Models for Out Fu1 X graph Show to HECX E OuttaX2 X handlebody

Show to Homeo X 37 Out tixA DehntwistonDX extends to a homeomorphism

ofX Showthis is in the kernel ofthemapabove3 X puncturedsurface

A homeomorphism ofX preserves the intersection

farm on HLX so the eigenvalues of themap on HelX ane't or come in pairs 1 I

Use this to find an automorphismofFz thatcannot be realized onanysurface

Stallings folds1 If f X Rn is a graphmorphism that

is locally injective provethat f is a homeomorphismDo not assume X is a rose

2 Use Stallings folds to factor the automorphisma Cba b s a E ba ens ca

of Fz Fla b o into a product of peg's and Xxg's

1 Suppose Au An are spherescuttingMn into a 2n punctured 3 ballandIan can the dual basis of it Mnlets be a sphere in Mn UAi separating A andAi

Ef A O0 0An

A O 0 O OAIs

let f be a diffeomorphism exchanging A and Sandfixingall other Ai What is the automorphism

ofqMn has au inducedby f

2 Use Whitehead'salgorithm to decide whethera ab a abb s b a bi and b i b ca bc 1 cab c b a c

ace automorphismsChoose a random set of words and check whetherit's a basis

Lecture comprehension questions1 why does a Dehn twist on a 2 sphere

in Mn S x 5 have order 2 in toHomeoMn2 Describe an inner automorphism in termsoffolding3 Prove the cut vertex lemma