LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS
|
|
- Hollie Dawson
- 6 years ago
- Views:
Transcription
1 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS I the previous sectio we used the techique of adjoiig cells i order to costruct CW approximatios for arbitrary spaces Here we will see that the same techique allows us to modify spaces by killig all homotopy groups above a certai dimesio This will be used to approximate a coected space by a tower of spaces which have oly o-trivial homotopy groups below or above a fixed dimesio where they are isomorphic to the oes of the give space The first case gives rise to the Postikov tower ad the secod oe to the Whitehead tower Moreover, the homotopy groups of two subsequet levels i these towers oly differ i oe dimesio I fact, the maps belogig to the towers are fibratios ad the fibers have precisely oe o-trivial homotopy group 1 The Postikov tower We kow that if α: e +1 represets a elemet [α] π (, x 0 ), the [α] 0 if ad oly if α exteds to a map e +1 Thus if we elarge to a space α e +1 by adjoiig a ( + 1)-cell with α as attachig map, the the iclusio i: iduces a map i : π (, x 0 ) π (, x 0 ) with i [α] 0 We say that [α] has bee killed (Naively, we thik of as a smallest extesio of that does that Some justificatio for this thikig will be provided i the exercises) The followig lemma expresses what happes to the homotopy groups i lower dimesios The proof is similar to the oe that the iclusio of the -skeleto of a CW complex is a -equivalece ad will hece ot be give Lemma 101 Let (, x 0 ) be a poited space, ad let α e +1 be obtaied from by adjoiig a ( + 1)-cell The the iclusio i: iduces a map π k (, x 0 ) π k (, x 0 ) which is a isomorphism for k < ad surjective for k It is difficult to cotrol what happes to the higher homotopy groups For example, sice π 3 (S 2 ) is o-trivial, addig a 2-cell to a elemet i π 1 may well add elemets i π 3 However, we ca kill all of π without chagig π k for k <, by iteratig the procedure of Lemma 101 Lemma 102 Let (, x 0 ) be a poited space The there exists a relative CW complex i: Y, costructed by adjoiig (+1)-cells oly, such that i : π k (, x 0 ) π k (Y, y 0 ) is bijective for k < ad such that π (Y, y 0 ) 0 Proof Let A be a set of represetatives α of geerators [α] of the group π (, x 0 ) Let Y be obtaied from by attachig a ( + 1)-cell e +1 α alog α: e +1 α for each α A: A e +1 A e +1 Y The by a iterated applicatio of Lemma 101, the map i: Y iduces isomorphisms i π k for k <, ad iduces the zero-map o π Sice this map is also surjective, we coclude that π (Y ) has to vaish 1 i
2 2 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS For the proof of the ext theorem, recall that ay map f : U V ca be factored as f p φ, f : U φ p P (f) V, where p is a Serre fibratio ad φ is a homotopy equivalece ( mappig fibratio, see Sectio 5) We say that (up to homotopy), ay map ca be tured ito a fibratio Theorem 103 (Postikov tower) For ay coected space, there is a tower of fibratios P 1 () ψ 1 ψ 2 P 2 () P 3 () ad compatible maps f i : P i () (compatible i the sese that ψ f +1 f : P ()), with the followig properties: (i) π k (P ()) 0 for k > (ii) π k () π k (P ()) is a isomorphism for k (ad hece so is π k P () π k P 1 () for k < ) (iii) The fiber F of ψ 1 has the property that π (F ) π () ad π k (F ) 0 for all k Remark 104 A space like this fiber F with oly oe o-trivial homotopy group is called a Eileberg-MacLae space If Z is such a space with π k (Z) 0 for all k ad π (Z) A, oe says that Z is a K(A, )-space (strictly speakig oe always meas the space Z together with a chose isomorphism π (Z) A) We will discuss these spaces i more detail i a later lecture With this termiology the situatio of the theorem ca be depicted as follows f 3f2 P 3 () ψ 2 P 2 () F 3 K(π 3 (), 3) F 