Math 754 Chapter III: Fiber bundles. Classifying spaces. Applications
|
|
- Loreen Fleming
- 5 years ago
- Views:
Transcription
1 Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle Bundles 9 3 Classiication o principal G-bundles 14 4 Exercises 18 1
2 1 Fiber bundles Let G be a topological group (i.e., a topological space endowed with a group structure so that the group multiplication and the inversion map are continuous), acting continuously (on the let) on a topological space F. Concretely, such a continuous action is given by a continuous map ρ: G F F, (g, m) g m, which satisies the conditions (gh) m = g (h m)) and e G m = m, or e G the identity element o G. Any continuous group action ρ induces a map Ad ρ : G Homeo(F ) given by g ( g ), with g G, F. Note that Ad ρ is a group homomorphism since (Ad ρ)(gh)() := (gh) = g (h ) = Ad ρ (g)(ad ρ (h)()). Note that or nice spaces F (such as CW complexes), i we give Homeo(F ) the compact-open topology, then Ad ρ : G Homeo(F ) is a continuous group homomorphism, and any such continuous group homomorphism G Homeo(F ) induces a continuous group action G F F. We assume rom now on that ρ is an eective action, i.e., that Ad ρ is injective. Deinition 1.1 (Atlas or a iber bundle with group G and iber F ). Given a continuous map : E B, an atlas or the structure o a iber bundle with group G and iber F on consists o the ollowing data: a) an open cover {U α } α o B, b) homeomorphisms (called trivializing charts or local trivializations) h α : 1 (U α ) U α F or each α so that the diagram 1 (U α ) h α U α F U α pr 1 commutes, c) continuous maps (called transition unctions) g αβ : U α U β G so that the horizontal map in the commutative diagram 1 (U α U β ) (U α U β ) F h α h β h 1 α h β (U α U β ) F is given by (x, m) (x, g βα (x) m). (By the eectivity o the action, i such maps g αβ exist, they are unique.) 2
3 Deinition 1.2. Two atlases A and B on are compatible i A B is an atlas. Deinition 1.3 (Fiber bundle with group G and iber F ). A structure o a iber bundle with group G and iber F on : E B is a maximal atlas or : E B. Example When G = {e G } is the trivial group, : E B has the structure o a iber bundle i and only i it is a trivial iber bundle. Indeed, the local trivializations h α o the atlas or the iber bundle have to satisy h β h 1 α : (x, m) (x, e G m) = (x, m), which implies h β h 1 α = id, so h β = h α on U α U β. This allows us to glue all the local trivializations h α together to obtain a global trivialization h: 1 (B) = E = B F. 2. When F is discrete, Homeo(F ) is also discrete, so G is discrete by the eectiveness assumption. So or the atlas o : E B we have 1 (U α ) = U α F = m F U α {m}, so is in this case a covering map. 3. A locally trivial iber bundle, as introduced in earlier chapters, is just a iber bundle with structure group Homeo(F ). Lemma 1.5. The transition unctions g αβ satisy the ollowing properties: (a) g αβ (x)g βγ (x) = g αγ (x), or all x U α U β U γ. (b) g βα (x) = g 1 αβ (x), or all x U α U β. (c) g αα (x) = e G. Proo. On U α U β U γ, we have: (h α h 1 β ) (h β h 1 γ ) = h α h 1 γ. Thereore, since Ad ρ is injective (i.e., ρ is eective), we get that g αβ (x)g βγ (x) = g αγ (x) or all x U α U β U γ. Note that (h α h 1 β ) (h β h 1 α ) = id, which translates into (x, g αβ (x)g βα (x) m) = (x, m). So, by eectiveness, g αβ (x)g βα (x) = e G or all x U α U β, whence g βα (x) = g 1 αβ (x). Take γ = α in Property (a) to get g αβ (x)g βα (x) = g αα (x). So by Property (b), we have g αα (x) = e G. Transition unctions determine a iber bundle in a unique way, in the sense o the ollowing theorem. Theorem 1.6. Given an open cover {U α } o B and continuous unctions g αβ : U α U β G satisying Properties (a)-(c), there is a unique structure o a iber bundle over B with group G, given iber F, and transition unctions {g αβ }. 3
4 Proo Sketch. Let Ẽ = α U α F {α}, and deine an equivalence relation on Ẽ by (x, m, α) (x, g αβ (x) m, β), or all x U α U β, and m F. Properties (a)-(c) o {g αβ } are used to show that is indeed an equivalence relation on Ẽ. Speciically, symmetry is implied by property (b), relexivity ollows rom (c) and transitivity is a consequence o the cycle property (a). Let E = Ẽ/ be the set o equivalence classes in E, and deine : E B locally by [(x, m, α)] x or x U α. Then it is clear that is well-deined and continuos (in the quotient topology), and the iber o is F. It remains to show the local triviality o. Let p : Ẽ E be the quotient map, and let p α := p Uα F {α} : U α F {α} 1 (U α ). It is easy to see that p α is a homeomorphism. We deine the local trivializations o by h α := p 1 α. Example Fiber bundles with iber F = R n and group G = GL(n, R) are called rank n real vector bundles. For example, i M is a dierentiable real n-maniold, and T M is the set o all tangent vectors to M, then : T M M is a real vector bundle on M o rank n. = More precisely, i ϕ α : U α R n are trivializing charts on M, the transition unctions or T M are given by g αβ (x) = d(ϕ α ϕ 1 β ) ϕ β (x). 2. I F = R n and G = O(n), we get real vector bundles with a Riemannian structure. 3. Similarly, one can take F = C n and G = GL(n, C) to get rank n complex vector bundles. For example, i M is a complex maniold, the tangent bundle T M is a complex vector bundle. 4. I F = C n and G = U(n), we get real vector bundles with a hermitian structure. We also mention here the ollowing act: Theorem 1.8. A iber bundle has the homotopy liting property with respect to all CW complexes (i.e., it is a Serre ibration). Moreover, iber bundles over paracompact spaces are ibrations. Deinition 1.9 (Bundle homomorphism). Fix a topological group G acting eectively on a space F. A homomorphism between bundles E B and E B with group G and iber F is a pair (, ˆ) o continuous maps, with : B B and ˆ : E E, such that: 1. the diagram E ˆ E commutes, i.e., ˆ =. B B 4
