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The Mod P Cohomology of the Projective Unitary Group DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2018 The mod p Cohomology of the Projective Unitary Group ISAK JOHANSSON KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES The mod p Cohomology of the Projective Unitary Group ISAK JOHANSSON Degree Projects in Mathematics (30 ECTS credits) Degree Programme in Mathematics (120 credits) KTH Royal Institute of Technology year 2018 Supervisor at KTH: Tilman Bauer Examiner at KTH: Tilman Bauer TRITA-SCI-GRU 2018:243 MAT-E 2018:46 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract We begin with an introduction to spectral sequences, in particular, we present how a spectral sequence can arise from an exact couple, state and construct the Serre spectral sequence, mention some of the properties of the mod p cohomology and state the dual Eilenberg-Moore spectral sequence. Fiber bundles, together with the concept of pullback bundles, principal bundles, classifying spaces and Chern classes are also discussed to lay a foundation for our results. We compute the mod p cohomology of the projective unitary group. Finally, we compute the mod 3 cohomology of the classifying space of the projective unitary group of order 3. ii Sammanfattning Denna uppsats inleds med en introduktion till spektralsekvenser. Vi visar hur spektrala sekvenser uppkommer fr˚anexakta par. Vidare presenteras Serres spektralsekvens, egenskaper hos mod p kohomologin och den duala verisionen av Eilenberg-Moores spektralsekvens. Fiberknippen, huvudknippen, klassifi- cerande rum och Cherns klasser diskuteras ¨aven och ligger till grund f¨orv˚ara resultat. Vi ber¨aknar mod p kohomologin av den projektiva unit¨ara gruppen. Slutligen ber¨aknar vi mod 3 kohomologin av det klassificerande rummet av den projektiva unit¨ara gruppen av ordning 3. iii Acknowledgements First and foremost, I would like to thank my supervisor, Tilman Bauer, for nice discussions and for always being a source of inspiration. I could not have done it without you. I would also like to thank my family for their continuous support and my partner Annika for always being there. Contents Introduction 1 1 Spectral Sequences 2 1.1 ExactCouples ............................ 4 1.2 TheSerreSpectralSequence. 6 1.2.1 The Construction of the Serre Spectral Sequence . 8 1.2.2 Examples . 9 1.2.3 TheGysinSequence . 11 1.3 The mod p Cohomology . 14 1.4 The Dual Eilenberg-Moore Spectral Sequence . 15 2 Fiber Bundles 18 2.1 ThePullback-Bundle. 20 2.2 ClassifyingSpaces .......................... 21 2.3 ChernClasses............................. 23 3 The mod p Cohomology of the Projective Unitary Group 26 3.1 Preview ................................ 26 3.2 Determining the Total Chern Class . 27 3.3 Determining the Transgressions . 29 3.4 MainResults ............................. 29 3.5 Wheretogonext........................... 31 4 The Classifying Space of the Projective Unitary Group 32 4.1 If p does not divide n ........................ 32 4.2 mod 3 Cohomology of BPU(3) . 33 Bibliography 36 iv Introduction The story about how spectral sequences were invented is quite fascinating. The french born mathematician Jean Leray invented them during his time as a pris- oner in a concentration camp during the second world war in his studies of sheaves. However, it was not until 1951 when Jean-Pierre Serre in his doc- torial thesis[18] applied spectral sequences to algebraic topology that spectral sequences became a popular tool for computations. Today, spectral sequences is one of the most powerful tools in algebraic topology. The spectral sequence that we in this thesis call the Serre spectral sequence sometimes go by the name Leray-Serre spectral sequence to acknowledge the earlier work done by Leray. First, an introduction to spectral sequence is given in chapter 1. In chapter 2, we discuss fiber bundles which we wish to apply spectral sequences to. In chapter 3 and 4 we use the theory from the previous chapters to come up with results on the mod p cohomology of the projective unitary group and the mod 3 cohomology of its classifying space. R will denote a commutative ring with unit. 1 Chapter 1 Spectral Sequences We will here introduce the important concept of a spectral sequence that we will use throughout the thesis. One should pay close attention to the Serre spectral sequence, for which we will use in most of our computations. The main reference to this