8 ); Q ), H 11 ( Aut ( F 7 ); Q ) � = 0 (Ohashi, C.-Kassabov-Vogtmann, Gray) 2 Orbifold Euler characteristic (If G 0 < G is of index n and is of finite cohomological dimension, then χ orb ( G ) := 1 n χ ( BG 0 )) 1 n 2 3 4 5 6 − 1 − 1 − 161 − 367 − 120257 χ orb ( Out ( F n )) 24 48 5760 5760 580608 A generating function for these orbifold Euler characteristics is known. 2 (Kontsevich, Smillie-Vogtmann) Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 3 / 29

H k ( Aut ( F n ); Q ) k \ n 2 3 4 5 6 7 8 9 10 11 2 0 0 | | | | | | vcd stable 3 0 0 0 0 | | | | stable 4 µ 1 0 0 0 | | | vcd stable 5 0 0 0 0 0 0 | stable 6 0 0 0 0 0 0 vcd stable 7 ǫ 1 0 0 0 0 0 0 8 µ 2 ? ? ? ? ? vcd 9 ? ? ? ? ? ? 10 ? ? ? ? ? vcd 11 ǫ 2 ? ? ? ? 12 µ 3 ? ? ? vcd Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 4 / 29

H k ( Out ( F n ); Q ) k \ n 2 3 4 5 6 7 8 9 10 11 2 0 0 0 0 0 0 | | stable 3 0 0 0 0 0 0 0 vcd stable 4 µ 1 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 vcd 6 0 0 0 0 0 0 0 7 0 0 0 0 0 0 vcd 8 µ 2 ? ? ? ? ? 9 ? ? ? ? ? vcd 10 ? ? ? ? ? 11 ? ? ? ? vcd 12 µ 3 ? ? ? Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 5 / 29

1 µ k ∈ H 4 k ( Out ( F 2 k +2 ); Q ) (Morita). It is unknown if µ k � = 0 unless k = 1 , 2 , 3. Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 6 / 29

1 µ k ∈ H 4 k ( Out ( F 2 k +2 ); Q ) (Morita). It is unknown if µ k � = 0 unless k = 1 , 2 , 3. 2 ǫ k ∈ H 4 k +3 ( Aut ( F 2 k +3 ); Q ) (CKV). It is unknown if ǫ k � = 0 unless k = 1 , 2. We could call these Eisenstein classes. Jim Conant Univ. of Tennessee joint w/ Martin Kassabov and Karen Vogtmann Hairy Graphs and the Homology of Out ( F n ) () July 11, 2012 6 / 29