5 2. i {(U α, h α )} α is a trivializing atlas o and {(V β, H β )} β is a trivializing atlas o, then the ollowing diagram commutes: (V β 1 (U α )) F H β 1 (V β 1 (U α )) ˆ 1 (U α ) h α U α F pr 1 V β 1 (U α ) U α pr 1 and there exist unctions d αβ : V β 1 (U α ) G such that or x V β 1 (U α ) and m F we have: h α ˆ H 1 β (x, m) = ((x), d αβ(x) m). An isomorphism o iber bundles is a bundle homomorphism (, ˆ) which admits a map (g, ĝ) in the reverse direction so that both composites are the identity. Remark Gauge transormations o a bundle : E B are bundle maps rom to itsel over the identity o the base, i.e., corresponding to continuous map g : E E so that g =. By deinition, such g restricts to an isomorphism given by the action o an element o the structure group on each iber. The set o all gauge transormations orms a group. Proposition Given unctions d αβ : V β 1 (U α ) G and d α β : V β 1 (U α ) G as in (2) above or dierent trivializing charts o and resp., then or any x V β V β 1 (U α U α ), we have d α β (x) = g α α((x)) d αβ (x) g ββ (x) (1.1) in G, where g α α are transition unctions or and g ββ are transition unctions or, Proo. Exercise. The unctions {d αβ } determine bundle maps in the ollowing sense: Theorem Given a map : B B and bundles E B, E B, a map o bundles (, ˆ) : exists i and only i there exist continuous maps {d αβ } as above, satisying (1.1). Proo. Exercise. Theorem Every bundle map ˆ over = id B is an isomorphism. In particular, gauge transormations are automorphisms. Proo Sketch. Let d αβ : V β U α G be the maps given by the bundle map ˆ : E E. So, i d α β : V β U α G is given by a dierent choice o trivializing charts, then (1.1) holds on V β V β U α U α, i.e., d α β (x) = g α α(x) d αβ (x) g ββ (x) (1.2) 5
6 in G, where g α α are transition unctions or and g ββ are transition unctions or. Let us now invert (1.2) in G, and set d βα (x) = d 1 αβ (x) to get: d β α (x) = g β β(x) d βα (x) g αα (x). So {d βα } are as in Deinition 1.9 and satisy (1.1). Theorem 1.12 implies that there exists a bundle map ĝ : E E over id B. We claim that ĝ is the inverse ˆ 1 o ˆ, and this can be checked locally as ollows: (x, m) ˆ (x, dαβ (x) m) ĝ (x, d βα (x) (d αβ (x) m)) = (x, d βα (x)d αβ (x) m) = (x, m). }{{} e G So ĝ ˆ = id E. Similarly, ˆ ĝ = id E One way in which iber bundle homomorphisms arise is rom the pullback (or the induced bundle) construction. Deinition 1.14 (Induced Bundle). Given a bundle E B with group G and iber F, and a continuous map : X B, we deine E := {(x, e) X E (x) = (e)}, with projections : E X, (x, e) x, and ˆ : E E, (x, e) e, so that the ollowing diagram commutes: E E e X x B (x) is called the induced bundle under or the pullback o by, and as we show below it comes equipped with a bundle map (, ˆ) :. The above deinition is justiied by the ollowing result: Theorem (a) : E X is a iber bundle with group G and iber F. (b) (, ˆ) : is a bundle map. 6
7 Proo Sketch. Let {(U α, h α )} α be a trivializing atlas o, and consider the ollowing commutative diagram: ( ) 1 ( 1 (U α )) 1 (U α ) U α F h α 1 (U α ) U α We have Deine by ( ) 1 ( 1 (U α )) = {(x, e) 1 (U α ) 1 (U α ) (x) = (e)}. }{{} =U α F k α : ( ) 1 ( 1 (U α )) 1 (U α ) F (x, e) (x, pr 2 (h α (e))). Then it is easy to check that k α is a homeomorphism (with inverse kα 1 (x, m) = (x, h 1 α ((x), m)), and in act the ollowing assertions hold: (i) {( 1 (U α ), k α )} α is a trivializing atlas o. (ii) the transition unctions o are g αβ := g αβ, i.e., g αβ (x) = g αβ ((x)) or any x 1 (U α U β ). Remark It is easy to see that ( g) = g ( ) and (id B ) =. Moreover, the pullback o a trivial bundle is a trivial bundle. As we shall see later on, the ollowing important result holds: Theorem Given a ibre bundle : E B with group G and iber F, and two homotopic maps g : X B, there is an isomorphism = g o bundles over X. (In short, induced bundles under homotopic maps are isomorphic.) As a consequence, we have: Corollary A iber bundle over a contractible space B is trivial. Proo. Since B is contractible, id B is homotopic to the constant map ct. Let b := Image(ct) i B, 7
8 so i ct id B. We have a diagram o maps and induced bundles: ct i E i E E ct i B ct id B i {b} i B Theorem 1.17 then yields: = (id B ) = ct i. Since any iber bundle over a point is trivial, we have that i = {b} F is trivial, hence = ct i = B F is also trivial. Proposition I E E B is a bundle map, then = as bundles over B. Proo. Deine h : E E by e ( (e ), (e )) B E. This is well-deined, i.e., h(e ) E, since ( (e )) = ( (e )). It is easy to check that h provides the desired bundle isomorphism over B. B E h E ˆ E B B Example We can now show that the set o isomorphism classes o bundles over S n with group G and iber F is isomorphic to n 1 (G). Indeed, let us cover S n with two contractible sets U + and U obtained by removing the south, resp., north pole o S n. Let i ± : U ± S n be the inclusions. Then any bundle over S n is trivial when restricted to U ±, that is, i ± = U ± F. In particular, U ± provides a trivializing cover (atlas) or, and any such bundle is completely determined by the transition unction g ± : U + U S n 1 G, i.e., by an element in n 1 (G). More generally, we aim to classiy iber bundles on a given topological space. Let B (X, G, F, ρ) denote the isomorphism classes (over id X ) o iber bundles on X with group G 8
