Algebraic Topology

Notes:

• Extracted Annotations (2/8/2019, 5:59:48 PM)

  "the" (Hatcher :199)

  "Homology groups Hn„X… are the result of a two-stage process: First one forms a" (Hatcher :199)

  "chain complex" (Hatcher :199)

  "then one takes the homology groups" (Hatcher :199)

  "G…" (Hatcher :199)

  "To obtain" (Hatcher :199)

  "cohomology groups Hn„X; G… we interpolate an intermediate step, replacing the" (Hatcher :199)

  "chain groups Cn by the dual groups Hom„Cn ; G…" (Hatcher :199)

  "If we dualize this free resolution by applying Hom„−; G… , we may lose exactness, but at least we get a chain comple" (Hatcher :202)

  "x" (Hatcher :202)

  All free resolutions of abelian groups are isomorphic (note on p.203)

   

  Constructing a free resolution for an abelian group (note on p.204)

   

  Universal coefficient theorem for cohomology (note on p.204)

   

  "Every abelian group H has a free resolution of the form 0→F →F0 →H→0 , with" (Hatcher :204)

  "obtainable in the following way" (Hatcher :204)

  "0 -→ Ext„Hn−1 „C …; G… -→ Hn„C ; G… h→ Hom„Hn „C …; G… -→ 0" (Hatcher :204)

  "Ext„H ⊕ H′; G… ≈ Ext„H; G… ⊕ Ext„H′; G… ." (Hatcher :204)

  "Ext„H; G… ƒ 0 if H is free." (Hatcher :204)

  "Ext„Zn ; G… ≈ G=nG ." (Hatcher :204)

  "Ext„H; Z… is isomorphic to the torsion subgroup of H if H is finitely generated" (Hatcher :205)

  Universal Coefficient Theorem for Cohomology in the nice case that everything is a finitely generated abelian group (note on p.205)

   

  Chain maps that induce isomorphisms on integral homology also induce isomorphisms in cohomology with coefficients in any group (note on p.205)

   

  "Hn„C ; Z… ≈ „Hn =Tn… ⊕ Tn−1 ." (Hatcher :205)

  "However, the splitting in the universal coefficient theorem is not natural" (Hatcher :205)

  "The key fact about abelian groups that was needed was that subgroups of free abelian groups are free. Submodules of free R modules are free if R is a principal ideal domain," (Hatcher :205)

  How to construct a free resolution for any module (note on p.206)

   

  "Moreover, every R module H has a free resolution, which can be constructed in the following way." (Hatcher :206)

  Formula relating cohomology to homology (note on p.207)

   

  Cohomology is the actual dual if the coefficients are from a field (note on p.207)

   

  "0 -→ Ext„Hn−1 „X…; G… -→ Hn„X; G… -→ Hom„Hn„X…; G… -→ 0" (Hatcher :207)

  Cup Product (note on p.215)

   

  Formula for the boundary of a cup product (note on p.215)

   

  "For cochains φ ∈ C k „X; R… and ψ ∈ C ℓ„X; R… , the cup product φ ' ψ ∈ C k‚ℓ „X; R… is the cochain whose value on a singular simplex σ : ∆k‚ℓ →X is given by the formula" (Hatcher :215)

  "δ„φ'ψ… ƒ δφ'ψ‚„−1…k φ'δψ" (Hatcher :215)

  "„φ ' ψ…„σ … ƒ φσ | †v0 ; · · · ; vk ]ψσ | †vk ; · · · ; vk‚ℓ ] " (Hatcher :215)

  Detailed examples of geometrically determining the cup product (note on p.216)

   

  "H1 „M…× H1 „M…→H2 „M…" (Hatcher :216)

  "a basis for H1 „M… is formed by the edges ai and bi" (Hatcher :216)

  "he edges ai give a basis for H1 „N ; Z2 …" (Hatcher :217)

  "αi ' αi is the nonzero element of H2 „N ; Z2 … ≈ Z2 and αi ' αj ƒ 0 for i ≠ j ." (Hatcher :217)

  "when g ƒ 1 we have N ƒ RP2 , and the cup product of a generator of H1 „RP2 ; Z2 … with itself is a generator of H2 „RP2 ; Z2 …" (Hatcher :217)

  Ring and Algebra structure on cohomology (note on p.221)

   

  Polynomial Rings (note on p.222)

   

  Exterior Algebras (note on p.222)

   

  Using the cup product to distinguish spaces (note on p.223)

   

  Relating cup product to Cartesian product of spaces (note on p.223)

   

  "homology or just the additive structure of cohomology it is impossible to conclude that CP2 is not homotopy equivalent to S 2 ∨ S 4" (Hatcher :223)

  Properties of tensor product (note on p.224)

   

  "A ⊗R B ƒ A ⊗ B when R is Zm or Q . But in general A ⊗ B is not the same as A ⊗ B" (Hatcher :224)

