GEOMETRY AND GROUP THEORY
1 MANIFOLDS 1 2 LIE GROUPS AND ALGEBRAS 71 CONTENTS 1 MANIFOLDS ONE OF THE MOST FUNDAMENTAL CONSTRUCTS IN GEOMETRY IS THE NOTION OF A MANIFOLD A MANIFOLD IS IN COLLOQUIAL LANGUAGE THE ARENA WHERE THINGS HAPPEN FAMILIAR EXAMPLES ARE THE THREE DIMENSIONAL SPACE THAT WE INHABIT AND EXPERIENCE IN EVERYDAY LIFE THE SURFACE OF BALL VIEWED AS A TWO DIMENSIONAL CLOSED SURFACE ON WHICH FOR EXAMPLE AN ANT MAY WALK AND THE FOUR DIMENSIONAL MINKOWSKI SPACETIME THAT IS THE ARENA WHERE SPECIAL RELATIVITY MAY BE FORMULATED IN ORDER TO GIVE A REASONABLY PRECISE DESCRIPTION OF A MANIFOLD IT IS HELPFUL RST TO GIVE A FEW RATHER FORMAL DE NITIONS IT IS NOT THE INTENTION IN THIS COURSE TO MAKE EVERYTHING TOO FORMAL AND RIGOROUS SO WE SHALL TRY TO STRIKE A BALANCE BETWEEN FORMALITY AND PRACTICAL UTILITY AS WE PROCEED IN PARTICULAR IF THINGS SEEM TO BE GETTING TOO ABSTRACT AND RIGOUROUS AT ANY STAGE THERE IS NO NEED TO PANIC BECAUSE IT WILL PROBABLY JUST BE A BRIEF INTERLUDE BEFORE RETURNING TO A MORE INTUITIVE AND INFORMAL DISCUSSION IN THIS SPIRIT LET US BEGIN WITH SOME FORMAL DE NITIONS 1 1 SOME SET THEORETIC CONCEPTS A SET IS A COLLECTION OF OBJECTS OR ELEMENTS TYPICALLY FOR US THESE OBJECTS WILL BE POINTS IN A MANIFOLD A SET U IS A SUBSET OF A SET V IF EVERY ELEMENT OF U IS ALSO AN ELEMENT OF V IF THERE EXIST ELEMENTS IN V THAT ARE NOT IN THE SUBSET U THEN U IS CALLED A PROPER SUBSET OF V IF U IS A SUBSET OF V THEN THE COMPLEMENT OF U IN V DENOTED BY V U IS THE SET OF ALL ELEMENTS OF V THAT ARE NOT IN U IF U IS A SUBSET BUT NOT A PROPER SUBSET THEN V U CONTAINS NO ELEMENTS AT ALL THIS SET CONTAINING NO ELEMENTS IS CALLED THE EMPTY SET AND IS DENOTED BY BY DE NITION THEREFORE IS A SUBSET OF EVERY SET THE NOTION OF THE COMPLEMENT CAN BE EXTENDED TO DE NE THE DI ERENCE OF SETS V AND U EVEN WHEN U IS NOT A SUBSET OF V THUS WE HAVE V U X X V AND X U 1 1 IF U IS A SUBSET OF V THEN THIS REDUCES TO THE COMPLEMENT DE NED PREVIOUSLY TWO SETS U AND V ARE EQUAL U V IF EVERY ELEMENT OF V IS AN ELEMENT OF U AND VICE VERSA THIS IS EQUIVALENT TO THE STATEMENT THAT U IS A SUBSET OF V AND V IS A SUBSET OF U FROM TWO SETS U AND V WE CAN FORM THE UNION DENOTED BY U V WHICH IS THE SET OF ALL ELEMENTS THAT ARE IN U OR IN V THE INTERSECTION DENOTED BY U V IS THE SET OF ALL ELEMENTS THAT ARE IN U AND IN V THE TWO SETS U AND V ARE SAID TO BE DISJOINT IF U V I E THEY HAVE NO ELEMENTS IN COMMON 1 SOME STRAIGHTFORWARDLY ESTABLISHED PROPERTIES ARE A B B A A B B A A B C A B C A B C A B C 1 2 A B C A B A C A B C A B A C IF A AND B ARE SUBSETS OF C THEN C C A A C C B B A A B A B C A B C A C B C A B C A C B 1 3 1 2 TOPOLOGICAL SPACES BEFORE BEING ABLE TO DE NE A MANIFOLD WE NEED TO INTRODUCE THE NOTION OF A TOPOLOGICAL SPACE THIS CAN BE DE NED AS A POINT SET S WITH OPEN SUBSETS OI FOR WHICH THE FOLLOWING PROPERTIES HOLD 1 THE UNION OF ANY NUMBER OF OPEN SUBSETS IS AN OPEN SET 2 THE INTERSECTION OF A NITE NUMBER OF OPEN SUBSETS IS AN OPEN SET 3 BOTH S ITSELF AND THE EMPTY SET ARE OPEN IT WILL BE OBSERVED THAT THE NOTION OF AN OPEN SET IS RATHER IMPORTANT HERE ESSENTIALLY A SET X IS OPEN IF EVERY POINT X INSIDE X HAS POINTS ROUND IT THAT ARE ALSO IN X IN OTHER WORDS EVERY POINT IN AN OPEN SET HAS THE PROPERTY THAT YOU