9 and iber F, and G-action on F given by ρ. I : X X is a continuous map, the pullback construction deines a map so that (id X ) = id and ( g) = g. : B (X, G, F, ρ) B (X, G, F, ρ) 2 Principle Bundles As we will see later on, the iber F doesn t play any essential role in the classiication o iber bundle, and in act it is enough to understand the set P (X, G) := B (X, G, G, m G ) o iber bundles with group G and iber G, where the action o G on itsel is given by the multiplication m G o G. Elements o P (X, G) are called principal G-bundles. O particular importance in the classiication theory o such bundles is the universal principal G-bundle G EG BG, with contractible total space EG. Example 2.1. Any regular cover p : E X is a principal G-bundle, with group G = 1 (X) p 1 (E). Here G is given the discrete topology. In particular, the universal covering X X is a principal 1 (X)-bundle. Example 2.2. Any ree (right) action o a inite group G on a (Hausdor) space E gives a regular cover and hence a principal G-bundle E E/G. More generally, we have the ollowing: Theorem 2.3. Let : E X be a principal G-bundle. Then G acts reely and transitively on the right o E so that E G = X. In particular, is the quotient (orbit) map. Proo. We will deine the action locally over a trivializing chart or. Let U α be a trvializing open in X with trivializing homeomorphism h α : 1 (U α ) = U α G. We deine a right action on G on 1 (U α ) by 1 (U α ) G 1 (U α ) = U α G (e, g) e g := h 1 α ( (e), pr 2 (h α (e)) g) Let us show that this action can be globalized, i.e., it is independent o the choice o the trivializing open U α. I (U β, h β ) is another trivializing chart in X so that e 1 (U α U β ), we need to show that e g = h 1 β ( (e), pr 2 (h β (e)) g), or equivalently, h 1 α ( (e), pr 2 (h α (e)) g) = h 1 β ( (e), pr 2 (h β (e)) g). (2.1) 9
10 Ater applying h α and using the transition unction g αβ or (e) U α U β, (2.1) becomes ( (e), pr 2 (h α (e)) g) = h α h 1 β ( (e), pr 2 (h β (e)) g) = ( (e), g αβ ((e)) (pr 2 (h β (e)) g)), (2.2) which is guaranteed by the deinition o an atlas or. It is easy to check locally that the action is ree and transitive. Moreover, E G is locally given as U α G G = Uα, and this local quotient globalizes to X. The converse o the above theorem holds in some important cases. Theorem 2.4. Let E be a compact Hausdor space and G a compact Lie group acting reely on E. Then the orbit map E E/G is a principal G-bundle. Corollary 2.5. Let G be a Lie group, and let H < G be a compact subgroup. Then the projection onto the orbit space : G G/H is a principal H-bundle. Let us now ix a G-space F. We deine a map P (X, G) B (X, G, F, ρ) as ollows. Start with a principal G bundle : E X, and recall rom the previous theorem that G acts reely on the right on E. Since G acts on the let on F, we have a let G-action on E F given by: g (e, ) (e g 1, g ). Let E G F := E F G be the corresponding orbit space, with projection map ω : E G F E G = X itting into a commutative diagram E F (2.3) E pr 1 X ω E F G Deinition 2.6. The projection ω := G F : E G F X is called the associated bundle with iber F. The terminology in the above deinition is justiied by the ollowing result. Theorem 2.7. ω : E G F X is a iber bundle with group G, iber F, and having the same transition unctions as. Moreover, the assignment ω := G F deines a one-to-one correspondence P (X, G) B (X, G, F, ρ). 10
11 Proo. Let h α : 1 (U α ) U α G be a trivializing chart or. Recall that or e 1 (U α ), F and g G, i we set h α (e) = ((e), h) U α G, then G acts on the right on 1 (U α ) by acting on the right on h = pr 2 (h α (e)). Then we have by the diagram (2.3) that Let us deine by ω 1 (U α ) = 1 (U α ) F (e, ) (e g 1, g ) = U α G F (u, h, ) (u, hg 1, g ). k α : ω 1 (U α ) U α F [(u, h, )] (u, h ). This is a well-deined map since [(u, hg 1, g )] (u, hg 1 g ) = (u, h ). It is easy to check that k a l is a trivializing chart or ω with inverse induced by U α F U α G F, (u, ) (u, id G, ). It is clear that ω and have the same transition unctions as they have the same trivializing opens. The associated bundle construction is easily seen to be unctorial in the ollowing sense. Proposition 2.8. I E X E X is a map o principal G-bundles (so is a G-equivariant map, i.e., (e g) = (e) g), then there is an induced map o associated bundles with iber F, E G F X G id F E G F X Example 2.9. Let : S 1 S 1, z z 2 be regarded as a principal Z/2-bundle, and let F = [ 1, 1]. Let Z/2 = {1, 1} act on F by multiplication. Then the bundle associated to with iber F = [ 1, 1] is the Möbius strip S 1 Z/2 [ 1, 1] = S1 [ 1, 1] (x, t) (a(x), t), with a : S 1 S 1 denoting the antipodal map. Similarly, the bundle associated to with iber F = S 1 is the Klein bottle. Let us now get back to proving the ollowing important result. Theorem Let : E Y be a iber bundle with group G and iber F, and let g : X Y be two homotopic maps. Then = g over id X. 11
12 It is o course enough to prove the theorem in the case o principal G-bundles. The idea o proo is to construct a bundle map over id X between and g : E? g E X So we irst need to understand maps o principal G-bundles, i.e., to solve the ollowing problem: given two principal G-bundles bundles E 1 1 X and E2 2 Y, describe the set maps( 1, 2 ) o bundle maps E 1 1 ˆ E 2 X Y Since G acts on the right o E 1 and E 2, we also get an action on the let o E 2 by g e 2 := e 2 g 1. Then we get an associated bundle o 1 with iber E 2, namely We have the ollowing result: ω := 1 G E 2 : E 1 G E 2 X. Theorem Bundle maps rom 1 to 2 are in one-to-one correspondence to sections o ω. Proo. We work locally, so it suices to consider only trivial bundles. Given a bundle map (, ) : 1 2, let U Y open, and V 1 (U) open, so that the ollowing diagram commutes (this is the bundle maps in trivializing charts) 2 V G U G V 1 2 U We deine a section σ in (V G) G (U G) σ ω as ollows. For e 1 V G, with x = 1 (e 1 ) V, we set V σ(x) = [e 1, (e 1 )]. 12