  When the cross product is an isomorphism (note on p.225)

   

  Division algebras over the real numbers (note on p.232)

   

  Division algebras over the complex numbers (note on p.232)

   

  Algebra structure of cohomology of some classifying spaces (note on p.236)

   

  Poincare Duality (note on p.239)

   

  Definition of manifold (note on p.240)

   

  "Poincar ́ duality in its most primitive form asserts that for a closed orientable manifold M of dimension n , there are isomorphisms Hk „M; Z… ≈ Hn−k „M; Z…" (Hatcher :240)

  "Without the orientability hypothesis there is a weaker statement that Hk „M; Z2 … ≈ Hn−k „M; Z2 …" (Hatcher :240)

  "in terms of homology:" (Hatcher :240)

  "Modulo their torsion subgroups, Hk „M; Z… and Hn−k „M; Z… are isomorphic, and the torsion subgroups of Hk „M; Z… and Hn−k−1 „M; Z… are isomorphic" (Hatcher :240)

  "Hi „X × S 1 ; Z… ≈ Hi „X; Z… ⊕ Hi−1 „X; Z…" (Hatcher :240)

  Dual cells (note on p.241)

   

  Definition of orientation for manifolds (note on p.242)

   

  "Whatever an orientation of Rn is, it should have the property that it is preserved under rotations and reversed by reflections" (Hatcher :242)

  "An orientation of Rn at a point x is a choice of generator of the infinite cyclic group Hn„Rn ; Rn − {x}… ," (Hatcher :242)

  "Hn„Rn ; Rn − {x}… ≈ Hn−1 „Rn − {x}… ≈ Hn−1 „S n−1 … where S n−1 is a sphere centered at x" (Hatcher :242)

  "Since these isomorphisms are natural, and rotations of S n−1 have degree 1 , being homotopic to the identity, while reflections have degree −1 , we see that a rotation ρ of Rn fixing x takes a generator α of Hn„Rn ; Rn − {x}… to itself, ρ∗ „α… ƒ α , while a reflection takes α to −α ." (Hatcher :242)

  "Note that with this definition, an orientation of Rn at a point x determines an orientation at every other point y via the canonical isomorphisms Hn„Rn ; Rn −{x}… ≈ Hn„Rn ; Rn − B… ≈ Hn„Rn ; Rn − {y }…" (Hatcher :242)

  "where B is any ball containing both x and y ." (Hatcher :242)

  Local homology (note on p.243)

   

  Full definition of orientation for a manifold (note on p.243)

   

  Definition of orientable manifold (note on p.243)

   

  Orientable covering space construction for a manifold (note on p.243)

   

  Orientability in terms of orientation double cover (note on p.243)

   

  "To simplify notation we will write Hn„X; X − A… as Hn„X | A… , or more generally Hn„X | A; G…" (Hatcher :243)

  "v" (Hatcher :243)

  "view Hn„X | A… as local homology of X at A ." (Hatcher :243)

  "a global orientation ought to be 'a consistent choice of local orientations at all points.'" (Hatcher :243)

  "Every manifold M has an orientable two-sheeted covering space f" (Hatcher :243)

  "f μx || x ∈ M and μx is a local orientation of M at x" (Hatcher :243)

  "If M is connected, then M is orientable iff f has two components" (Hatcher :243)

  "M is orientable if it is simply-connected, or more generally if π „M… has no subgroup of index two." (Hatcher :243)

  Orientation as a section of a covering space (note on p.244)

   

  "covering space f→M can be embedded in a larger covering space M" (Hatcher :244)

  "The co" (Hatcher :244)

  Fundamental class (note on p.245)

   

  Torsion subgroup of homology in dimension n-1 is either trivial or integers mod 2 for a manifold (note on p.247)

   

  Cap product (note on p.248)

   

  "define an R bilinear cap product a : Ck „X; R…× C ℓ„X; R…→Ck−ℓ „X; R… for k ≥ ℓ by setting" (Hatcher :248)

  "for σ : ∆k →X and φ ∈ C ℓ„X; R…" (Hatcher :248)

  "σ a φ ƒ φ σ | †v0 ; · · · ; vℓ] σ | †vℓ; ··· ; vk ]" (Hatcher :248)

  Statement of Poincare duality theorem (note on p.250)

   

  Example: Computing cap products from simplicial homology (note on p.250)

   

  Diagram for cap product (note on p.250)

   

  Formula relating cup and cap product (note on p.258)

   

  Cup product pairing for Poincare duality (note on p.258)

   

  "ψ„α a φ… ƒ „φ ' ψ…„α…" (Hatcher :258)

  "α ∈ Ck‚ℓ „X; R… , φ ∈ C k „X; R… , and ψ ∈ C ℓ„X; R…" (Hatcher :258)