CAN WIGGLE IT AROUND A LITTLE AND IT IS STILL INSIDE X CONSIDER FOR EXAMPLE THE SET OF ALL REAL NUMBERS R IN THE INTERVAL 0 R 1 THIS IS CALLED AN OPEN INTERVAL AND IS DENOTED BY 0 1 AS ITS NAME IMPLIES THE OPEN INTERVAL DE NES AN OPEN SET INDEED WE CAN SEE THAT FOR ANY REAL NUMBER R SATISFYING 0 R 1 WE CAN ALWAYS ND REAL NUMBERS BIGGER THAN R AND SMALLER THAN R THAT STILL THEMSELVES LIE IN THE OPEN INTERVAL 0 1 BY CONTRAST THE INTERVAL 0 R 1 IS NOT OPEN THE POINT R 1 LIES INSIDE THE SET BUT IF IT IS WIGGLED TO THE RIGHT BY ANY AMOUNT NO MATTER HOW TINY IT TAKES US TO A POINT WITH R 1 WHICH IS NOT INSIDE THE SET GIVEN THE COLLECTION OI OF OPEN SUBSETS OF S WE CAN DE NE THE NOTION OF THE LIMIT POINT OF A SUBSET AS FOLLOWS A POINT P IS A LIMIT POINT OF A SUBSET X OF S PROVIDED THAT GEOMETRY AND GROUP THEORY 2 EVERY OPEN SET CONTAINING P ALSO CONTAINS A POINT IN X THAT IS DISTINCT FROM P THIS DE NITION YIELDS A TOPOLOGY FOR S AND WITH THIS TOPOLOGY S IS CALLED A TOPOLOGICAL SPACE SOME FURTHER CONCEPTS NEED TO BE INTRODUCED FIRST WE DE NE A BASIS FOR THE TOPOLOGY OF S AS SOME SUBSET OF ALL POSSIBLE OPEN SETS IN S SUCH THAT BY TAKING INTERSECTIONS AND UNIONS OF THE MEMBERS OF THE SUBSET WE CAN GENERATE ALL POSSIBLE OPEN SUBSETS IN S AN OPEN COVER UI OF S IS A COLLECTION OF OPEN SETS SUCH THAT EVERY POINT P IN S IS CONTAINED IN AT LEAST ONE OF THE UI THE TOPOLOGICAL SPACE S IS SAID TO BE COMPACT IF EVERY OPEN COVERING UI HAS A NITE SUB COLLECTION UI1 UIN THAT ALSO COVERS S FINALLY WE MAY DE NE THE NOTION OF A HAUSDOR SPACE THE TOPOLOGICAL SPACE S IS SAID TO OBEY THE HAUSDOR AXIOM AND HENCE TO BE AN HAUSDOR SPACE IF FOR ANY PAIR OF DISTINCT POINTS P1 AND P2 IN S THERE EXIST DISJOINTS OPEN SETS O1 AMD O2 EACH CONTAINING JUST ONE OF THE TWO POINTS IN OTHER WORDS FOR ANY DISTINCT PAIR OF POINTS P1 AND P2 WE CAN ND A SMALL OPEN SET AROUND EACH POINT SUCH THAT THE TWO OPEN SETS DO NOT OVERLAP 1 WE ARE NOW IN A POSITION TO MOVE ON TO THE DE NITION OF A MANIFOLD 1 3 MANIFOLDS BEFORE GIVING A FORMAL DE NITION OF A MANIFOLD IT IS USEFUL TO INTRODUCE WHAT WE WILL RECOGNISE SHORTLY AS SOME VERY SIMPLE BASIC EXAMPLES FIRST OF ALL CONSIDER THE REAL LINE RUNNING FROM MINUS TO PLUS IN NITY SLIGHTLY MORE PRECISELY WE CONSIDER THE OPEN INTERVAL I E THE SET OF POINTS X SUCH THAT X WE DENOTE THIS BY THE SYMBOL IR THE LETTER R REPRESENTING THE REAL NUMBERS IN FACT IR IS THE PROTOTYPE EXAMPLE OF A MANIFOLD IT IS A ONE DIMENSIONAL TOPOLOGICAL SPACE PARAMETERISED BY THE POINTS ON THE REAL LINE A SIMPLE EXTENSION OF THE ABOVE IS TO CONSIDER THE N DIMENSIONAL SPACE CONSISTING OF N COPIES OF THE REAL LINE WE DENOTE THIS BY IRN A FAMILIAR EXAMPLE IS THREE DIMENSIONAL EUCLIDEAN SPACE WITH CARTESIAN COORDINATES X Y Z THUS OUR FAMILIAR THREE DIMENSIONAL SPACE CAN BE CALLED THE 3 MANIFOLD IR3 WE CAN NOW GIVE A FORMAL DE NITION OF A SMOOTH N MANIFOLD WITH A SMOOTH ATLAS OF CHARTS AS 1 A TOPOLOGICAL SPACE S 2 AN OPEN COVER UI WHICH ARE KNOWN AS PATCHES 1THE GREAT MATHEMATICIAN AND GEOMETER MICHAEL ATIYAH GAVE A NICE COLLOQUIAL DE NITION A TOPOLOGICAL SPACE IS HAUSDOR IF THE POINTS CAN BE HOUSED O ONE SHOULD IMAGINE THIS BEING SPOKEN IN A RATHER PLUMMY ENGLISH ACCENT IN WHICH THE WORD O IS PRONOUNCED OR GEOMETRY AND GROUP THEORY 3