Algebraic Topology PDF
Preview Algebraic Topology
Clark Bray Adrian Butscher Simon Rubinstein-Salzedo Algebraic Topology Algebraic Topology Clark Bray Adrian Butscher (cid:129) (cid:129) Simon Rubinstein-Salzedo Algebraic Topology 123 ClarkBray Adrian Butscher Department ofMathematics Autodesk Research Duke University Toronto, ON,Canada Durham, NC,USA Simon Rubinstein-Salzedo EulerCircle PaloAlto, CA,USA ISBN978-3-030-70607-4 ISBN978-3-030-70608-1 (eBook) https://doi.org/10.1007/978-3-030-70608-1 MathematicsSubjectClassification: 55-01 ©SpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword Stanford University Mathematics Camp (SUMaC) was founded in the 1994–95 academic year, when Stanford mathematics professors Rafe Mazzeo and Ralph Cohen successfully secured a four-year grant from the Howard Hughes Medical Institute to fund a mathematics summer program for high school students. Ijoined the founding team to help design the program and teach the course in the first summer. Thestudentswerewonderful,andtheoverallexperiencewas delightfully rewardingforeveryoneinvolved,inspiringmetocontinuewithSUMaCeversince. From the beginning, we recognized great value in showing mathematically curious and talented high school students advanced topics from the undergraduate curriculum. Although many of these students could have developed their talents through mathematics competitions, there were few opportunities for them to explorepuremathematicsinadeepway.Asweputitinourfirstprogrammaterials, our aim was to “excite and inspire students by exposing them to the beauty of mathematics.” We recognized the value of creating a friendly environment for interaction among students with shared interest in mathematics. Along with that goal, we also sought to reach students from communities traditionally under-represented in mathematics or who did not have opportunities for advanced academics generally. Overtheyears,SUMaChasbeensuccessfulatdrawinghighschoolstudentsatthe highest level of mathematical talent, and each year SUMaC creates a community wherethesestudentscanengageinmathematicswithsimilarlytalentedandcurious peers. These students are immersed in a social-academic environment that shapes their educational path and leads to long-lasting friendships. In1995,SUMaChadjustadozenparticipants,primarilyfromtheSanFrancisco BayArea,andallwithinatwo-hourdriveoftheStanfordcampus.Inthisfirstyear, theprogramwas three weeks long, andthecourse wasa streamlined version ofan introductorycourseinabstractalgebraattheundergraduatelevelthatalsoincluded topics from numbertheoryand geometry. Theprogramconsisted of lectures along withproblem-solvingsessionsthatallowedparticipantstoengagemorefullyinthe v vi Foreword course material. Additionally, the program included an opportunity to explore topicsofthestudents’choiceingreaterdepth,andtheygotpracticecommunicating mathematics by giving presentations to their peers. These features continue to be the essential ingredients of the SUMaC program. Buildingonthesuccessofthefirstsummer,SUMaCexpandedto28studentsin the second summer, and then 35 students in the third. Although the demand could easily sustain more growth, we found having 35–42 students was an optimal size for the style of program that we had developed, and it has remained in that range over the years. From the beginning, we secured a single campus residence that we couldmakeourown,furtheringourgoalofestablishingasocialenvironmentwhere theparticipantsandresidentialstaffwouldfeellikefamily.Startingin1997,wehad participants from outside of California, and in the following year students joined from outside the US. Now the program draws an international mix of students, representing a diversity of backgrounds and experiences, who share a common passion for mathematics. From early on, we were interested in opportunities for students to return to SUMaCfor asecond summer. One of thefirst twelve students from 1995returned in 1996 to explore Galois theory and other topics through guided independent study.In1997,fourstudentsfromthepreviousyearreturnedforaspecialone-week program in Real Analysis led by Rafe Mazzeo. In 1997, we launched the first version of “Program 2,” a course designed to run concurrently with the original course, which then became known as “Program 1.” This allowed students the potential to return for a second summer, if they had participated in Program 1 following their sophomore year in high school. While the Program 1 maintains a focus on abstract algebra and number theory, Program2hasvariedintopic.From1998through2000,Program2wasacoursein complex analysis designed and taught by Dr. Marc Sanders, who had received his Ph.D. in Mathematics atStanfordin1994. In2001,Dr. Clark Bray,who had been working for SUMaC while in the Ph.D. program in mathematics at Stanford, designed a course in algebraic topology that became the program 2 course from 2001 through 2004. From 2005 through 2007, Prof. Rafe Mazzeo and his student Dr. Pierre Albin taughttheProgram2course.Theykeptthefocusonalgebraictopologywhilealso includingideasfromgeometrictopology,wheremethodsfromalgebraandcalculus have proved to be effective tools. Starting in 2008, Adrian Butscher took over Program 2 and further developed thecourseworkonalgebraictopology, buildingonthecoursedesignthathadbeen usedpreviously.In2009,mathematicsPh.D.studentSimonRubinstein-Salzedo,an alumnus of SUMaC 2001, joined the instructional team of SUMaC as a TA for Adrian’s algebraic topology course. Adrian continued as the SUMaC Program 2 instructoruntil2013,andSimonremainedoneoftheTAsforthecourseevenafter receiving his Ph.D. from Stanford in 2012. In2014,Simontookoverteaching theSUMaCProgram 2course.Hehadbeen working with Adrian to refine, expand, and improve the course materials, and that collaboration continued for several years. Simon has now been