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Vorlesung: Introduction to Homotopy Theory

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Vorlesung: Introduction to Homotopy Theory Vorlesung: Introduction to homotopy theory Ulrich Bunke Contents 1 Covering theory 3 1.1 Paths and path components . .3 1.2 Lifting properties . .4 1.3 The fundamental groupoid . .7 1.4 The fundamental group . .9 1.5 Coverings and representations of the fundamental groupoid . 11 1.6 Specialization to the fundamental group . 14 1.7 Properties of coverings . 16 1.8 The analogy with Galois theory . 20 1.9 Van Kampen . 22 1.10 Flat vector bundles . 25 1.11 Lifting of maps . 29 2 Die Homotopiekategorie 30 2.1 Basic constructions . 30 2.2 Pairs and pointed spaces . 33 2.3 Suspension and loops . 37 2.4 The homotopy category . 38 2.5 H and co-H-spaces . 40 3 Cofibre sequences 45 3.1 The mapping cone . 45 3.2 The mapping cone sequence . 49 3.3 Functoriality properties of the mapping cone . 50 3.4 The long mapping cone sequence . 53 4 Cohomology 56 4.1 Spectra . 56 4.2 CW-complexes and the AHSS . 61 4.3 Calculations of cohomology . 65 1 5 Fibre sequences 67 5.1 The homotopy fibre . 67 5.2 The fibre sequence . 70 5.3 The long exact homotopy sequence . 73 6 Homotopy groups 74 6.1 Calculation of homotopy groups . 74 6.2 The Blakers-Massey theorem and applications . 77 6.3 Homotopy of classical groups . 79 6.4 Quotients . 80 6.5 Proof of the Blakers-Massey theorem . 83 7 Diverse Constructions 87 7.1 The Ω1Σ1-construction . 87 7.2 Simplicial sets, singular complex and geometric realization . 89 7.3 Simplicial spaces and classifying spaces of topological groups . 94 7.4 Classification of principal bundles . 101 7.5 Topological abelian groups . 107 7.6 Homology . 111 7.7 Hurewicz . 118 8 Aufgaben 120 2 1 Covering theory 1.1 Paths and path components We let I := [0; 1] ⊂ R denote the standard interval with its induced topology. Let Y be a topological space and y0; y1 2 Y be two points. A path in Y from y0 to y1 is a map γ : I ! Y such that γ(0) = y0 and γ(1) = y1. We consider three points y0; y1; y2 2 Y . If γ is a path from y0 to y1 and µ is a path from y1 to y2, then we can define a new path µ ◦ γ from y0 to y1 called the concatenation of γ and µ. It is given by γ(2t) t 2 [0; 1=2) (µ ◦ γ)(t) := : µ(2t − 1) t 2 [1=2; 1] We say that y0 and y1 belong to the same path component of Y if there exists a path from y0 to y1. The relation between points y0 and y1 of Y y0 and y1 belong to the same path component of Y is an equivalence relation on Y . Problem 1.1. Show this assertion. The equivalence classes with respect to this equivalence relation are called the path components of Y . The set of path components will be denoted by π0(Y ). The symbol [y] denotes the path component of Y which contains y. A space Y is called path-connected, if π0(Y ) has at most one element. Example 1.2. The standard interval I is path-connected. Example 1.3. The space Rn is path connected. Example 1.4. More generally, a manifold M is path connected if and only if the following cohomological condition is satisfied: 0 dimR HdR(M) ≤ 1 : Problem 1.5. Show this assertion. Problem 1.6. Show that for general topological spaces the condition of being path con- nected is strictly stronger than the condition of being connected. 3 Example 1.7. The subset Q ⊂ R with the induced topology is not path-connected. Every path in Q is constant. In fact, Q 3 q 7! [q] 2 π0(Q) is a bijection. Example 1.8. Let p 2 N be a prime. Then the space of p-adic numbers Zp is not path-connected. Again, every path in Zp is constant and Zp 3 x 7! [x] 2 π0(Zp) is a bijection. Example 1.9. If G is a topological group, then π0(G) is a group with operations defined on representatives by [g][h] := [gh] ; g; h 2 G: For example, ∼ π0(GL(n; R)) = Z=2Z ; where the isomorphism is given by the sign of the determinant. Problem 1.10. Show this assertion 1.2 Lifting properties We consider the following diagram of topological spaces: A / X : > i f B / Y We understand the bold part as given data. If the dotted arrow exists for all