  "Hk „M; R… × Hn−k „M; R… --------→ R; „φ; ψ… ֏ „φ ' ψ…†M]" (Hatcher :258)

  When the cup product pairing for Poincare Duality is nonsingular (note on p.259)

   

  Short proof of the algebra structure of complex projective space using cup product results (note on p.259)

   

  "Such a bilinear pairing A× B→R is said to be nonsingular if the maps A→Hom „B; R… and B→Hom „A; R… ," (Hatcher :259)

  "are both isomorphisms." (Hatcher :259)

  Relation between middle dimensional cup product pairing and classification of nonsingular bilinear forms (note on p.261)

   

  "the middle-dimensional cup product pairing Hn nonsingular bilinear form on Hn f r ee „M…× Hn r ee „M…→Z is a f r ee „M…" (Hatcher :261)

  "nr ee „M…× Hn r ee „M…→Z is a nr ee „M… . This form is symmetric or skew-symmetric according to whether n is even or odd" (Hatcher :261)

  "One can ask whether all these forms actually occur as cup product pairings in closed manifolds M4k for a given k . The answer is yes for 4k ƒ 4; 8; 16 but seems to be unknown in other dimensions." (Hatcher :261)

  "In dimension 4 there are at most two nonhomeomorphic simply-connected closed 4 manifolds with the same bilinear form" (Hatcher :261)

  "there are two manifolds with the same form if the square α ' α of some α ∈ H2 „M4 … is an odd multiple of a generator of H4 „M4 … , for example for CP2 , and otherwise the M4 is unique, for example for S 4 or S 2 × S 2" (Hatcher :261)

  "the homotopy type of a simply-connected closed 4 manifold is uniquely determined by its cup product structure." (Hatcher :261)

  "The algebra in the skew-symmetric case is rather simple: With a suitable choice of basis, the matrix of a skew-symmetric nonsingular bilinear form over Z can be put into the standard form consisting of 2× 2 blocks 0 along the diagonal and zeros elsewhere," (Hatcher :261)

  Collar neighborhoods (note on p.262)

   

  Lefschetz duality (note on p.263)

   

  Conditions in homology for embedding a manifold in real n-space (note on p.265)

   

  "One reason for this is Whitehead's theorem that a map between CW complexes which induces isomorphisms on all homotopy groups is a homotopy equivalence." (Hatcher :346)

  "One of the rare cases when a CW complex does have its homotopy type uniquely determined by its homotopy groups is when it has just a single nontrivial homotopy group" (Hatcher :346)

  "Cohomology classes in a CW complex correspond bijectively with homotopy classes of maps from the complex into an Eilenberg-MacLane space." (Hatcher :346)

  "A more elementary and direct connection between homotopy and homology is the Hurewicz theorem, asserting that the first nonzero homotopy group πn„X… of a simply-connected space X is isomorphic to the first nonzero homology group eHn „X… ." (Hatcher :347)

  "Though the excision property does not always hold for homotopy groups, in some important special cases there is a range of" (Hatcher :347)

  "f dimensions in which it does hold. This leads to the idea of stable homotopy group" (Hatcher :347)

  "Among other things, fibrations allow one to describe, in theory at least, how the homotopy type of an arbitrary CW complex is built up from its homotopy groups by an inductive procedure of forming 'twisted products' of Eilenberg-MacLane spaces" (Hatcher :347)

  "This is the notion of a Postnikov tower." (Hatcher :347)

  Table of homotopy groups of spheres (note on p.348)

   

  Covering spaces yield isomorphisms on higher homotopy groups (note on p.351)

   

  "roposition 4.1. A covering space projection p : „ e ; ex0 …→„X; x0 … induces isomorphisms p∗ : πn„ e ; ex0 …→πn„X; x0 … for all n ≥ 2 ." (Hatcher :351)

  Whitehead's theorem: a map inducing isomorphisms on all homotopy groups of a CW complex is a homotopy equivalence (note on p.355)

   

  Homotopy groups as obstructions to lifting maps of CW complexes (note on p.357)

   

  Cellular approximation theorem (note on p.358)

   

  Definition: weak homotopy equivalence (note on p.361)

   

  A weak homotopy equivalence that is not a homotopy equivalence (note on p.361)

   

  CW approximation to a space (note on p.361)

   

  Construction of a CW approximation of a space (note on p.361)

   

  "Whitehead's theorem can be restated as saying that a weak homotopy equivalence between CW complexes is a homotopy equivalence" (Hatcher :361)

  "there exist noncontractible spaces whose homotopy groups are all trivial" (Hatcher :361)

  "and for such spaces" (Hatcher :361)

  "map to a point is a weak homotopy equivalence that is not a homotopy equivalence" (Hatcher :361)

  Postnikov Tower construction (note on p.363)

   

  "πi „Xn … ƒ 0 for i > n" (Hatcher :363)

