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Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere

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Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere Finite Group Actions on the Four-Dimensional Sphere By Sacha Breton, B.Sc. A Thesis Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Master of Science McMaster University c Copyright by Sacha Breton, August 30th 2011 Master of Science (2011) McMaster University Hamilton, Ontario Department of Mathematics and Statistics TITLE: Finite Group Actions on the Four-Dimensional Sphere AUTHOR: Sacha Breton B.Sc. (McGill University) SUPERVISOR: Ian Hambleton NUMBER OF PAGES: 41 + v ii Abstract Smith theory provides powerful tools for understanding the geometry of singular sets of group actions on spheres. In this thesis, tools from Smith theory and spectral sequences are brought together to study the singular sets of elementary abelian groups acting locally linearly on S4. It is shown that the singular sets of such actions are homeomorphic to the singular sets of linear actions. A short review of the literature on group actions on S4 is included. iii Acknowledgements I would like to thank Professor Ian Hambleton for his suggestion of thesis topic, for the breadth and depth of his advice, and for his patience and encouragement. I would also like to thank Val´erieTremblay, Barnaby and Sue Ore, my parents, and my friends for their continued support. iv Contents Descriptive Note ii Abstract iii Acknowledgements iv 1 Introduction 1 2 Introductory Material 1 2.1 Basic definitions and conventions . .1 2.2 The Projection and the Transfer homomorphisms . .4 2.3 Smith Theory . .7 2.4 Group cohomology . 10 2.4.1 Classifying spaces . 10 2.5 Introduction to spectral sequences . 12 2.5.1 What is a spectral sequence . 12 2.5.2 The Borel construction . 13 3 Literature review 15 4 The shape of singular sets of elementary abelian group ac- tions on S4 23 4.1 Groups of type Z=p × Z=p, for p odd . 23 4.2 G = Z=2 × Z=2 .......................... 24 4.2.1 G acts linearly on S4 ................... 24 4.2.2 G acts locally linearly and preserving orientation on S4 25 4.3 G = Z=2 × Z=2 × Z=2 ...................... 28 4.3.1 G acts linearly on S4 ................... 28 4.3.2 G acts locally linearly and preserving orientation on S4 30 4.4 G = (Z=2)4 ............................ 32 4.4.1 The linear case . 32 4.5 G acts locally linearly and preserving orientation on S4 .... 33 5 Conclusion 36 v Finite Group Actions on the Four-Dimensional Sphere 1 Introduction The study of group actions on manifolds brings together many different tech- niques. Tools from the algebra of the group itself, from the algebraic topology of the space, from the geometry of low dimensional manifolds, from represen- tation theory, from moduli spaces and other domains are brought together to answer questions about which groups act on a space, how those groups can be classified, in which ways a group can act, how to distinguish between group actions, what geometric information is preserved by a group action, and others. Manifolds of dimension four, and in particular the 4-sphere, are well-suited for study, because their geometry and topology are both struc- tured (due to constructions such as the intersection form) and exotic. Here the focus will be on the singular sets of group actions on 4-spheres. Smith theory, a tool for decoding information on the homology or cohomology of the fixed point sets of groups, will be the main theory exploited to extract information about these sets. The body of this paper will be divided into three sections. In the first section, I will address preliminary concerns, such as definitions and founda- tional theorems. In the second, the literature regarding group actions on the 4-sphere will be surveyed. In the third section, I will prove the main result: Theorem 1. The singular sets of locally linear or smooth orientation pre- serving actions of elementary abelian groups on the four-dimensional sphere are homeomorphic to the singular sets of the standard linear actions. 