choices of horizontal arrows, then we say that f has the right lifting property (RLP) with respect to i or, equivalently, i has the left lifting property (RLP) with respect to f. We add the adjective unique if the dotted arrow is unique and abbreviate this by ULLP or URRP, respectively. We now consider the lifting of paths. Definition 1.11. A map of spaces f : X ! Y has the (unique) path lifting property, if it has the RLP (URLP) with respect to the inclusion of the beginning f0g ! I of the interval. We spell this out. We consider the diagram x0 f0g / X : = γ~ f γ I / Y 4 In this situation the pathγ ~ is called a lift of γ with beginning in x0 2 X. The map f : X ! Y has the (unique) path lifting property, if for every datum (γ; x0) as above a (unique) liftγ ~ exists. Example 1.12. Let X ! Y be a real vector bundle over a smooth manifold Y . Then X ! Y has the path lifting property. If the path is smooth, then a lift can be found using differential geometry. Indeed, one can choose a connection and defineγ ~ using the parallel transport along γ. For continuous path we use Lemma 1.14. Example 1.13. If f : X ! Y is a smooth proper submersion between manifolds, then f has the path lifting property. For smooth paths the argument is similar as in Example 1.12 using a (non-linear) con- nection. In order to lift continuous paths we observe that f : X ! Y is a locally trivial fibre bundle and apply Lemma 1.14. The condition that f is proper can not be dropped. As a counterexample let f : (−1; 1=2) ! (−1; 2) be the inclusion. The path γ : I ! (−1; 2) given by the obvious inclusion has no lift with beginning in 0. Lemma 1.14. A locally trivial fibre bundle f : X ! Y has the path lifting property. Proof. Let a diagram x0 f0g / X = γ~ f γ I / Y be given. Then the image γ can be covered by a finite number of open subsets over which the bundle is trivial. We can write the path γ as a multiple concatenation of paths which are contained in such open subsets. Since we can concatenate the lifts as well, we can reduce the problem to the case of a trivial bundle. We assume that X = Y × F and write x0 = (γ(0); f0). Then can define a lift by t 7! γ~(t) := (γ(t); f0). 2 If φ is a path in F starting in f0, then we can consider the lift t 7! (γ(t); φ(t)). This accounts for the non-uniqueness of the lift. Definition 1.15. A locally trivial fibre bundle with discrete fibres is called a covering. Example 1.16. Let X be a topological space an G be a group which acts freely and properly on X. Then the projection X ! X=G is a covering. More concrete examples of coverings are 5 n n ∼ n n 1. R ! T = R =Z 3 ∼ 2 p 2. S ! L(p; q) = S =Z (L(p; q) is the lense space) n n ∼ n 3. S ! RP = S =(Z=2Z) Lemma 1.17. If f : X ! Y is a covering, then f has the unique path lifting property. Proof. x0 f0g / X = γ~ f γ I / Y be given. As in the proof of Lemma 1.14 we can reduce to the case where the cover- ing is trivial X = Y × F ! Y for a discrete space F . The only way to lift γ is as γ~(t) := (γ(t); f0). 2 Remark 1.18. Note that the argument works if the fibre of f just has the property that every path in it is necessarily constant. An example is Y × Zp ! Y . Hence the unique path lifting property does not imply that our map is a covering. For a space A we consider URLP for the inclusion i0 : A ! I × A induced by 0 2 I.A map f : X ! Y which has the URLP with respect to this map is said to have the unique homotopy lifting property for A. Lemma 1.19. A covering has the unique homotopy lifting property for all spaces. Proof. We consider a diagram h~0 A / X : ; h~ i0 f h I × A / Y ~ For every a 2 A we can define hjI×{ag using the unique path lifting property. It remains to check that h~ is continuous. This can be done locally. Using concatenation we reduce ~ ~ to the case that f : Y × F ! Y is the projection. But then h(t; a) = (h(t; a); prF h0(a)). This map is obviously continuous. 2 6 1.3 The fundamental groupoid We consider a space X and a pair of points x0; x1 2 X. By Px0;x1 (X) we denote the set of all paths from x0 to x1.
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