  "starting at the stage of attaching cells of dimension n ‚" (Hatcher :363)

  "2" (Hatcher :363)

  "Thus we attach „n ‚ 2… cells to X" (Hatcher :363)

  "form a space with πn‚1 trivial" (Hatcher :363)

  "isomorphism on πi for i ≤ n" (Hatcher :363)

  "The result is a CW complex Xn" (Hatcher :363)

  Whitehead tower (note on p.365)

   

  Homotopy classes of maps into spaces can't distinguish weak homotopy equivalences (note on p.366)

   

  "using the notations †X; Y ] for the set of homotopy classes of maps X→Y and hX; Y i for the set of basepointpreserving-homotopy classes of basepoint-preserving maps X→Y ." (Hatcher :366)

  Freudenthal Suspension Theorem (note on p.369)

   

  Proof: pi_n S^n = Z (note on p.370)

   

  Eilgenberg-MacLane Spaces (note on p.374)

   

  K(Z, n) and K(G, n) for a general group G (note on p.374)

   

  Hurewicz Theorem (note on p.375)

   

  Definition: fiber bundles (note on p.384)

   

  Definition: fibration (note on p.384)

   

  Long exact sequence of a fibration (note on p.385)

   

  The Hopf Bundle and an illustration (note on p.386)

   

  Fibrations involving Stiefel and Grassman manifolds (note on p.390)

   

  Bott periodicity (note on p.392)

   

  Stable homotopy groups of spheres (note on p.393)

   

  "isomorphism Hn„X; G… ≈ hX; K„G; n…i" (Hatcher :402)

  "r" (Hatcher :402)

  "alizing an arbitrar" (Hatcher :402)

  "alizing" (Hatcher :402)

  "ealiz" (Hatcher :402)

  "y CW complex as a sort of twisted product of Eilenberg-MacLane spaces, up to homotopy equivalence" (Hatcher :402)

  "The most geometric interpretation of the phrase 'twisted product' is the notion of fiber bundle introduced in the previous section, but here we need the more homotopy-theoretic notion of a fibration" (Hatcher :402)

  "ostnikov" (Hatcher :402)

  "P" (Hatcher :402)

  "towers can be expressed as sequences of fibrations with fibers Eilenberg-MacLane spaces" (Hatcher :402)

  "k invariants , which describe, at least in principle, how Postnikov towers for a broad class of spaces are determined by a sequence of cohomology classes" (Hatcher :402)

  "There are natural bijections T : hX; K„G; n…i→Hn„X; G…" (Hatcher :402)

  "Such a T has the form T „†f ]… ƒ f ∗„α… for a certain distinguished class α ∈ Hn„K„G; n…; G…" (Hatcher :402)

  "A class α ∈ Hn„K„G; n…; G… with the property stated in the theorem is called a fundamental class." (Hatcher :403)

  "Concretely, if we choose K„G; n… to be a CW complex with „n − 1… skeleton a point" (Hatcher :403)

  "T" (Hatcher :403)

  "ointe" (Hatcher :403)

  "he functors hn„X… ƒ hX; K„G; n…i define a reduced cohomology theory on the category of basepointed CW complexes. (2) If a reduced coho" (Hatcher :403)

  "2" (Hatcher :403)

  "2" (Hatcher :403)

  "mology theory h∗" (Hatcher :403)

  "∗ defined on CW complexes has coefficient groups hn„S 0 … which are" (Hatcher :403)

  "zero for n ≠ 0 , then there are natural isomorphisms hn„X… ≈ eHn„X; h0 „S 0 …… for all CW complexes X and all n ." (Hatcher :403)

  "When does a sequence of spaces Kn define a cohomology theory by setting hn„X… ƒ hX; Kni ? Note that this will be a reduced cohomology theory since hX; Kni is trivial when X is a point." (Hatcher :403)

  "there is a problem with where to choose the basepoint in S X . If x0 is a basepoint of X , the basepoint of S X should be somewhere along the segment {x0 } × I ⊂ S X , most likely either an endpoint or the" (Hatcher :403)

  "midpoint, but no single choice of such a basepoint gives a well-defined sum." (Hatcher :404)

  "ΣX ƒ S X=„{x0 } × I …" (Hatcher :404)

  "The space ΣX is called the reduced suspension of X" (Hatcher :404)

  "hΣX; Ki ƒ hX; ΩKi where ΩK is the space of loops in K at its chosen basepoint and the constant loop is taken as the basepoint of ΩK ." (Hatcher :404)

  "X ≃ Y implies ΩX ≃ ΩY" (Hatcher :404)

  "the loopspace of a CW complex has the homotopy type of a CW complex" (Hatcher :404)

  "Thus for a sequence of spaces Kn to define a cohomology theory hn„X… ƒ hX; Kni we have been led to the assumption that each Kn should be a loopspace and in fact a double loopspace" (Hatcher :405)

