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Group Actions on Products of Spheres

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Group Actions on Products of Spheres Group Actions on Products of Spheres Kenneth Plante 1 Introduction In this paper, we explore free group actions using the techniques of group coho- mology. In particular, we are interested in free actions of finite abelian groups on products of spheres. We start by providing some basic examples which motivate our general course of study. Proposition 1. Let G =(Z/p)r.IfG acts freely on S1, then r =1. Proof. Assume G acts freely on S1.Thenwehaveacoveringp : S1 S1/G.It 1 1 ! follows from covering space theory that G = ⇡1(S /G)/p (⇡1(S )). But since 1 ⇠1 ⇤ G is a finite group, we know that S /G ⇠= S .Thus,wehave 1 1 G = ⇡1(S )/p (⇡1(S )) = Z/ G ⇠ ⇤ ⇠ | | The result follows. More generally, if G acts freely on Sn, then it is known that we must have r = 1. This fact has led to the following conjecture: k Conjecture 1. Let G =(Z/p)r.IfG acts freely on Sni , then r k. i=1 This conjecture remains unproven, but many partialQ results are known. For example, in this paper, we will prove it in the case of an elementary p-group act- ing freely and homologically trivially on a product of equidimensional spheres, i.e. ni = nj for all i, j. We will then explore some additional work on the conjecture in other cases. The primary tool for studying these actions will be group cohomology, specifically Tate cohomology. 2 Algebraic Foundations Let G be a group. We wish to construct the notion of a G-module. In order to do this, we impose a natural ring structure on G: Let ZG be the free Z-module generated by the elements of G, i.e. the abelian group consisting of elements a(g)g where a(g) Z and a(g) = 0 for almost all g, with multiplication g G 2 induced2 by multiplication in G. We call ZG the group ring of G. P The naturality of the construction G ZG is characterized by the following adjunction: 7! 1 Theorem 1. Let U : Rng Grp be the functor which maps each ring (with ! unity) to its group of units. Then the functor Z( ):Grp Rng, G ZG, which maps each group to its group ring, is a left− adjoint of!U. 7! Proof. Let G be a group and R a ring with unity. Let f : G U(R) be a group ! homomorphism. Define f 0 : ZG R by extending f linearly, i.e. ! f 0 cix = cif(x) i ! i X X for ci Z (only finitely many nonzero) and x G. Multiplication in ZG is 2 2 given by the operation in G,sof 0 preserves multiplication by assumption. Thus, f 0 is a ring homomorphism. Thus, we can extend each group homomorphism f : G U(R) to a ring homomorphism f 0 : ZG R. It is clear that we ! ! can also restrict ring homomorphisms f : Z(G) R to group homomorphisms ! f 00 : G U(R), and that these processes are inverses, i.e. f 0 f 00 =id and ! ◦ ZG f 00 f 0 = id . Thus, we have a bijection ◦ G Hom(ZG, R) ⇠= Hom(G, U(R)) Naturality follows from the fact that Z(f(G)) = f 0(Z(G)) and U(f(R)) = f 00(U(R)). This establishes the desired adjunction. We refer to modules over ZG, for any group G, as G-modules. In particular, a (left) G-module is an abelian group M with a (left) G-action on M,i.e.a ring homomorphism ZG End M. For example, we may consider Z a trivial ! Gmodule, namely with the trivial action of ZG on Z. Throughout this paper, we will use G as a subscript to refer to the group ring ZG, e.g. HomG(A, B) will refer to the group of ZG-module homomorphisms from A to B. We will also refer to a complex of G-modules as a G-complex. With these foundations in place, we are ready to define the cohomology of a group. Definition 1. Let G be a group and A a G-module. Let n n n H (G, M)=Ext (Z,M)=R (HomG( ,M)) G − n n Hn(G, M) = Tor (Z,M)=L ( G M) G ⌦ where Z denotes the trivial G-module. We call Hn(G, M) the nth cohomol- ogy of G with coefficients in M,andHn(G, M) the nth homology of G with coefficients in M. Explictly, the construction of the cohomology groups proceeds as follows: Form a projective resolution P of Z over ZG P : P1 P0 Z 0 ···! ! ! ! 