2 K(π 2 (), 2) ψ 1 P 1 () f1 where we used to deote a fibratio Proof of Theorem 103 Let i : Y be a space obtaied from by killig π k () for all k >, ie, such that (i) (i ) : π k () π k (Y ) is a isomorphism for all k (ii) π k (Y ) 0 for all k > Such a space Y ca be obtaied by repeated applicatio of the procedure of Lemma 102, where Y (+1) Y (+1) Y (+2) kills π +1 () by adjoiig ( + 2)-cells, Y (+2), ad so o The resultig space Y m> Y (m) kills π +2 (Y (+1) ) by adjoiig, the uio edowed with ( + 3)-cells to Y (+1) the weak topology, has the desired property, as is immediate from the fact that ay map K Y
3 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 3 with K compact (eg, K S k or K S k [0, 1]) must factor through some Y (m) If you see what this costructio does, the it is clear that there is a caoical iclusio φ : Y +1 Y makig the followig diagram commute (we eed to adjoi more cells for Y tha for Y +1 ): i +1 Y +1 φ Y i Thus, is approximated by smaller ad smaller relative CW complexes Y +1 Y Y 2 Y 1 Now let P 1 () Y 1, ad let f 1 : P 1 () be i 1 : P 1 () Let P 2 () be the space fittig ito a factorizatio of Y 2 φ 1 Y 1 id P 1 () ito a homotopy equivalece j 2 followed by a fibratio ψ 1 Next factor j 2 φ 2 i a similar way as ψ 2 j 3, ad so o, all fittig ito a diagram i 3 j 3 Y 3 P 3 () φ 2 ψ 2 i 2 j 2 Y 2 P 2 () φ 1 ψ 1 i1 Y 1 P 1 () Write f : P () for the compositio j i, ad deote the fiber of ψ 1 : P () P 1 () by F P () Now let us look at the homotopy groups By costructio we have (i) ad (ii) above, ad hece the same is true for P () istead of Y : (i) (f ) : π k () π k (P ()) is a isomorphism for all k (ii) π k (P ()) 0 for all k > We ca feed this iformatio i the log exact sequece of the fibratio F P () ψ 1 P 1 (), a part of which looks like π k+1 (P ) π k+1 (P 1 ) π k (F ) π k (P ) π k (P 1 ) where for simplicity we write P for P (), ad omit all base poits from the otatio So, we clearly have: (i) For k >, the group π k (F ) lies betwee two zero groups, hece is itself the zero-group
4 4 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS (ii) For k <, the group π k (F ) lies betwee a surjectio ad a isomorphism, π k (F ), hece is zero agai (iii) For k, the relevat part of the sequece looks like 0 0 π (F ) π (P ) 0 whece π (F ) is isomorphic to π (P ) π () This tells us that F is a K(π (), )-space ad hece proves the theorem Remark 105 Much more ca be said about these Postikov towers: uder some coditios, the fibratio P P 1 is eve a fiber budle 2 The Whitehead tower The Postikov tower builds up the homotopy groups of (together with all relatios betwee them, such as the actio of π 1 o π ) from below, by costructig for each a space with homotopy groups π 1,, π oly There is also a costructio from above, called the Whitehead tower of, as described i the followig theorem Theorem 106 (Whitehead tower) Let be a coected space There exists a tower W 1 () W 2 () W 3 () with the followig properties: (i) π k (W ()) 0 for k (ii) The map π k (W ()) π k () is a isomorphism for all k > (iii) The map W () W 1 () is a fibratio whose fiber is a K(π (), 1)-space Proof As i the proof of the Postikov tower, ca be approximated by extesios Y +1 Y Y 2 Y 1, where π k (Y ) 0 for k > ad π k () π k (Y ) is a isomorphism for k For Y, let W () be the space of paths i Y from the base poit to, as i the pullback W () Y [0,1] 1 x0 i Y Y So W () is a fibratio (Remember we used this fibratio to describe relative homotopy groups of the pair (Y, ) i the exercises to Sectio 4) These spaces fit aturally ito a sequece W1 () W2 () W3 ()