  "Actually we do not need Kn to be literally a loopspace since it would suffice for it to be homotopy equivalent to a loopspace, as hX; Kn i depends only on the homotopy type of" (Hatcher :405)

  "c" (Hatcher :405)

  "ivalent to a loopspace, as hX; Kn i depends only on the homotopy type of Kn . In fact" (Hatcher :405)

  "In" (Hatcher :405)

  "it would suffice to have just a weak homotopy equivalence Kn" (Hatcher :405)

  "ype of Kn . In fact it would suffice to have just a weak homotopy equivalence Kn →ΩLn for some space Ln" (Hatcher :405)

  Brown representability theorem (note on p.406)

   

  "coho" (Hatcher :406)

  "a general cohomology theory hn„X… need not vanish for negative" (Hatcher :406)

  "n" (Hatcher :406)

  "A space Kn in an Ω spectrum is sometimes called an infinite loopspace" (Hatcher :406)

  Puppe sequence (note on p.407)

   

  Homology of K(Z, 3) (note on p.413)

   

  Pullback of a fibration (note on p.415)

   

  Fibration over a contractible base is a product (note on p.415)

   

  Replace any map with a fibration (path space) (note on p.416)

   

  "There is a simple but extremely useful way to turn arbitrary mappings into fibrations." (Hatcher :416)

  Fibers are weakly homotopy equivalent to loop spaces of the base (note on p.417)

   

  Postnikov Towers (note on p.419)

   

  How to decompose a space as a postnikov tower (note on p.419)

   

  "πi „Xn … ƒ 0 for i > n ." (Hatcher :419)

  "on πi f" (Hatcher :419)

  "o" (Hatcher :419)

  "≤" (Hatcher :419)

  "≤" (Hatcher :419)

  "n" (Hatcher :419)

  "induces an isomorphism on πi for i ≤ n . (2) πi „Xn … ƒ 0 for" (Hatcher :419)

  ">" (Hatcher :419)

  "convert the map Xn →Xn−1 into a fibration, its fiber Fn is a K„πn X; n…" (Hatcher :419)

  "is apparent from" (Hatcher :419)

  "πi‚1 „Xn …→πi‚1 „Xn−1 …→πi „Fn…→πi „Xn…→πi „Xn−1 …" (Hatcher :419)

  How to build the postnikov tower for a space, given the homotopy groups (note on p.420)

   

  Conditions on when a Postnikov tower is composed of principal fibrations (note on p.421)

   

  "I" (Hatcher :421)

  "I" (Hatcher :421)

  "n general, a fibration F →E→B is called principal if there is a commutative diag" (Hatcher :421)

  "ra" (Hatcher :421)

  "m" (Hatcher :421)

  "where the second row is a fibration sequence and the vertical maps are weak homotopy equivalences." (Hatcher :421)

  "or example, if all the kn 's are zero, X is just the product of the spaces K„πnX; n… , and in the general case X is some sort of twisted product" (Hatcher :421)

  "of K„πn X; n… 's." (Hatcher :421)

  "F" (Hatcher :421)

  "However, in the situation of Postnikov towers, the homotopy fiber is a K„π ; n… with π abelian since n ≥ 2 , so it is a loopspace." (Hatcher :422)

  Diagrams for lifting and extension problems (note on p.424)

   

  Cohomology of SU and Sp (note on p.443)

   

  Cohomology of Grassman manifolds (note on p.444)

   

  Definition: n flag (note on p.445)

   

  Poincare Series (note on p.446)

   

  Euler class (note on p.447)

   

  Obstruction to existence of a section of a fiber bundle (note on p.447)

   

  When the suspension map is an isomorphism on homotopy (note on p.482)

   

  pi_n+1 S^n (note on p.483)

Attachments

• Hatcher - Algebraic Topology.pdf

Abstract Algebra

Notes:

• Extracted Annotations (2/8/2019, 5:55:45 PM)

  Proof that injective is equivalent to left inverse and surjective to right inverse (note on p.16)

  
  

  "( 1) The map f is injective if and only iff has a left inverse. (2) The map f is surjective if and only iff has a right inverse." (Dummit and Foote :16)

  The Euclidean algorithm (note on p.19)

  
  

  Example of applying the Euclidean algorithm (note on p.19)

  
  

  Euler phi function: how to compute (note on p.21)

  
  

Attachments

• Dummit and Foote - Abstract Algebra.pdf

Real and complex analysis

Tags:

• Mathematical analysis

Notes:

• Extracted Annotations (2/8/2019, 6:02:43 PM)

  "The Riesz representation theorem and the Hahn-Banach theorem allow one to "guess" the Poisson integral formula. They team up in the proof of Runge's theorem. They combine with Blaschke's theorem on the zeros of bounded holomorphic functions to give a pro" (Rudin 1987:14)