teaching the Foreword vii Program 2 course for six years. Given his engaging teaching style, his passion for mathematics, and his wonderful presentation of the course material, Simon has inspiredhisstudentsandhelpedthemtaketheirmathematicaltalentandcuriosityto a higher level. All have left the course with a deeper understanding and greater appreciationofmathematics,andmany havebecomesuccessfulmathematiciansin their own right. Dr. Rick Sommer Director, Stanford University Mathematics Camp Stanford, California, USA Introduction This book is based on a four-week class that we have taught many times at the Stanford University Mathematics Camp (SUMaC). Students attending this camp have just finished grades 10 and 11 and are selected from among the strongest mathematics students of that age in the world. Still, we do not assume that they have seen typical material that students would be familiar with before taking an algebraictopologyclass,suchasabstractalgebraorpoint-settopology(or,forthat matter, multivariable calculus or linear algebra). Thus we include background on these subjects as needed. Asinanymathematicsbook,theproblemsareveryimportant.Theyareintended tobedoablebutchallenging,andideallyseveralpeoplewillworkontheproblems togetherand share ideas. When comparedwithcompetition problems that students ofthisageareoftenfamiliarwith,thedifficultyinmostoftheproblemsinthisbook lieselsewhere:mostofthemdonotrequireclevertricksinordertosolve,butrather the challenge is in unraveling the definitions and theorems and becoming accus- tomed to a deeper level of abstraction. Thepresentationofmaterialinthisbookdifferstosomeextentfromotherbooks on algebraic topology due to our different audience. While we aim to present the materialrigorouslywhenreasonable,therearetimeswhenwefeelthatthetechnical detailsofthesubjectareoverwhelming,soweskipcertainchallengingstepsinour arguments. This isespecially true inour discussion of homology. We have chosen towork with simplicial orDhomology,sothat we candohands-oncomputations. This is opposed to singular homology, where the proofs are much easier but computations are very difficult. Given our target audience, this feels like the right decision. We also occasionally take short detours to discuss other interesting and tan- gentiallyrelated topicsinmathematics.At leastoneofusfeels thathewouldhave learned many more interesting things as a student, had more authors not been so disciplined about staying on topic! Thus we have been as undisciplined as we feel we can get away with. Each chapter of the book corresponds to one day of class at SUMaC. Each morning, the instructor presents material in the chapter in a 150-minute lecture ix x Introduction (with a break). In the afternoons, students work on the problems for at least 150 minutes, then possibly more in the evenings if they choose to do so. During this time, students also discuss problems from the previous chapter one-on-one with a teaching assistant. It is difficult to learn this amount of material in such a short amount of time. Some students manage to learn nearly all of it, and some students struggle more with certain topics depending on their mathematical background, geometricintuition,andotherfactors.Buteveryonewhoattendsgetsalotoutofit and learns a tremendous amount of new mathematics that they wouldn’t have learned otherwise. We believe that, at a less blistering pace, this book can also be used either for self-study or as a textbook for an introductory undergraduate topology course. For students who aren’t studying this material full-time, learning a chapter or two a week is probably a more reasonable goal. Wehopeyouenjoyreadingthisbookasmuchaswehaveenjoyedwritingitand teachingtheclasses.Bothoftheseactivitieshavebeenexceptionallyrewardingand exciting for us. Wewouldliketothankmanypeoplewhohavereadearlierversionsofthisbook andmadesuggestionsandcorrections.Thesepeopleinclude,butarenotlimitedto, Porter Adams, Neil Makur, Nicholas Scoville, Lynn Sokei, Peterson Tretheway, Enrique Treviño, Nina Zubrilina, the anonymous referees, and all the TAs and students who have been part of the SUMaC community. This book also benefited fromthecontributionsofPierreAlbinandRafeMazzeo,whohavetaughttheclass basedonsomeearlierversionsofthismaterial.WewouldalsoliketothankDahlia Fisch and the Springer production team for making this book a reality. Contents 1 Surface Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Open Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Functions and Their Properties. . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1 The Definition of a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Examples of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Spheres as Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Surfaces with Boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Closed, Bounded, and Compact Surfaces . . . . . . . . . . . . . . . . 24 2.6 Equivalence Relations and Topological Equivalence . . . . . . . . 24 2.7 Homeomorphic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 The Euler Characteristic and Identification Spaces. . . . . . . . . . . . . 31 3.1 Triangulations and the Euler Characteristic. . . . . . . . . . . . . . . 31 3.2 Invariance of the Euler Characteristic. . . . . . . . . . . . . . . . . . . 35 3.3 Identification Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 ID Spaces as Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.5 Abstract Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.6 The Quotient Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.7 Further Examples of ID Spaces . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Triangulations of ID Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.9 The Connected Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 xi
The list of books you might like