5 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 5 Now tur these iclusios ito fibratios (by factorig ito a homotopy equivalece followed by a fibratio as before) to obtai a diagram W 1 () W2 () W3 () W 1 () W 2 () W 3 () where the lower horizotal maps are all fibratios ad the vertical oes are homotopy equivaleces Now let us look at the homotopy groups: We kow π k ( W ) π k (W ), ad there are two fibratios to play with, viz W () ad W () W 1 () The fiber of the first oe is the loop space ΩY of Y, ad the fiber of the secod oe will be deoted G The the log exact sequece of W () looks like or equivaletly π k (ΩY ) π k ( W ) π k () π k 1 (ΩY ) π k+1 (Y ) π k ( W ) π k () π k (Y ) But π k (Y ) 0 for k > ad π k () π k (Y ) is a isomorphism for k, so π k ( W ()) π k (), k >, ad π k ( W ) 0, k, ad hece the same is true for W istead of W Next, the log exact sequece associated to W () W 1 () looks like π k+1 (W ) π k+1 (W 1 ) π k (G ) π k (W ) π k (W 1 ) (where we write W for W (), etc), ad we otice: (i) if k > the π k (G ) is squeezed i betwee two isomorphisms, so π k (G ) 0 (ii) if k 2 the π k (G ) sits betwee two zero groups hece is zero itself (iii) for k we obtai π +1 (W ) π +1 (W 1 ) π (G ) 0 ad the first map is a isomorphism so that π (G ) 0 (iv) i the remaiig case k 1 the sequece looks like 0 π (W 1 ) π 1 (G ) 0, so that we have a isomorphism π () π (W 1 ) π 1 (G ) Thus, this tells us that G is a K(π (), 1)-space Note that the spaces W () used i the proof of the Whitehead tower are precisely the homotopy fibers of the maps i : Y costructed i the proof of the Postikov tower The remaiig work i the proof of Theorem 106 the cosists of turig a certai sequece of maps betwee the homotopy fibers i a sequece of fibratios ad aalyzig what happes at the level of homotopy groups This observatio is sometimes referred to by sayig that the Whitehead tower is obtaied from the Postikov tower by passig to homotopy fibers 3 Examples of Eileberg Mac Lae spaces I the costructio of the Postikov ad Whitehead towers approximatig a give space, K(π, )- spaces aturally came up We will coclude this lecture by givig a few of the most elemetary examples of K(π, )-spaces
6 6 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS Remark 107 Recall that a K(π, )-space, or a Eileberg Mac Lae space of type (π, ), is a space (, x 0 ) such that π i (, x 0 ) for all i together with a isomorphism π (, x 0 ) π Here π ca be a poited set if 0, a group is 1, or a abelia group if 2 It ca be show that for such a π, a K(π, )-space always exists, ad is uique up to homotopy, although we will ot give the geeral costructio i this lecture Example 108 (Examples of K(π, )-spaces) (i) The circle S 1 is a K(Z, 1)-space Ideed, it is a coected space with fudametal group Z, ad oe way to see that the higher homotopy groups vaish is to cosider the uiversal coverig space R S 1 This is a fiber budle with discrete fiber F ad cotractible total space, so the log exact sequece gives us isomorphisms 0 π i (F ) π i+1 (S 1 ) for i > 0 (ii) The same argumet applies to wedges of spheres Cosider for example the figure eight S 1 S 1 Its fudametal group is the free group o two geerators Z Z The uiversal cover of S 1 S 1 ca be explicitly described i terms of the grid i the plae, G (Z R) (R Z) R 2 The map w : G S 1 S 1 ca be described by wrappig each edge of legth 1 i the grid aroud oe of the circles (i a way respectig orietatios): say the vertical edges to the left had circle ad the horizotal edges to the right had oe The uiversal cover E of S 1 S 1 is the space of homotopy classes of paths i G which start i the origi, ad E S 1 S 1 is the compositio E ɛ1 G w S 1 S 1 (where ɛ 1 is evaluatio at the edpoit) The fiber of ɛ 1 : E G over a give grid poit (, m) with, m Z is the set of combiatorial paths from (0, 0) to (, m): a sequece of alteratig decisios: go left or go right, go up or go dow, where successios of up-dow ad left-right cacel each other Sice each homotopy class of paths i E has a uique such combiatorial descriptio, the space E