  "of of the Miintz-Szasz theorem, which concerns approximation on an interval. The fact that 13 is a Hilbert space is used in the proof of the Radon-Nikodym theorem, which leads to the theorem about differentiation of indefinite integrals, which in turn yields the existence of radial limits of bounded harmonic functions. The theorems of Plancherel and Cauchy combined give a theorem of Paley and Wiener which, in turn, is used in the Denjoy-Carleman theorem about infinitely differentiable functions on the real line. The maximum modulus theorem gives information about linear transform­ ations on I! -spaces." (Rudin 1987:14)

  "the first 15 chapters in two semesters" (Rudin 1987:15)

  Interesting change of variables trick for integration. Example: 1/(1+x^2). (note on p.19)

  
  

  Definition of a topology. (note on p.23)

  
  

  Definition of a sigma algebra. (note on p.23)

  
  

  Definition of metric space (note on p.24)

  
  

Attachments

• Rudin - 1987 - Real and complex analysis.pdf

Principles of mathematical analysis

Tags:

• Mathematical analysis

Notes:

• Extracted Annotations (2/8/2019, 6:02:34 PM)

  Square root of 2 is irrational (note on p.11)

  
  

  Ordered sets (note on p.12)

  
  

  Proof that square roots exist (note on p.19)

  
  

  Properties of conjugation (note on p.23)

  
  

  Properties of complex modulus (note on p.23)

  
  

  Cauchy-Schwarz for complex series (note on p.24)

  
  

  Norm and inner product (note on p.25)

  
  

  Inequalities involving norms (note on p.25)

  
  

  Big list of topological definitions (note on p.41)

  
  

  List of closed, open, perfect, and bounded sets (note on p.42)

  
  

  Definition of compactness (note on p.45)

  
  

  Definition of the Cantor set (note on p.50)

  
  

  Equivalent definitions of convergence of a sequence (note on p.57)

  
  

  "If {p,} is a sequence in a compact metric space X, then some subsequence of {p,} converges to a point of X." (Rudin 1976:60)

  "Every bounded sequence in R¥ contains a convergent subsequence." (Rudin 1976:60)

  Definition of a cauchy sequence (note on p.61)

  
  

  "A sequence {p,} in a metric space X is said to be a Cauchy sequence if for every & > 0 there is an integer N such that d(p,, p,,) N." (Rudin 1976:61)

  Convergent implies cauchy in metric spaces (note on p.62)

  
  

  "A metric space in which every Cauchy sequence converges is said to be complete." (Rudin 1976:63)

  "3.14 Theorem Suppose {s,} is monotonic. Then {s,} converges if and only if it is bounded." (Rudin 1976:64)

  Definition of series (note on p.68)

  
  

  "3.23 Theorem If Za, converges, then lim,_, . a, = 0." (Rudin 1976:69)

  Convergence of series if terms are dominated by a convergent sum (note on p.69)

  
  

  Sum of a geometric series (note on p.70)

  
  

  The p-series (note on p.71)

  
  

  Definition of the exponential (note on p.72)

  
  

  Proof of the root test (note on p.74)

  
  

  Proof of the ratio test (note on p.75)

  
  

  "The ratio test is frequently easier to apply than the root test, since it is usually easier to compute ratios than nth roots. However, the root test has wider scope. More precisely: Whenever the ratio test shows convergence, the root test does too; whenever the root test is inconclusive, the ratio test is too." (Rudin 1976:77)

  Testing power series convergence (note on p.78)

  
  

  Summation by parts (note on p.79)

  
  

Attachments

• Rudin - 1976 - Principles of mathematical analysis.pdf

Rational Homotopy Theory and Differential Forms

Notes:

• Extracted Annotations (2/8/2019, 5:55:59 PM)

  "boundary is" (Griffiths and Morgan 2013:23)

  LES Homotopy (note on p.23)

  
  

  "Theorem 2.8 (Universal Coefficient Theorem). 1. There is a short exact sequence: 0 ! Ext.Hn1 .X/;Z/ ! Hn .X/ ! Hom.Hn .X/;Z/ ! 0: 2. For any abelian group G, there are short exact sequences: f0g ! Hn .X/ ̋ G ! Hn .XI G/ ! Tor.Hn1 .X/; G/ ! f0g; 0 ! Ext.Hn1 .X/; G/ ! Hn .XI G/ ! Hom.Hn .X/; G/ ! 0:" (Griffiths and Morgan 2013:28)

  Statement of UCT (note on p.28)

  
  

  "Theorem 2.9. Let M be a closed, oriented manifold of dimension n. Then there is a fundamental class –M 2 Hn .M/ and cap product with this class induces an isomorphism \–MW Hq .M/ ! Hnq .M/:" (Griffiths and Morgan 2013:28)