13 This map is well-deined, since or any g G we have: [e 1 g, (e 1 g)] = [e 1 g, (e 1 ) g] = [e 1 g, g 1 (e 1 )] = [e 1, (e 1 )]. Now, it is an exercise in point-set topology (using the local deinition o a bundle map) to show that σ is continuous. Conversely, given a section o E 1 G E ω 2 X, we deine a bundle by (, ) by (e 1 ) = e 2, where σ( 1 (e 1 )) = [(e 1, e 2 )]. Note that this is an equivariant map because [e 1 g, e 2 g] = [e 1 g, g 1 e 2 ] = [e 1, e 2 ], hence (e 1 g) = e 2 g = (e 1 ) g. Thus descends to a map : X Y on the orbit spaces. We leave it as an exercise to check that (, ) is indeed a bundle map, i.e., to show that locally (v, g) = ((v), d(v)g) with d(v) G and d : V G a continuous unction. The ollowing result will be needed in the proo o Theorem Lemma Let : E X I be a bundle, and let 0 := i 0 : E 0 X be the pullback o under i 0 : X X I, x (x, 0). Then = (pr 1 ) 0 = 0 id I, where pr 1 : X I X is the projection map. Proo. It suices to ind a bundle map (pr 1, pr 1 ) so that the ollowing diagram commutes î 0 E 0 E pr 1 E 0 0 X i 0 X I pr 1 X By Theorem 2.11, this is equivilant to the existence o a section σ o ω : E G E 0 X I. Note that there exists a section σ 0 o ω 0 : E 0 G E 0 X = X {0}, corresponding to the bundle map (id X, id E0 ) : 0 0. Then composing σ 0 with the top inclusion arrow, we get the ollowing diagram X {0} σ 0 E G E 0 X I s id ω X I Since ω is a ibration, by the homotopy liting property one can extend sσ 0 to a section σ o ω. We can now inish the proo o Theorem
14 Proo o Theorem Let H : X I Y be a homotopy between and g, with H(x, 0) = (x) and H(x, 1) = g(x). Consider the induced bundle H over X I. Then we have the ollowing diagram. E H E Ĥ E g E H X {0} g i 0 X I X {1} Since = H (, 0), we get = i 0H. By Lemma 2.12, H = pr 1 ( ) = pr 1 (g ), and thus = i 0H = i 0 pr 1 g = g. We conclude this section with the ollowing important consequence o Theorem 2.11 Corollary A principle G-bundle : E X is trivial i and only i has a section. Proo. The bundle is trivial i and only i = ct, with ct : X point the constant map, and : G point the trvial bundle over a point space. This is equivalent to saying that there is a bundle map E G X i 1 ct point or, by Theorem 2.11, to the existence o a section o the bundle ω : E G G X. On the other hand, ω =, since E G G X looks locally like H pr 1 Y X 1 (U α ) G = U α G G (u, g1, g 2 ) (u, g 1 g 1, gg 2 ) = Uα G, with the last homeomorphism deined by [(u, g 1, g 2 )] (u, g 1 g 2 ). Altogether, is trivial i and only i : E X has a section. 3 Classiication o principal G-bundles Let us assume or now that there exists a principal G-bundle G : EG BG, with contractible total space EG. As we will see below, such a bundle plays an essential role in the classiication theory o principal G-bundles. Its base space BG turns out to be unique up to homotopy, and it is called the classiying space or principal G-bundles due to the ollowing undamental result: 14
15 Theorem 3.1. I X is a CW-complex, there exists a bijective correspondence Φ : P(X, G) = [X, BG] G Proo. By Theorem 2.10, Φ is well-deined. Let us next show that Φ is onto. Let P(X, G), : E X. We need to show that = G or some map : X BG, or equivalently, that there is a bundle map (, ) : G. By Theorem 2.11, this is equivalent to the existence o a section o the bundle E G EG X with iber EG. Since EG is contractible, such a section exists by the ollowing: Lemma 3.2. Let X be a CW complex, and : E X B (X, G, F, ρ) with i (F ) = 0 or all i 0. I A X is a subcomplex, then every section o over A extends to a section deined on all o X. In particular, has a section. Moreover, any two sections o are homotopic. Proo. Given a section σ 0 : A E o over A, we extend it to a section σ : X E o over X by using induction on the dimension o cells in X A. So it suices to assume that X has the orm X = A φ e n, where e n is an n-cell in X A, with attaching map φ : e n A. Since e n is contractible, is trivial over e n, so we have a commutative diagram e n σ 0 1 (e n ) e n h pr 1 = e n F σ with h : 1 (e n ) e n F the trivializing chart or over e n, and σ to be deined. Ater composing with h, we regard the restriction o σ 0 over e n as given by σ 0 (x) = (x, τ 0 (x)) e n F, with τ 0 : e n = S n 1 F. Since n 1 (F ) = 0, τ 0 extends to a map τ : e n F which can be used to extend σ 0 over e n by setting σ(x) = (x, τ(x)). Ater composing with h 1, we get the desired extension o σ 0 over e n. Let us now assume that σ and σ are two sections o. To ind a homotopy between σ and σ, it suices to construct a section Σ o id I : E I X I. Indeed, i such Σ exists, then Σ(x, t) = (σ t (x), t), and σ t provides the desired homotopy. Now, by regarding σ as a section o id I over X {0}, and σ as a section o id I over X {1}, the question reduces to constructing a section o id I, which extends the section over X {0, 1} deined by (σ, σ ). This can be done as in the irst part o the proo. 15
16 In order to inish the proo o Theorem 3.1, it remains to show that Φ is a one-to-one map. I 0 = G = g G = 1, we will show that g. Note that we have the ollowing commutative diagrams: E 0 = E G 0 X = X {0} E G G B G E 0 = E1 = g E G E G 0 X = X {1} ĝ G g B G where we regard ĝ as deined on E 0 via the isomorphism 0 = 1. By putting together the above diagrams, we have a commutative diagram E 0 I 0 Id α=(,0) (ĝ,1) E 0 {0, 1} E G 0 {0,1} G X I X {0, 1} α=(,0) (g,1) B G Thereore, it suices to extend (α, α) to a bundle map (H, Ĥ) : 0 Id G, and then H will provide the desired homotopy g. By Theorem 2.11, such a bundle map (H, Ĥ) corresponds to a section σ o the iber bundle ω : (E 0 I) G E G X I. On the other hand, the bundle map (α, α) already gives a section σ 0 o the iber bundle ω 0 : (E 0 {0, 1}) G E G X {0, 1}, which under the obvious inclusion (E 0 {0, 1}) G E G (E 0 I) G E G can be regarded as a section o ω over the subcomplex X {0, 1}. Since EG is contractible, Lemma 3.2 allows us to extend σ 0 to a section σ o ω deined on X I, as desired. Example 3.3. We give here a more conceptual reasoning or the assertion o Example By Theorem 3.1, we have B(S n, G, F, ρ) = P(S n, G) = [S n, BG] = n (BG) = n 1 (G), where the last isomorphism ollows rom the homotopy long exact sequence or G, since EG is contractible. Back to the universal principal G-bundle, we have the ollowing 16