2 i.e. an exact sequence in which Pk is projective as a G-module for all k. Now, form the cochain complex Hom (P, M): 0 Hom (P ,M) Hom (P ,M) ... G ! G 0 ! G 1 ! where we dualize P and remove the Z term. The nth homology of this n complex is the ExtG(Z,M), i.e. the nth cohomology of G with coefficients in M. Note that any two projective resolutions of a module are chain homotopic, and chain homotopies induce isomorphisms in homology and cohomology, hence this construction is well-defined. Similarly, we can compute the homology groups by forming the chain com- plex P M : P M P M 0 ⌦G ···! 1 ⌦G ! 0 ⌦G ! and taking its nth homology. However, this construction is still too abstract to use to compute the ho- mology and cohomology groups in specific cases. We provide an example of one computational method called the bar resolution, which gives a concrete proce- dure for constructing projective resolutions over ZG. For each n N,letFn be 2 the free G-module generated by the (n + 1)-tuples (g0,g1,...,gn) of elements of G.DefinealeftG-action on Fn by g ((g ,g ,...,g ) (gg ,gg ,...,gg )) 7! 0 1 n ! 0 1 n and a boundary operator @n : Fn Fn 1 given by ! − n @ (g ,g ,...,g )= ( 1)k(g ,...,gˆ ,...,g ) n 0 1 n − 0 k n kX=0 Finally, define the augmentation ✏ : F0 Z by ✏ = 1. With these definitions, ! @3 @2 @1 ✏ ... F2 F1 F0 Z ! ! ! ! forms a projective resolution (in fact, a free resolution) of Z over ZG.This gives us a useful algebraic procedure for computing the cohomology of a group directly from the algebraic definition given above. 3 Classifying Spaces We now give a topological interpretation of group cohomology, which provides an alternative method for computing the cohomology of a group in terms of cell complexes. First, we introduce the notion of a topological group and fiber bundle. Definition 2. A topological group is a set G which is both a group and a topolog- 1 ical space, such that the group multiplication (a, b) ab and inversion a a− are both continuous functions. 7! 7! 3 Definition 3. Let E and B be topological spaces, B connected, and p : E B 1 ! be a continuous surjection. Choose a point b B and let F = p− (b).Iffor 2 any point b0 B there is a neighborhood Ub0 B of b0 and a homeomorphism 2 1 ⇢ φ : U F p− (U ), such that the following diagram commutes b0 ⇥ ! b0 φ 1 U F p− (U ) b0 ⇥ b0 ⇡ p Ub0 commutes, where ⇡ : U F U is the natural projection, then b0 ⇥ ! b0 p F E B ! ! is called a fiber bundle, with base space B, total space, E, and fiber F . 1 Note that, under these conditions, p− (x) ⇠= F for all x B,soF is inde- pendent of the choice of base point, up to homeomorphisms.2 We now introduce the notion of a principle bundle, which combines the notions of a topological group and a fiber bundle. p Definition 4. Let G be a topological group and G E B be a fiber bundle. Assume the following conditions hold: ! ! 1. G acts freely on the total space E via : E G E in such a way that the following diagram commutes ⇥ ! E G E ⇥ p 1 p ⇥ B 1 id B ⇥ 1 1 2. The action on the fibers p− (b) G p− (b) induced by psi is free and transitive. ⇥ ! 3. We have the same commutative diagram as above, namely φ 1 U G p− (U ) b ⇥ b ⇡ p Ub with the added condition that φ(xg)= (φ(x),g)=φ(x)g,forallg G and x U G. 2 2 b ⇥ p Then we call G E B a principle G-bundle. ! ! 4 1 Note that the added condition on the homeomorpshism φ : Ub G p− (Ub) e↵ectively asserts that φ commutes with group actions. When⇥ this! condition holds of a continuous map, we call it an equivarient map. A particularly interesting and important case of principle bundles in eqivari- ent topology is the classifying space BG of a group G. These spaces are defined p in terms of pullback squares of fiber bundles. If F E B is a fiber bundle and f : X B is any continuous function, we can form! the! pullback square ! f ⇤(E) E p f X B Then the map f ⇤(E) X induces a fiber bundle over X in terms of f and p ! E B. According to the following theorem due to Milnor, principles G-bundles can! be classified up to pullback squares as follows: Theorem 2. Let G be a group. Then there is a space BG and a principle G- bundle G EG BG, such that EG is weakly contractible, and if G E X is any other! principle! G-bundle, then there is a unique map f : X !BG,up! ! to homotopy, such that f ⇤(EG)=E.
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