2 Introductory Material 2.1 Basic definitions and conventions Before we begin, it is necessary to state certain definitions and conventions which we will be using throughout. Definition 1. A group G is said to act on an set X if there is a homomor- phism from G to Aut(X), the group of bijections from X to itself. In the case where the set X has added structure, such as if X is a smooth manifold, then a group G acts on X if there is a homomorphism from G to the group of bijective self-maps which preserve that structure, in this case diffeomorphisms. If a group G acts on a space X, then X is called a G-space. 1 Finite Group Actions on the Four-Dimensional Sphere Definition 2. A group action is effective if the homomorphism G ! Aut(X) is injective. In other words, no elements of G act as the identity on X It will not be necessary to consider ineffective group actions. This is because if a group acts ineffectively, it factors through an effective one. Definition 3. A group action is free if no non-identity elements of the group fix any point of the space. Definition 4. The stabilizer (or isotropy group) of a point p 2 X, Gp, is the subgroup of G which fixes p. The stabilizer of a subset Y ⊂ X, GY is the subgroup of G which fixes every point of Y. Gp = fg 2 G j g · p = pg GY = fg 2 G j 8p 2 Y; g · p = pg For every subgroup H of G acting on X, the fixed point set of H will be denoted XH . XH = fx 2 X j 8h 2 H; h · x = xg S H Definition 5. f1g6=H⊂G X is called the singular set of the group action. It is the set of all points which are fixed by some non-identity element of the group. Definition 6. A group action is pseudo-free if its singular set is a union of isolated points. Definition 7. The orbit of a point p 2 X is the set S fg · pg, and will be g2G S denoted G · p. The orbit of a subset Y ⊂ X, G · Y, is the union p2Y G · p. There is a bijection between the orbit of a point and the cosets of the isotropy subgroup of that point: for a point y = g · x 2 G · x, assign the coset gGx. Thus, for G a finite group, jG · pj = jG=Gpj −1 The stabilizers of points in the same orbit are conjugate, with Gp = g Gg·pg. −1 −1 −1 If h 2 Gp,(ghg ) · g · p = g · p, so ghg 2 Gg·p, and h 2 g Gg·pg. If −1 −1 0 0 h 2 g Gg·pg, h · p = g · g · g · p with g 2 Gg·p, so h · p = p, which is to say h 2 Gp. 2 Finite Group Actions on the Four-Dimensional Sphere Definition 8. A map between G-sets φ : X ! Y is G-equivariant if g·φ(x) = φ(g · x) for every g 2 G and every x 2 X. Definition 9. Two spaces are G-equivalent if there exists an equivariant homeomorphism between them. There is a bijective map (in the case of finite groups, actually a homeo- morphism) α: G=Gx ! G · x gGx 7! g · x This map is equivariant, therefore G=Gx and G · x are G-equivalent spaces. We say an orbit which is equivalent to G=H is of type G=H. Definition 10. Let X and Y be sets on which G acts. G acts diagonally on the product X × Y. We define the equivalence relation (x; y) ∼ (g · x; g · y); 8g 2 G The set of equivalence classes of this relation is X ×G Y. Definition 11. Let P ⊂ X be an orbit of type G=H. A tube about P is an equivariant homeomorphism into an open neighborhood of P φ: G ×H A ! X where A is a space on which G acts. Definition 12. For x 2 S ⊂ M, a smooth manifold, such that GxS = S, S is a slice at x if the map G ×Gx S ! M (g; s) 7! g · s is a a tube about G · x. Equivalently, a slice at x is defined as a subspace S of X with the following properties: 1. S is closed in G · S 2. G · S is an open neighborhood of G · x 3 Finite Group Actions on the Four-Dimensional Sphere 3. Gx · S = S 4. If g · S \ S is non-empty, then g is an element of Gx Definition 13. Let P be an orbit of type G=H, and let V be a Euclidean space on which H acts orthogonally. A linear tube about P is a G-equivariant embedding onto a neighborhood of P of the form φ : G ×H V ! M Definition 14. S is a linear slice at x if S is a slice at x and the canonically associated tube is G-equivalent to a linear tube. Definition 15. The action of G on X is locally smooth if there is a linear tube about each orbit. P Definition 16. Given a ring R, the group ring R[G] = f agg j ag 2 Rg. P P g2G For two elements of R[G], r = g2G agg and s = g2G bgg P r + s = g2G(ag + bg)g rs = P P a b g g2G g1g2=g g1 g2 If X is a set on which G acts, Z[X] is a Z[G] module which admits a ∼ decomposition as Z[X] = ×Z[G=Gxi ], where the direct sum is indexed by the orbits of the G action, and each xi is a representative of a distinct orbit.
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