17 Theorem 3.4. Let G be a locally compact topological group. Then a universal principal G- bundle G : EG BG exists (i.e., satisying i (EG) = 0 or all i 0), and the construction is unctorial in the sense that a continuous group homomorphism µ : G H induces a bundle map (Bµ, Eµ) : G H. Moreover, the classiying space B G is unique up to homotopy. Proo. To show that BG is unique up to homotopy, let us assume that G : E G B G and G : E G B G are universal principal G-bundles. By regarding G as the universal principal G-bundle or G, we get a map : B G B G such that G = G, i.e., a bundle map: E G G B G ˆ E G G B G Similarly, y regarding G as the universal principal G-bundle or G, there exists a map g : B G B G such that G = g G. Thereore, G = g G = g G = ( g) G. On the other hand, we have G = (id BG ) G, so by Theorem 3.1 we get that g id BG. Similarly, we get g id B G, and hence : B G B G is a homotopy equivalence. We will not discuss the existence o the universal bundle here, instead we will indicate the universal G-bundle, as needed, in speciic examples. Example 3.5. Recall that we have a iber bundle O(n) V n (R ) G n (R ), (3.1) with V n (R ) contractible. In particular, the uniqueness part o Theorem 3.4 tells us that BO(n) G n (R ) is the classiying space or rank n real vector bundles. Similarly, there is a iber bundle U(n) V n (C ) G n (C ), (3.2) with V n (C ) contractible. Thereore, BU(n) G n (C ) is the classiying space or rank n complex vector bundles. Beore moving to the next example, let us mention here without proo the ollowing useul result: Theorem 3.6. Let G be an abelian group, and let X be a CW complex. There is a natural bijection T : [X, K(G, n)] H n (X, G) [] (α) where α H n (K(G, n), G) = Hom(H n (K(G, n), Z), G) is given by the inverse o the Hurewicz isomorphism G = n (K(G, n)) H n (K(G, n), Z). 17
18 Example 3.7 (Classiication o real line bundles). Let G = Z/2 and consider the principal Z/2-bundle Z/2 S RP. Since S is contractible, the uniqueness o the universal bundle yields that BZ/2 = RP. In particular, we see that RP classiies the real line (i.e., rank-one) bundles. Since we also have that RP = K(Z/2, 1), we get: P(X, Z/2) = [X, BZ/2] = [X, K(Z/2, 1)] = H 1 (X, Z/2) or any CW complex X, where the last identiication ollows rom Theorem 3.6. Let now be a real line bundle on a CW complex X, with classiying map : X RP. Since H (RP, Z/2) = Z/2[w], with w a generator o H 1 (RP, Z/2), we get a well-deined degree one cohomology class w 1 () := (w) = called the irst Stieel-Whitney class o. The bijection P(X, Z/2) H 1 (X, Z/2) is then given by w 1 (), so real line bundles on X are classiied by their irst Stieel-Whitney classes. Example 3.8 (Classiication o complex line bundles). Let G = S 1 and consider the principal S 1 -bundle S 1 S CP. Since S is contractible, the uniqueness o the universal bundle yields that BS 1 = CP. In particular, as S 1 = GL(1, C), we see that CP classiies the complex line (i.e., rank-one) bundles. Since we also have that CP = K(Z, 2), we get: P(X, S 1 ) = [X, BS 1 ] = [X, K(Z, 2)] = H 2 (X, Z) or any CW complex X, where the last identiication ollows rom Theorem 3.6. Let now be a complex line bundle on a CW complex X, with classiying map : X CP. Since H (CP, Z) = Z[c], with c a generator o H 2 (CP, Z), we get a well-deined degree two cohomology class c 1 () := (c) called the irst Chern class o. The bijection P(X, S 1 = ) H 2 (X, Z) is then given by c 1 (), so complex line bundles on X are classiied by their irst Chern classes. Remark 3.9. I X is any orientable closed oriented surace, then H 2 (X, Z) = Z, so Example 3.8 shows that isomorphism classes o complex line bundles on X are in bijective correspondence with the set o integers. On the other hand, i X is a non-orientable closed surace, then H 2 (X, Z) = Z/2, so there are only two isomorphism classes o complex line bundles on such a surace. 4 Exercises 1. Let p : S 2 RP 2 be the (oriented) double cover o RP 2. Since RP 2 is a non-orientable surace, we know by Remark 3.9 that there are only two isomorphism classes o complex line bundles on RP 2 : the trivial one, and a non-trivial complex line bundle which we denote by 18
19 : E RP 2. On the other hand, since S 2 is a closed orientable surace, the isomorphism classes o complex line bundles on S 2 are in bijection with Z. Which integer corresponds to complex line bundle p : p E S 2 on S 2? 2. Consider a locally trivial iber bundle S 2 E S 2. Recall that such can be regarded as a iber bundle with structure group G = Homeo(S 2 ) = SO(3). By the classiication Theorem 3.1, SO(3)-bundles over S 2 correspond to elements in [S 2, BSO(3)] = 2 (BSO(3)) = 1 (SO(3)). (a) Show that 1 (SO(3)) = Z/2. (Hint: Show that SO(3) is homeomorphic to RP 3.) (b) What is the non-trivial SO(3)-bundle over S 2? 3. Let : E X be a principal S 1 -bundle over the simply-connected space X. Let a H 1 (S 1, Z) be a generator. Show that c 1 () = d 2 (a), where d 2 is the dierential on the E 2 -page o the Leray-Serre spectral sequence associated to, i.e., E p,q 2 = H p (X, H q (S 1 )) H p+q (E, Z). 4. By the classiication Theorem 3.1, (isomorphism classes o) S 1 -bundles over S 2 are given by [S 2, BS 1 ] = 2 (BS 1 ) = 1 (S 1 ) = Z and this correspondence is realized by the irst Chern class, i.e., c 1 (). (a) What is the irst Chern class o the Hop bundle S 1 S 3 S 2? (b) What is the irst Chern class o the sphere (or unit) bundle o the tangent bundle T S 2? (c) Construct explicitely the S 1 -bundle over S 2 corresponding to n Z. (Hint: Think o lens spaces, and use the above Exercise 3.) 19
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6
CLASS NOTES MATH 527 (SPRING 2011) WEEK 6 BERTRAND GUILLOU 1. Mon, Feb. 21 Note that since we have C() = X A C (A) and the inclusion A C (A) at time 0 is a coibration, it ollows that the pushout map i
More information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
More informationMath 216A. A gluing construction of Proj(S)
Math 216A. A gluing construction o Proj(S) 1. Some basic deinitions Let S = n 0 S n be an N-graded ring (we ollows French terminology here, even though outside o France it is commonly accepted that N does
More informationMath 248B. Base change morphisms
Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback
More informationIn the index (pages ), reduce all page numbers by 2.
Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by
More informationLECTURE 6: FIBER BUNDLES
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class o ibrations given by iber bundles. Fiber bundles lay an imortant role in many geometric contexts. For examle, the Grassmaniann
More informationTangent Categories. David M. Roberts, Urs Schreiber and Todd Trimble. September 5, 2007
Tangent Categories David M Roberts, Urs Schreiber and Todd Trimble September 5, 2007 Abstract For any n-category C we consider the sub-n-category T C C 2 o squares in C with pinned let boundary This resolves
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationNOTES ON FIBER BUNDLES
NOTES ON FIBER BUNDLES DANNY CALEGARI Abstract. These are notes on fiber bundles and principal bundles, especially over CW complexes and spaces homotopy equivalent to them. They are meant to supplement
More informationL E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S
L A U R E N T I U M A X I M U N I V E R S I T Y O F W I S C O N S I N - M A D I S O N L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S i Contents 1 Basics of Homotopy
More informationLecture 19: Equivariant cohomology I
Lecture 19: Equivariant cohomology I Jonathan Evans 29th November 2011 Jonathan Evans () Lecture 19: Equivariant cohomology I 29th November 2011 1 / 13 Last lecture we introduced something called G-equivariant
More informationFiber bundles and characteristic classes
Fiber bundles and characteristic classes Bruno Stonek bruno@stonek.com August 30, 2015 Abstract This is a very quick introduction to the theory of fiber bundles and characteristic classes, with an emphasis
More informationDescent on the étale site Wouter Zomervrucht, October 14, 2014
Descent on the étale site Wouter Zomervrucht, October 14, 2014 We treat two eatures o the étale site: descent o morphisms and descent o quasi-coherent sheaves. All will also be true on the larger pp and
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN The notions o separatedness and properness are the algebraic geometry analogues o the Hausdor condition and compactness in topology. For varieties over the
More informationTHE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS
THE HOMOTOPY THEORY OF EQUIVALENCE RELATIONS FINNUR LÁRUSSON Abstract. We give a detailed exposition o the homotopy theory o equivalence relations, perhaps the simplest nontrivial example o a model structure.
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
More information2 IGOR BELEGRADEK More generally, Kapovich [Kap2] proved that there is a universal unction C(?;?) such that or any incompressible singular suraces g1,
INTERSECTION PAIRING IN HYPERBOLIC MANIFOLDS, VECTOR BUNDLES AND CHARACTERISTIC CLASSES Igor Belegradek November 10, 1996 Abstract. We present some constraints on the intersection pairing in hyperbolic
More informationSEPARATED AND PROPER MORPHISMS
SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Last quarter, we introduced the closed diagonal condition or a prevariety to be a prevariety, and the universally closed condition or a variety to be complete.
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationChern Classes and the Chern Character
Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the
More informationLIMITS AND COLIMITS. m : M X. in a category G of structured sets of some sort call them gadgets the image subset
5 LIMITS ND COLIMITS In this chapter we irst briely discuss some topics namely subobjects and pullbacks relating to the deinitions that we already have. This is partly in order to see how these are used,
More informationVALUATIVE CRITERIA BRIAN OSSERMAN
VALUATIVE CRITERIA BRIAN OSSERMAN Intuitively, one can think o separatedness as (a relative version o) uniqueness o limits, and properness as (a relative version o) existence o (unique) limits. It is not
More informationLecture on Equivariant Cohomology
Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationFiber Bundles, The Hopf Map and Magnetic Monopoles
Fiber Bundles, The Hopf Map and Magnetic Monopoles Dominick Scaletta February 3, 2010 1 Preliminaries Definition 1 An n-dimension differentiable manifold is a topological space X with a differentiable
More informationThe Ordinary RO(C 2 )-graded Cohomology of a Point
The Ordinary RO(C 2 )-graded Cohomology of a Point Tiago uerreiro May 27, 2015 Abstract This paper consists of an extended abstract of the Master Thesis of the author. Here, we outline the most important
More informationIntroduction to higher homotopy groups and obstruction theory
Introduction to higher homotopy groups and obstruction theory Michael Hutchings February 17, 2011 Abstract These are some notes to accompany the beginning of a secondsemester algebraic topology course.
More informationTHE H-PRINCIPLE, LECTURE 14: HAEFLIGER S THEOREM CLASSIFYING FOLIATIONS ON OPEN MANIFOLDS
THE H-PRINCIPLE, LECTURE 14: HAELIGER S THEOREM CLASSIYING OLIATIONS ON OPEN MANIOLDS J. RANCIS, NOTES BY M. HOYOIS In this lecture we prove the following theorem: Theorem 0.1 (Haefliger). If M is an open
More informationStable complex and Spin c -structures
APPENDIX D Stable complex and Spin c -structures In this book, G-manifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.
More informationGeometry of homogeneous spaces. Andreas Čap
Geometry of homogeneous spaces rough lecture notes Spring Term 2016 Andreas Čap Faculty of Mathematics, University of Vienna E-mail address: Andreas.Cap@univie.ac.at Contents 1. Introduction 1 2. Bundles
More informationVariations on a Casselman-Osborne theme
Variations on a Casselman-Osborne theme Dragan Miličić Introduction This paper is inspired by two classical results in homological algebra o modules over an enveloping algebra lemmas o Casselman-Osborne
More informationVALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS
VALUATIVE CRITERIA FOR SEPARATED AND PROPER MORPHISMS BRIAN OSSERMAN Recall that or prevarieties, we had criteria or being a variety or or being complete in terms o existence and uniqueness o limits, where
More informationAlgebraic Topology exam
Instituto Superior Técnico Departamento de Matemática Algebraic Topology exam June 12th 2017 1. Let X be a square with the edges cyclically identified: X = [0, 1] 2 / with (a) Compute π 1 (X). (x, 0) (1,
More informationLecture 2. x if x X B n f(x) = α(x) if x S n 1 D n
Lecture 2 1.10 Cell attachments Let X be a topological space and α : S n 1 X be a map. Consider the space X D n with the disjoint union topology. Consider further the set X B n and a function f : X D n
More informationMath 752 Week s 1 1
Math 752 Week 13 1 Homotopy Groups Definition 1. For n 0 and X a topological space with x 0 X, define π n (X) = {f : (I n, I n ) (X, x 0 )}/ where is the usual homotopy of maps. Then we have the following
More informationCHOW S LEMMA. Matthew Emerton
CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there
More informationEQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES
EQUIVARIANT COHOMOLOGY IN ALGEBRAIC GEOMETRY LECTURE TWO: DEFINITIONS AND BASIC PROPERTIES WILLIAM FULTON NOTES BY DAVE ANDERSON 1 For a Lie group G, we are looking for a right principal G-bundle EG BG,
More informationMath 6530: K-Theory and Characteristic Classes
Math 6530: K-Theory and Characteristic Classes Taught by Inna Zakharevich Notes by David Mehrle dmehrle@math.cornell.edu Cornell University Fall 2017 Last updated December 12, 2017. The latest version
More informationThe Clifford algebra and the Chevalley map - a computational approach (detailed version 1 ) Darij Grinberg Version 0.6 (3 June 2016). Not proofread!