is clearly cotractible (iii) Recall that RP, the real projective space of dimesio, is the space of lies i R +1 It ca be costructed as S / Z 2 where the group Z 2 {0, 1} acts by the atipodal map o the uit sphere S {(x 0,, x ) R +1 x x 2 1} The embeddig S S +1 sedig (x 0,, x ) to (x 0,, x, 0) seds S to the equator iside S +1, ad is compatible with this atipodal actio so that we get a commutative diagram S 0 S 1 S 2 RP 0 RP 1 RP 2 There is a atural CW decompositio of S +1, give iductively by a CW decompositio of S with two ( + 1)-cells attached to it: the orther ad the souther hemispheres This makes S ito a CW complex with exactly two k-cells i each dimesio k Oe
7 LECTURE 11: POSTNIKOV AND WHITEHEAD TOWERS 7 ca also take the uio alog the upper row of the diagram (with the weak topology) to obtai the ifiite-dimesioal sphere S S, a CW complex with exactly two -cells i each dimesio Note that sice π i (S ) 0 for i < k we also obtai π i (S ) 0 for all i 0 I other words, S is a weakly cotractible CW complex, ad hece by Whitehead s theorem is cotractible I a similar way, we ca take the uio alog the lower row i the above diagram to obtai RP RP, a CW complex, the ifiite-dimesioal real projective space, with exactly oe -cell i each dimesio The log exact sequece of the coverig projectio S RP with discrete fiber Z 2 shows that π i (RP ) 0, 1 < i <, or i 0, π 1 (RP ) Z 2, ad by passig to the limit, oe cocludes that RP is a K(Z 2, 1)-space (Alteratively, oe ca show that S RP is still a coverig projectio with fiber Z 2 to coclude that RP is a K(Z 2, 1)) (iv) Recall that CP, the complex projective space of (complex) dimesio, is the space of (complex) lies i C +1 It ca be costructed as (C +1 {0})/ C where C C {0} acts by multiplicatio; or, by choosig poits o the lie of orm 1, as the quotiet of the uit sphere i C +1, CP S 2+1 /S 1, where S 1 C agai acts by multiplicatio The quotiet S 2+1 CP has eough local sectios (check this!), hece is a fiber budle with fiber S 1 The embeddig iduces maps C +1 C +2 : (z 0,, z ) (z 0,, z, 0) S 2+1 S 2+3 CP CP +1 ad oe ca agai take the uio, to obtai a map S CP with CP the ifiitedimesioal complex projective space The space CP is a quotiet of S by S 1, ad the map is agai a fiber budle The spaces CP have compatible CW complex structures, give by exactly oe k-cell i each dimesio k Oe way to see this is to represet a lie i C +1 by a poit z (z 0,, z ), z R, z 0, ad z z z 2 1 There is a uique way of doig this The the last coordiate t z is uiquely determied by z (z 0,, z 1 ) (sice t 1 z ), ad these (z 0,, z 1 ) form a disk of dimesio 2 The boudary of this disk is give by z 1, i other words t 0, ad this is exactly the part already i CP 1 I ay case, either of the two argumets at the ed of the previous example shows that CP is a K(Z, 2)-space
Math Homotopy Theory Spring 2013 Homework 6 Solutions
Math 527 - Homotopy Theory Sprig 2013 Homework 6 Solutios Problem 1. (The Hopf fibratio) Let S 3 C 2 = R 4 be the uit sphere. Stereographic projectio provides a homeomorphism S 2 = CP 1, where the North
More informationLECTURE 12: THE DEGREE OF A MAP AND THE CELLULAR BOUNDARIES
LECTURE 12: THE DEGREE OF A MAP AD THE CELLULAR BOUDARIES I this lecture we will study the (homological) degree of self-maps of spheres, a otio which geeralizes the usual degree of a polyomial. We will
More informationLecture XVI - Lifting of paths and homotopies
Lecture XVI - Liftig of paths ad homotopies I the last lecture we discussed the liftig problem ad proved that the lift if it exists is uiquely determied by its value at oe poit. I this lecture we shall
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationSeunghee Ye Ma 8: Week 5 Oct 28
Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value
More informationA COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP
A COUNTABLE SPACE WITH AN UNCOUNTABLE FUNDAMENTAL GROUP JEREMY BRAZAS AND LUIS MATOS Abstract. Traditioal examples of spaces that have ucoutable fudametal group (such as the Hawaiia earrig space) are path-coected
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationProduct measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationFIXED POINTS OF n-valued MULTIMAPS OF THE CIRCLE
FIXED POINTS OF -VALUED MULTIMAPS OF THE CIRCLE Robert F. Brow Departmet of Mathematics Uiversity of Califoria Los Ageles, CA 90095-1555 e-mail: rfb@math.ucla.edu November 15, 2005 Abstract A multifuctio
More informationLecture 4: Grassmannians, Finite and Affine Morphisms
18.725 Algebraic Geometry I Lecture 4 Lecture 4: Grassmaias, Fiite ad Affie Morphisms Remarks o last time 1. Last time, we proved the Noether ormalizatio lemma: If A is a fiitely geerated k-algebra, the,
More informationN n (S n ) L n (Z) L 5 (Z),
. Maifold Atlas : Regesburg Surgery Blocksemiar 202 Exotic spheres (Sebastia Goette).. The surgery sequece for spheres. Recall the log exact surgery sequece for spheres from the previous talk, with L +
More information1 Approximating Integrals using Taylor Polynomials
Seughee Ye Ma 8: Week 7 Nov Week 7 Summary This week, we will lear how we ca approximate itegrals usig Taylor series ad umerical methods. Topics Page Approximatig Itegrals usig Taylor Polyomials. Defiitios................................................
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationA brief introduction to linear algebra
CHAPTER 6 A brief itroductio to liear algebra 1. Vector spaces ad liear maps I what follows, fix K 2{Q, R, C}. More geerally, K ca be ay field. 1.1. Vector spaces. Motivated by our ituitio of addig ad
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 11
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract We will itroduce the otio of reproducig kerels ad associated Reproducig Kerel Hilbert Spaces (RKHS). We will cosider couple
More informationMath 220A Fall 2007 Homework #2. Will Garner A
Math 0A Fall 007 Homewor # Will Garer Pg 3 #: Show that {cis : a o-egative iteger} is dese i T = {z œ : z = }. For which values of q is {cis(q): a o-egative iteger} dese i T? To show that {cis : a o-egative
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationSome Examples of Limits of Kleinian Groups
Some Examples of Limits of Kleiia Groups Jeff Brock August 21, 2007 The goal of this lecture will be to exhibit the thick-thi decompositio for a collectio of examples of Kleiia groups. I this lecture,
More informationModel Theory 2016, Exercises, Second batch, covering Weeks 5-7, with Solutions
Model Theory 2016, Exercises, Secod batch, coverig Weeks 5-7, with Solutios 3 Exercises from the Notes Exercise 7.6. Show that if T is a theory i a coutable laguage L, haso fiite model, ad is ℵ 0 -categorical,
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationLecture 6: Integration and the Mean Value Theorem. slope =
Math 8 Istructor: Padraic Bartlett Lecture 6: Itegratio ad the Mea Value Theorem Week 6 Caltech 202 The Mea Value Theorem The Mea Value Theorem abbreviated MVT is the followig result: Theorem. Suppose
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationRoberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series
Roberto s Notes o Series Chapter 2: Covergece tests Sectio 7 Alteratig series What you eed to kow already: All basic covergece tests for evetually positive series. What you ca lear here: A test for series
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationSolutions to Homework 1
Solutios to Homework MATH 36. Describe geometrically the sets of poits z i the complex plae defied by the followig relatios /z = z () Re(az + b) >, where a, b (2) Im(z) = c, with c (3) () = = z z = z 2.