chapter has been [10], an introduction to spectral sequence can also be found in [4]. In section 1.1, we give a way of constructing a spectral sequence from an exact couple, but for now let us present some definitions without giving any further idea for how a spectral sequences arise. At a first glance, spectral sequences will seem messy and not very intuitive. We will here aim to demonstrate that spectral sequences are not so terrifying and once one has got his mind right about the indices, one can really enjoy working with them. Definition 1. A di↵erential bigraded module is a bigraded module Ep,q with a map d with bidegree ( r, r + 1) or (r, r 1) such that − − − p,q p r,q r+1 d : E E − − r 7! or p,q p+r,q+r 1 d : E E − . r 7! and d d =0, such a map d is called a di↵erential. ◦ Definition 2. A spectral sequence is a family of di↵erential bigraded modules , Er⇤ ⇤ together with maps dr of bidegree (r, r + 1) or ( r, r 1) such that { } , , − − − d d =0and E⇤ ⇤ = H(E⇤ ⇤,d ). r ◦ r r+1 r r We will mainly be interested in the second di↵erential with bidegree (r, 1 r), a spectral sequence with such a di↵erential is called a spectral sequence− 2 CHAPTER 1. SPECTRAL SEQUENCES 3 of cohomology type. A spectral sequence with bidegree ( r.r 1) is called a spectral sequence of homology type. − − The definition on its on may not give a lot of intuition for why spectral sequences is such a powerful tool, but it turns out that they are and in fact also easy to visualize. A spectral sequence can be thought of as a book with pages of planes. p,q 0,2 On page r and coordinate (p, q)wefindtheEr term. On page 3, we find E3 as 5 4 3 2 1 0 0 1 2 3 4 5 0,2 3,0 and a di↵erential from E3 to E3 can be seen as 5 4 3 2 1 0 0 1 2 3 4 5 Notice that the di↵erentials will always go from a white route to a grey route or vice versa. We should also note that the homology is well-defined at every object in our spectral sequences, this is implied by the condition that says that d d = 0. Therefore, it makes sense to define the next page as the homology r ◦ r of the previous page with respect to the map dr. Here it is time for a remark. , , We want to emphasize the fact that knowing Er⇤ ⇤ and dr determines Er⇤+1⇤ but not the di↵erentials dr+1. A natural question would be to ask: - what happens as we go to page infinity? Well, if the spectral sequence converges, that is that for some i and for all j>i, , , Ei⇤ ⇤ = Ej⇤ ⇤, we define the page at infinity to be just the same as page i, i.e , , E⇤ ⇤ = Ei⇤ ⇤. Spectral sequences that does not converge is out of scope of this thesis,1 all spectral sequences in this thesis converges. The Serre spectral sequence is a spectral sequence of algebras, such a spectral sequence is a spectral sequence with the extra structure of a multiplication. Definition 3. A spectral sequence of algebras over R is a spectral sequence, Er together with di↵erentials dr that satifies the Leibniz rule, and multipli- cations{ } φ : E E E such{ that} the φ can be written as the composition r r ⌦ r ! r r+1 H(φ ) φ : E E ⇠= H(E ) H(E ) H(E E ) r H(E ) ⇠= E r+1 r+1 ⌦ r+1 ! r ⌦ r ! r ⌦ r −− − ! r ! r+1 CHAPTER 1. SPECTRAL SEQUENCES 4 and where the middle map is given by [u] [v] [u v]. ⌦ ! ⌦ That d satisfies the Leibniz rule means that for d : Ep,q Er,s p+q=n ⇤ ! r+s=n+1 ⇤ and for x and y in Ep,q, ⇤ L L d(x y)=d(x) y +( 1)p+qx d(y). ⌦ ⌦ − ⌦ We are now ready to move on and see how a spectral sequence can arise from an exact couple. 1.1 Exact Couples Spectral sequences arise naturally in many di↵erent setups. The most common examples are via a filtration or an exact couple. In this section we treat the case of an exact couple and later we will show how we can use this machinery in the construction of Serre’s spectral sequence. Definition 4. An exact couple is a collection D, E, i, j, k , where D and E are R-modules and i, j, k are maps such that we{ have the following} diagram, D / D ` i k j ~ E with im(i)=ker(j), im(j)=ker(k) and im(k)=ker(i). We like to think of an exact couple as a long exact sequence that just goes around and around. From an exact couple it is possible to extend to something we will call a derived couple. Definition 5. A derived couple is obtained from an exact couple D, E, i, j, k 1 { } by defining: D0 = im(i), i0 = i , j0 = j i− , k0(x + j k(E)) = k(x) for x |i(D) ◦ ◦ in E, E0 = H(E,j k) ◦ D0 / D0 ` i0 k0 ~ j0 E0 It can be shown that the derived couple is also an exact couple.
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