The Cliord algebra and the Chevalley map - a computational approach detailed version 1 Darij Grinberg Version 0.6 3 June 2016. Not prooread! 1. Introduction: the Cliord algebra The theory o the Cliord
More informationCATEGORIES. 1.1 Introduction
1 CATEGORIES 1.1 Introduction What is category theory? As a irst approximation, one could say that category theory is the mathematical study o (abstract) algebras o unctions. Just as group theory is the
More information(1) Let π Ui : U i R k U i be the natural projection. Then π π 1 (U i ) = π i τ i. In other words, we have the following commutative diagram: U i R k
1. Vector Bundles Convention: All manifolds here are Hausdorff and paracompact. To make our life easier, we will assume that all topological spaces are homeomorphic to CW complexes unless stated otherwise.
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
More information1 Whitehead s theorem.
1 Whitehead s theorem. Statement: If f : X Y is a map of CW complexes inducing isomorphisms on all homotopy groups, then f is a homotopy equivalence. Moreover, if f is the inclusion of a subcomplex X in
More informationDivision Algebras and Parallelizable Spheres, Part II
Division Algebras and Parallelizable Spheres, Part II Seminartalk by Jerome Wettstein April 5, 2018 1 A quick Recap of Part I We are working on proving the following Theorem: Theorem 1.1. The following
More informationWe then have an analogous theorem. Theorem 1.2.
1. K-Theory of Topological Stacks, Ryan Grady, Notre Dame Throughout, G is sufficiently nice: simple, maybe π 1 is free, or perhaps it s even simply connected. Anyway, there are some assumptions lurking.
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationMATH 215B HOMEWORK 5 SOLUTIONS
MATH 25B HOMEWORK 5 SOLUTIONS. ( marks) Show that the quotient map S S S 2 collapsing the subspace S S to a point is not nullhomotopic by showing that it induces an isomorphism on H 2. On the other hand,
More informationTHE BORSUK-ULAM THEOREM FOR GENERAL SPACES
THE BORSUK-ULAM THEOREM FOR GENERAL SPACES PEDRO L. Q. PERGHER, DENISE DE MATTOS, AND EDIVALDO L. DOS SANTOS Abstract. Let X, Y be topological spaces and T : X X a free involution. In this context, a question
More informationThe Sectional Category of a Map
arxiv:math/0403387v1 [math.t] 23 Mar 2004 The Sectional Category o a Map M. rkowitz and J. Strom bstract We study a generalization o the Svarc genus o a iber map. For an arbitrary collection E o spaces
More informationOn Eilenberg-MacLanes Spaces (Term paper for Math 272a)
On Eilenberg-MacLanes Spaces (Term paper for Math 272a) Xi Yin Physics Department Harvard University Abstract This paper discusses basic properties of Eilenberg-MacLane spaces K(G, n), their cohomology
More informationHomework 3 MTH 869 Algebraic Topology
Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }
More informationTopological K-theory, Lecture 3
Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ
More informationDIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015
DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry
More informationINTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY
INTRODUCTION TO EQUIVARIANT COHOMOLOGY THEORY YOUNG-HOON KIEM 1. Definitions and Basic Properties 1.1. Lie group. Let G be a Lie group (i.e. a manifold equipped with differentiable group operations mult
More informationEQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms
EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be
More informationERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY
ERRATA FOR INTRODUCTION TO SYMPLECTIC TOPOLOGY DUSA MCDUFF AND DIETMAR A. SALAMON Abstract. These notes correct a few typos and errors in Introduction to Symplectic Topology (2nd edition, OUP 1998, reprinted
More informationFibre Bundles: Trivial or Not?
Rijksuniversiteit Groningen Bachelor thesis in mathematics Fibre Bundles: Trivial or Not? Timon Sytse van der Berg Prof. Dr. Henk W. Broer (first supervisor) Prof. Dr. Jaap Top (second supervisor) July
More informationWe have the following immediate corollary. 1
1. Thom Spaces and Transversality Definition 1.1. Let π : E B be a real k vector bundle with a Euclidean metric and let E 1 be the set of elements of norm 1. The Thom space T (E) of E is the quotient E/E
More informationTHE STEENROD ALGEBRA. The goal of these notes is to show how to use the Steenrod algebra and the Serre spectral sequence to calculate things.
THE STEENROD ALGEBRA CARY MALKIEWICH The goal of these notes is to show how to use the Steenrod algebra and the Serre spectral sequence to calculate things. 1. Brown Representability (as motivation) Let
More informationGroupoids and Orbifold Cohomology, Part 2
Groupoids and Orbifold Cohomology, Part 2 Dorette Pronk (with Laura Scull) Dalhousie University (and Fort Lewis College) Groupoidfest 2011, University of Nevada Reno, January 22, 2012 Motivation Orbifolds:
More information1.1 Definition of group cohomology
1 Group Cohomology This chapter gives the topological and algebraic definitions of group cohomology. We also define equivariant cohomology. Although we give the basic definitions, a beginner may have to
More informationDEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS
DEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS NORBIL CORDOVA, DENISE DE MATTOS, AND EDIVALDO L. DOS SANTOS Abstract. Yasuhiro Hara in [10] and Jan Jaworowski in [11] studied, under certain
More informationGENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS
GENERALIZED ABSTRACT NONSENSE: CATEGORY THEORY AND ADJUNCTIONS CHRIS HENDERSON Abstract. This paper will move through the basics o category theory, eventually deining natural transormations and adjunctions
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationLecture 6: Classifying spaces
Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E
More informationSMSTC Geometry & Topology 1 Assignment 1 Matt Booth
SMSTC Geometry & Topology 1 Assignment 1 Matt Booth Question 1 i) Let be the space with one point. Suppose X is contractible. Then by definition we have maps f : X and g : X such that gf id X and fg id.
More informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2-orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Path-lifting, covering homotopy Locally smooth actions Smooth actions
More informationRepresentation Theory of Hopf Algebroids. Atsushi Yamaguchi
Representation Theory o H Algebroids Atsushi Yamaguchi Contents o this slide 1. Internal categories and H algebroids (7p) 2. Fibered category o modules (6p) 3. Representations o H algebroids (7p) 4. Restrictions
More informationPICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
More informationMath 637 Topology Paulo Lima-Filho. Problem List I. b. Show that a contractible space is path connected.
Problem List I Problem 1. A space X is said to be contractible if the identiy map i X : X X is nullhomotopic. a. Show that any convex subset of R n is contractible. b. Show that a contractible space is
More informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
More informationMATH8808: ALGEBRAIC TOPOLOGY
MATH8808: ALGEBRAIC TOPOLOGY DAWEI CHEN Contents 1. Underlying Geometric Notions 2 1.1. Homotopy 2 1.2. Cell Complexes 3 1.3. Operations on Cell Complexes 3 1.4. Criteria for Homotopy Equivalence 4 1.5.
More informationTopological Methods in Algebraic Geometry
Topological Methods in Algebraic Geometry Lectures by Burt Totaro Notes by Tony Feng Michaelmas 2013 Preface These are live-texed lecture notes for a course taught in Cambridge during Michaelmas 2013 by
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationMATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS
Key Problems 1. Compute π 1 of the Mobius strip. Solution (Spencer Gerhardt): MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS In other words, M = I I/(s, 0) (1 s, 1). Let x 0 = ( 1 2, 0). Now
More informationJoseph Muscat Categories. 1 December 2012
Joseph Muscat 2015 1 Categories joseph.muscat@um.edu.mt 1 December 2012 1 Objects and Morphisms category is a class o objects with morphisms : (a way o comparing/substituting/mapping/processing to ) such
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationCONFIGURATION SPACES AND BRAID GROUPS
CONFIGURATION SPACES AND BRAID GROUPS FRED COHEN AND JONATHAN PAKIANATHAN Abstract. The main thrust of these notes is 3-fold: (1) An analysis of certain K(π, 1) s that arise from the connections between
More informationarxiv:math/ v1 [math.ag] 18 Oct 2003
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉS-GASTESI
More informationDEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS
DEGREE OF EQUIVARIANT MAPS BETWEEN GENERALIZED G-MANIFOLDS NORBIL CORDOVA 1, DENISE DE MATTOS 2, AND EDIVALDO L. DOS SANTOS 3 Abstract. Yasuhiro Hara in [10] and Jan Jaworowski in [11] studied, under certain
More informationThe Space of Negative Scalar Curvature Metrics
The Space o Negative Scalar Curvature Metrics Kyler Siegel November 12, 2012 1 Introduction Review: So ar we ve been considering the question o which maniolds admit positive scalar curvature (p.s.c.) metrics.
More informationCobordant differentiable manifolds
Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated
More informationCELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA. Contents 1. Introduction 1
CELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA PAOLO DEGIORGI Abstract. This paper will first go through some core concepts and results in homology, then introduce the concepts of CW complex, subcomplex
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationLECTURE 26: THE CHERN-WEIL THEORY
LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if
More informationIntroduction to. Riemann Surfaces. Lecture Notes. Armin Rainer. dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D
Introduction to Riemann Surfaces Lecture Notes Armin Rainer dim H 0 (X, L D ) dim H 0 (X, L 1 D ) = 1 g deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held
More informationNAP Module 4. Homework 2 Friday, July 14, 2017.
NAP 2017. Module 4. Homework 2 Friday, July 14, 2017. These excercises are due July 21, 2017, at 10 pm. Nepal time. Please, send them to nap@rnta.eu, to laurageatti@gmail.com and schoo.rene@gmail.com.
More informationMATH 215B. SOLUTIONS TO HOMEWORK (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements.
MATH 215B. SOLUTIONS TO HOMEWORK 2 1. (6 marks) Construct a path connected space X such that π 1 (X, x 0 ) = D 4, the dihedral group with 8 elements. Solution A presentation of D 4 is a, b a 4 = b 2 =
More informationMath 751 Week 6 Notes
Math 751 Week 6 Notes Joe Timmerman October 26, 2014 1 October 7 Definition 1.1. A map p: E B is called a covering if 1. P is continuous and onto. 2. For all b B, there exists an open neighborhood U of
More informationTCC Homological Algebra: Assignment #3 (Solutions)
TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate
More informationSubgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.
Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this
More informationBUNDLES, STIEFEL WHITNEY CLASSES, & BRAID GROUPS
BUNDLES, STIEFEL WHITNEY CLASSES, & BRAID GROUPS PETER J. HAINE Abstract. We explain the basics of vector bundles and principal G-bundles, as well as describe the relationship between vector bundles and
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationThe cohomology of orbit spaces of certain free circle group actions
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 1, No. 1, February 01, pp. 79 86. c Indian Academy of Sciences The cohomology of orbit spaces of certain free circle group actions HEMANT KUMAR SINGH and TEJ BAHADUR
More informationPart II. Algebraic Topology. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 Paper 3, Section II 18I The n-torus is the product of n circles: 5 T n = } S 1. {{.. S } 1. n times For all n 1 and 0
More informationLogarithm of a Function, a Well-Posed Inverse Problem
American Journal o Computational Mathematics, 4, 4, -5 Published Online February 4 (http://www.scirp.org/journal/ajcm http://dx.doi.org/.436/ajcm.4.4 Logarithm o a Function, a Well-Posed Inverse Problem
More informationQUALIFYING EXAM, Fall Algebraic Topology and Differential Geometry
QUALIFYING EXAM, Fall 2017 Algebraic Topology and Differential Geometry 1. Algebraic Topology Problem 1.1. State the Theorem which determines the homology groups Hq (S n \ S k ), where 1 k n 1. Let X S
More information