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationOn groups of diffeomorphisms of the interval with finitely many fixed points II. Azer Akhmedov
O groups of diffeomorphisms of the iterval with fiitely may fixed poits II Azer Akhmedov Abstract: I [6], it is proved that ay subgroup of Diff ω +(I) (the group of orietatio preservig aalytic diffeomorphisms
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationFundamental Theorem of Algebra. Yvonne Lai March 2010
Fudametal Theorem of Algebra Yvoe Lai March 010 We prove the Fudametal Theorem of Algebra: Fudametal Theorem of Algebra. Let f be a o-costat polyomial with real coefficiets. The f has at least oe complex
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid
More informationTENSOR PRODUCTS AND PARTIAL TRACES
Lecture 2 TENSOR PRODUCTS AND PARTIAL TRACES Stéphae ATTAL Abstract This lecture cocers special aspects of Operator Theory which are of much use i Quatum Mechaics, i particular i the theory of Quatum Ope
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationMathematical Induction
Mathematical Iductio Itroductio Mathematical iductio, or just iductio, is a proof techique. Suppose that for every atural umber, P() is a statemet. We wish to show that all statemets P() are true. I a
More informationRelations Among Algebras
Itroductio to leee Algebra Lecture 6 CS786 Sprig 2004 February 9, 2004 Relatios Amog Algebras The otio of free algebra described i the previous lecture is a example of a more geeral pheomeo called adjuctio.
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationFundamental Concepts: Surfaces and Curves
UNDAMENTAL CONCEPTS: SURACES AND CURVES CHAPTER udametal Cocepts: Surfaces ad Curves. INTRODUCTION This chapter describes two geometrical objects, vi., surfaces ad curves because the pla a ver importat
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationAlternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More information10.6 ALTERNATING SERIES
0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationProblems from 9th edition of Probability and Statistical Inference by Hogg, Tanis and Zimmerman:
Math 224 Fall 2017 Homework 4 Drew Armstrog Problems from 9th editio of Probability ad Statistical Iferece by Hogg, Tais ad Zimmerma: Sectio 2.3, Exercises 16(a,d),18. Sectio 2.4, Exercises 13, 14. Sectio
More informationLecture #20. n ( x p i )1/p = max
COMPSCI 632: Approximatio Algorithms November 8, 2017 Lecturer: Debmalya Paigrahi Lecture #20 Scribe: Yua Deg 1 Overview Today, we cotiue to discuss about metric embeddigs techique. Specifically, we apply
More informationRay-triangle intersection
Ray-triagle itersectio ria urless October 2006 I this hadout, we explore the steps eeded to compute the itersectio of a ray with a triagle, ad the to compute the barycetric coordiates of that itersectio.
More informationREVIEW FOR CHAPTER 1
REVIEW FOR CHAPTER 1 A short summary: I this chapter you helped develop some basic coutig priciples. I particular, the uses of ordered pairs (The Product Priciple), fuctios, ad set partitios (The Sum Priciple)
More information24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS
24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS Corollary 2.30. Suppose that the semisimple decompositio of the G- module V is V = i S i. The i = χ V,χ i Proof. Sice χ V W = χ V + χ W, we have:
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationSome examples of vector spaces
Roberto s Notes o Liear Algebra Chapter 11: Vector spaces Sectio 2 Some examples of vector spaces What you eed to kow already: The te axioms eeded to idetify a vector space. What you ca lear here: Some
More informationMath 451: Euclidean and Non-Euclidean Geometry MWF 3pm, Gasson 204 Homework 3 Solutions
Math 451: Euclidea ad No-Euclidea Geometry MWF 3pm, Gasso 204 Homework 3 Solutios Exercises from 1.4 ad 1.5 of the otes: 4.3, 4.10, 4.12, 4.14, 4.15, 5.3, 5.4, 5.5 Exercise 4.3. Explai why Hp, q) = {x
More informationProblem Set 2 Solutions
CS271 Radomess & Computatio, Sprig 2018 Problem Set 2 Solutios Poit totals are i the margi; the maximum total umber of poits was 52. 1. Probabilistic method for domiatig sets 6pts Pick a radom subset S
More informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationFUNDAMENTALS OF REAL ANALYSIS by. V.1. Product measures
FUNDAMENTALS OF REAL ANALSIS by Doğa Çömez V. PRODUCT MEASURE SPACES V.1. Product measures Let (, A, µ) ad (, B, ν) be two measure spaces. I this sectio we will costruct a product measure µ ν o that coicides
More information7.1 Convergence of sequences of random variables
Chapter 7 Limit Theorems Throughout this sectio we will assume a probability space (, F, P), i which is defied a ifiite sequece of radom variables (X ) ad a radom variable X. The fact that for every ifiite
More informationTopological Folding of Locally Flat Banach Spaces
It. Joural of Math. Aalysis, Vol. 6, 0, o. 4, 007-06 Topological Foldig of Locally Flat aach Spaces E. M. El-Kholy *, El-Said R. Lashi ** ad Salama N. aoud ** *epartmet of Mathematics, Faculty of Sciece,
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationSymmetrization maps and differential operators. 1. Symmetrization maps
July 26, 2011 Symmetrizatio maps ad differetial operators Paul Garrett garrett@math.um.edu http://www.math.um.edu/ garrett/ The symmetrizatio map s : Sg Ug is a liear surjectio from the symmetric algebra
More informationNotes for Lecture 11
U.C. Berkeley CS78: Computatioal Complexity Hadout N Professor Luca Trevisa 3/4/008 Notes for Lecture Eigevalues, Expasio, ad Radom Walks As usual by ow, let G = (V, E) be a udirected d-regular graph with
More informationSingular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine
Lecture 11 Sigular value decompositio Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaie V1.2 07/12/2018 1 Sigular value decompositio (SVD) at a glace Motivatio: the image of the uit sphere S
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More information(k) x n. n! tk = x a. (i) x p p! ti ) ( q 0. i 0. k A (i) n p
Math 880 Bigraded Classes & Stirlig Cycle Numbers Fall 206 Bigraded classes. Followig Flajolet-Sedgewic Ch. III, we defie a bigraded class A to be a set of combiatorial objects a A with two measures of
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationRecitation 4: Lagrange Multipliers and Integration
Math 1c TA: Padraic Bartlett Recitatio 4: Lagrage Multipliers ad Itegratio Week 4 Caltech 211 1 Radom Questio Hey! So, this radom questio is pretty tightly tied to today s lecture ad the cocept of cotet
More informationMA131 - Analysis 1. Workbook 2 Sequences I
MA3 - Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationSEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS
SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS STEVEN DALE CUTKOSKY Let (R, m R ) be a equicharacteristic local domai, with quotiet field K. Suppose that ν is a valuatio of K with valuatio rig (V, m
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationTopic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
More informationCHAPTER 5 SOME MINIMAX AND SADDLE POINT THEOREMS
CHAPTR 5 SOM MINIMA AND SADDL POINT THORMS 5. INTRODUCTION Fied poit theorems provide importat tools i game theory which are used to prove the equilibrium ad eistece theorems. For istace, the fied poit
More informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationMade-to-Order Weak Factorization Systems
Made-to-Order Weak Factorizatio Systems Emily Riehl 1 The Algebraic Small Obect Argumet For a cocomplete category M which satisies certai smalless coditio (such as beig locally presetable), the algebraic
More information(for homogeneous primes P ) defining global complex algebraic geometry. Definition: (a) A subset V CP n is algebraic if there is a homogeneous
Math 6130 Notes. Fall 2002. 4. Projective Varieties ad their Sheaves of Regular Fuctios. These are the geometric objects associated to the graded domais: C[x 0,x 1,..., x ]/P (for homogeeous primes P )
More informationOn Topologically Finite Spaces
saqartvelos mecierebata erovuli aademiis moambe, t 9, #, 05 BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol 9, o, 05 Mathematics O Topologically Fiite Spaces Giorgi Vardosaidze St Adrew the
More information