  Statement of Poincare Duality (note on p.28)

  
  

  "While" (Griffiths and Morgan 2013:29)

  Intersection Pairing for smooth manifolds (note on p.29)

  
  

  "is perfect on the quotients of these homology groups by their torsion subgroups. This is called the intersection pairing" (Griffiths and Morgan 2013:29)

  "[To calculate the higher homotopy groups of S 1 , recall thatR1 exp ! S1 is the universal cover. In general, the unique path lifting property of Q X ! X implies that  i . Q X/ —  i .X/ if i > 1 if Q X is a covering space of X. SinceR1 is contractible,  i .S1" (Griffiths and Morgan 2013:49)

  Higher homotopy groups of S1 using the universal cover (note on p.49)

  
  

  "Spectral Sequence of a Fibration" (Griffiths and Morgan 2013:51)

  Fibration Spectral Sequence (note on p.51)

  
  

  "//. Later we will show that any simply connected CW complex is homotopy equivalent to an iterated fibration of the K. ; n/." (Griffiths and Morgan 2013:70)

  Eilenberg Maclane (note on p.70)

  
  

  ") Let us consider the case X D S2 . Then X2 D K.Z; 2/ DCP1; H3 .CP1/ D 0 and H4 .CP1/ DZ. Hence,  3 .S2 / DZ. If we form K.Z; 3/ ! X3 ? ? y CP1 with k-invariant the identity H4 .CP1/ !Z, then H4 .X3 / D 0 and H5 .X3 / DZ=2." (Griffiths and Morgan 2013:80)

  Pi3 S2 (note on p.80)

  
  

Attachments

• Griffiths and Morgan - 2013 - Rational Homotopy Theory and Differential Forms.pdf

Characteristic Classes

Notes:

• Extracted Annotations (2/8/2019, 7:23:22 PM)

  Definition of smooth or C^\infty (note on p.7)

   

  Definition of a smooth manifold (note on p.8)

   

  Expression of tangent space as the span of partial derivatives (note on p.10)

   

  The derivative is an endofunctor on the category of smooth manifolds (note on p.12)

   

  Definition of a vector bundle (note on p.17)

   

  Local triviality condition (note on p.17)

   

  Definition: Local coordinate system (note on p.17)

   

  "DEFINITION. A real vector bundle £ over B consists of the following: 1) a topological space E = E(£) called the total space, 2) a (continuous) map 7: E » B called the projection map, and 3) for each b ¢ B the structure of a vector space" over the real numbers in the set 7~1(b)." (Milnor and Stasheff :17)

  "These must satisfy the following restriction: Condition of local triviality. For each point b of B there should exist a neighborhood U C B, an integer n> 0, and a homeomorphism h:Ux RP > 71) so that, for each b ¢ U, the correspondence x + h(b, x) defines an isomorphism between the vector space R™ and the vector space 7 1(b)." (Milnor and Stasheff :17)

  "Such a pair (U,h) will be called a local coordinate system for & about b. If it is possible to choose U equal to the entire base space, then & will be called a trivial bundle." (Milnor and Stasheff :17)

  "In Steenrod's terminology an R™-bundle is a fiber bundle with fiber R™ and with the full linear group GL (R) in n variables as structural group." (Milnor and Stasheff :18)

  Definition: Isomorphism of vector bundles (note on p.18)

   

  "Example 2. The tangent bundle ry; of a smooth manifold M." (Milnor and Stasheff :18)

  Definition: the normal bundle (note on p.19)

   

  Definition of the canonical line bundle in RP^n (note on p.19)

   

  "If ry is a trivial bundle, then the manifold M is called parallelizable." (Milnor and Stasheff :19)

  Theorem: The canonical line bundle is nontrivial for n >= 1 (note on p.20)

   

  Definition: Section of a vector bundle (note on p.20)

   

  Definition: a nowhere zero section (note on p.20)

   

  "THEOREM 2.1. The bundle Ve over P" is not trivial, for n> 1." (Milnor and Stasheff :20)

  "(A cross-section of the tangent bundle of a smooth manifold M is usually called a vector field on M.)" (Milnor and Stasheff :20)

  The canonical line bundle on RP^n does not have a nowhere zero section (note on p.20)

   

  Definition: Independent sections (note on p.22)

   

  An isomorphism of total spaces that isomorphically maps fibers to fibers is an isomorphism of bundles (note on p.22)

   

  "THEOREM 2.2. An R™-bundle ¢ is trivial if and only if & admits n cross-sections s,,...,S, which are nowhere dependent. n" (Milnor and Stasheff :22)

  Proof that S^3 is parallelizable. (note on p.24)

   

  "Hence S° is parallelizable." (Milnor and Stasheff :24)

  Definition: Quadratic function
  (note on p.25)

   

  Deriving an inner product from a quadratic map (note on p.25)

