The action on homology of finite groups of automorphisms of surfaces and graphs
The action on homology of finite groups of automorphisms of surfaces and graphs
复制标题
曲面和图的有限自同构群的同调作用
DOI：
--
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Andrew Putman
中科院分区：
文献类型：
--
作者：
Andrew Putman
我们证明，除了平凡的情况下，有限阶同胚的表面和图形必须采取行动非平凡的同源性。对于曲面，这个经典定理通常是从莱夫谢茨不动点定理推导出来的，而对于图，它通常是通过组合操作推导出来的。我们的证明是不同的，是在相同的精神作为原来的证明（由于赫尔维茨）这个定理的表面。本文证明了：若S是亏格至少为2的紧致定向曲面，f：S → S是周期同胚且f ∈ H = id，则导出映射f ∈ H：H1（S;Z）→ H1（S;Z）必非平凡.这是著名的Hurwitz [H]定理。出现在大多数现代教科书中的证明（由于塞尔）从莱夫谢茨不动点定理推导出来;见，例如，[FM].我们的目标是给一个证明，是更多的精神，赫尔维茨的原始证明。虽然时间长了一点，但我觉得这个证明很有启发性。此外，与使用Lefschetz不动点定理的证明不同，它可以很容易地适用于证明图的类似结果（我不知道该将这个类似结果归因于谁，尽管它是Baumslag-Taylor [BT]工作的简单结果）。关键是关于有限群作用的轨道空间的同调群的一个命题，我们在§1中证明了这个命题。然后，我们在§2中证明了曲面的主要定理，在§3中证明了图的主要定理。1商空间的同调如果G是群，M是域F上的G-表示（即G线性作用于其上的F-向量空间;这与F[G]-模相同），则M的不变量为M：= {m ∈ M| g（m）= m，对于所有g ∈ G}，M的共不变量为MG：= M/I，其中I = g（m）− m| g ∈ G，m ∈ M ∈ G.它们通过对偶性联系在一起：令M = HomF（M，F），我们有一个自然同构（M）G =（MG）。本节的目的是证明下面的简单命题。我不知道该把它归于谁，但它可以在例如[Br，定理III.2.4]中找到。
We prove that aside from trivial cases, finite-order homeomorphisms of surfaces and graphs must act nontrivially on homology. For surfaces, this classical theorem is usually deduced from the Lefschetz fixed point theorem, while for graphs it is usually deduced via combinatorial manipulations. Our proof is different and is in the same spirit as the original proof (due to Hurwitz) of this theorem for surfaces. In this note, we prove that if S is a compact oriented surface whose genus is at least 2 and f : S → S is a periodic homeomorphism with f ̸= id, then the induced map f∗ : H1(S;Z) → H1(S;Z) must be nontrivial. This is a well-known theorem of Hurwitz [H]. The proof that appears in most modern textbooks (due to Serre) deduces it from the Lefschetz fixed point theorem; see, e.g., [FM]. Our goal is to give a proof that is more in the spirit of Hurwitz’s original proof. While it is a little longer, I feel that this proof is quite instructive. Moreover, unlike the proof using the Lefschetz fixed point theorem, it can easily be adapted to prove the analogous result for graphs (I do not know who to attribute this analogous result to, though it is an easy consequence of work of Baumslag–Taylor [BT]). The key is a proposition concerning the homology groups of orbit spaces of finite group actions which we prove in §1. We then prove our main theorem for surfaces in §2 and for graphs in §3. 1 The homology of quotient spaces If G is a group and M is a G-representation over a field F (that is, an F-vector space on which G acts linearly; this is the same as a F[G]-module), then the invariants of M are M := {m ∈ M | g(m) = m for all g ∈ G} and the coinvariants of M are MG := M/I with I = ⟨g(m) − m | g ∈ G, m ∈ M⟩. These are related by duality: letting M∗ = HomF(M,F), we have a natural isomorphism (M∗)G ∼= (MG). The goal of this section is to prove the following simple proposition. I do not know who to attribute it to, but it can be found in e.g. [Br, Theorem III.2.4].