By Lizhen Ji; Scott A Wolpert; Shing-Tung Yau
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Additional info for Geometry of Riemann surfaces and their moduli spaces
Nding axioms) that would imply these stability theorems has lead to the discovery of new such theorems (see for example, , ). As for other general features, notice that some of the above stability theorems that addressed Stability Question 2 involved some variation of the Pontrjagin-Thom construction. This was true of the proofs of Theorems 8, 13, 29, 32 described above. It would certainly be of great value to understand under what conditions the Pontrjagin-Thom construction yields a (homology) equivalence, and therefore a stability theorem.
The paper gives a coherent way of studying the cobordisms that deﬁned the equivalence relation in Thom’s theory. This work has inspired considerable work by many people in algebraic and diﬀerential topology over the last few years. Unfortunately, the description of much of this new work is beyond the scope of this paper. 2. Automorphisms of free groups. One last stability phenomenon that we will discuss concerns automorphisms of the free group on ngenerators, Aut(Fn ), and the outer automorphism groups, Out(Fn ), deﬁned to be the quotient Out(Fn ) = Aut(Fn )/Inn(Fn ), where Inn(Fn ) < Aut(Fn ) is the subgroup of inner automorphisms.
We will describe one of their main results, and interpret it as a stability theorem for these moduli spaces. Let Σ be a closed Riemann surface of genus g, and let E → Σ be a principal G-bundle, where G is a compact Lie group. To make the statements of the following theorems easier, we will assume that G is semisimple. Let g be the Lie algebra, and Let ad(E) = E ×G g → Σ the corresponding “adjoint bundle”, where G acts on g by conjugation. Let A(E) be the space of connections on E, and let AF (E) be the subspace consisting of ﬂat connections.
Geometry of Riemann surfaces and their moduli spaces by Lizhen Ji; Scott A Wolpert; Shing-Tung Yau