   

  "DEFINITION. A Euclidean vector bundle is a real vector bundle ¢& together with a continuous function §: E() > R such that the restriction of pu to each fiber of ¢ is positive definite and quadratic. The function pu itself will be called a Euclidean metric on the vector bundle ¢£." (Milnor and Stasheff :25)

  "In the case of the tangent bundle ry; of a smooth manifold, a Euclidean metric yp: DM -> R" (Milnor and Stasheff :25)

  "is called a Riemannian metric, and M together with pu is called a Riemannian manifold." (Milnor and Stasheff :26)

  "ote. In Steenrod's terminology a Euclidean metric on & gives-rise to a reduction of the structural group of £ from the full linear group to the orthogonal group." (Milnor and Stasheff :26)

  "A priori there appear to be two different concepts of triviality for Euclidean vector bundles;" (Milnor and Stasheff :26)

  Trivial bundle iff there exist n independent orthonormal sections (note on p.26)

   

  "Show that the unit sphere S" admits a vector field which is nowhere zero, providing that n is odd. Show that the normal bundle of S$ c R™! is trivial for all n." (Milnor and Stasheff :27)

  "If S" admits a vector field which is nowhere zero, show that the identity map of S" is homotopic to the antipodal map." (Milnor and Stasheff :27)

  "For n even show that the antipodal map of S™ is homotopic to the reflection {CSP Xn41) = (=X, Xy, 000, Xp)" (Milnor and Stasheff :27)

  "and therefore has degree —1." (Milnor and Stasheff :27)

  "show that S? is not parallelizable for n even, n> 2." (Milnor and Stasheff :27)

  Definition of a map of bundles (note on p.30)

   

  Definition: Whitney Sums
  (note on p.31)

   

  Definition: subbundle (note on p.31)

   

  Bundles split as the Whitney sum of any subbundle and its perp (note on p.32)

   

  Definition: The Stiefel-Whitney numbers (note on p.55)

   

  "THEOREM 4.9 [Pontrjagin]. If B is a smooth compact (n+1)- dimensional manifold with boundary equal to M (compare §17), then the Stiefel-Whitney numbers of M are all zero." (Milnor and Stasheff :56)

  Stiefel-Whitney numbers classify manifolds up to cobordism
  (note on p.57)

   

  "THEOREM 4.10 [Thom]. If all of the Stiefel-Whitney numbers of M are zero, then M can be realized as the boundary of some smooth compact manifold." (Milnor and Stasheff :57)

  "DEFINITION. The Grassmann manifold G (Ro+K) is the set of all n-dimensional planes through the origin of the coordinate space ROK," (Milnor and Stasheff :60)

  "An n-frame in R™K is an n-tuple of linearly independent vectors of R™K," (Milnor and Stasheff :60)

  "The collection of all n-frames in R™¥ forms an open subset of the n-fold Cartesian product ROHK X eee X RO+K called the Stiefel manifold v (ROK)," (Milnor and Stasheff :60)

  Theorem: The cohomology ring of the infinite Grassmanian in Z/2Z coefficients is generated by the Stiefel-Whitney classes (note on p.85)

   

  "THEOREM 7.1. The cohomology ring H*G,; 2/2) is a polynomial algebra over 7/2 freely generated by the Stiefel-Whitney classes w, (YD, ery w,(y™)." (Milnor and Stasheff :85)

  Definition of the fundamental class (note on p.92)

   

  Definition: The Thom Isomorphism
  
  (Needed to define the Stiefel-Whitney class) (note on p.92)

   

  Definition: Stiefel-Whitney Class
  (Depends on Thom's identity) (note on p.93)

   

  Definition: Orientation of a bundle (note on p.98)

   

  Definition: The Euler class of an n-plane bundle (note on p.99)

   

  "PROPERTY 9.7. If the oriented vector bundle ¢ possesses a nowhere zero cross-section, then the Euler class e(£) must be zero." (Milnor and Stasheff :103)

  "LEMMA 14.1. If w is a complex vector bundle, then the underlying real vector bundle wg has a canonical preferred orientation." (Milnor and Stasheff :154)

  "Applying this lemma to the special case of a tangent bundle, it follows that any complex manifold has a canonical preferred orientation." (Milnor and Stasheff :154)

  "every orientation for the tangent bundle of a manifold gives rise to a unique orientation of the manifold." (Milnor and Stasheff :154)

Attachments

• Milnor and Stasheff - Characteristic Classes.pdf

Differential Forms in Algebraic Topology

Notes:

• Extracted Annotations (2/8/2019, 5:55:35 PM)

  Setup for spectral sequences (note on p.85)

   

  Spectral sequence of a filtered complex (note on p.86)

   

  The Grassmanian
  (note on p.154)

Attachments

• Bott and Tu - 1982 - Differential Forms in Algebraic Topology.pdf