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23 Introduction to Homotopy Theory 23 Introduction to homotopy theory The objects of homotopy theory are the categories of spaces and spectra Sp.Inthis overview I want to explain certain features and constructions withS these categories which will become relevant in the seminar. One needs -categories or model categories in order to capture theses objects on a technical level.1 In this overview I want to explain how one can work with spaces and spectra properly by simply obeying some rules of correct language. As a preparation I recall the notion of limits and colimits. For categories C and I (the latter assumed to be small) let CI be the functor category. There is a constant diagram functor c : C CI . ! The limit (colimit) is a (in general partially defined) right (left) adjoint of c: I I c : C ⌧ C : limI , colimI : C ⌧ C : c. We will illuminate this abstract definition by giving examples. Example 23.1. Atypicalexampleofalimitisapull-backorfibreproduct: X Y / X . ⇥Z ✏ ✏ Y / Z Atypicalexampleofacolimitisapush-out Z / X . ✏ ✏ Y / X Y tZ The quotient by an action a of a group G on X is the colimit of the diagram a G X 2, X / X/G . ⇥ pr2 2 We start with a discussion of spaces .Onecancharacterize as the presentable - category (this essentially means thatS it admits all limits and colimits)S generated by1 a point. There are many models for . S Example 23.2. One can start from topological spaces Top.Amorphismf in Top is called a weak equivalence if ⇡ (f)isabijectionand⇡ (f):⇡ (X, x) ⇡ (Y,y)are 0 n n ! n 29 isomorphisms for all n 1andx X. If we invert weak equivalences (technically this happens inside -categories),≥ then2 we get a model for the category of spaces 1 1 Top Top[W − ] . ! 'S 2 op Example 23.3. One can define a model for starting from simplicial sets sSet := Set∆ . Recall that ∆op is the category whose objectsS are the posets [n]= 0,...,n and whose morphisms are order preserving maps. For a category C we call {C∆op the} category of simplicial objects in C. In order to explain the notion of a weak equivalence in sSet we need the geometric realization. The standard simplices ∆n Rn+1 provide a functor ⇢ n ∆− :∆ Top , [n] ∆ . ! 7! Considering sets as topological spaces, a simplicial set X sSet provides a diagram of topological spaces 2 σ id m ⇤⇥ n σ:[m] [n] ∆ X[n] / [n] ∆ ∆ X[n] , (4) ! ⇥ 2 ⇥ id σ F ⇥ ⇤ F ✏ ✏ m [m] ∆ ∆ X[m] / X 2 ⇥ k k F whose colimit is the called the geometric realization of X.Amorecondensed,butequiv- alent, definition uses the notion of coend ∆op X := ∆ X. k k ⇥ Z By definition, a morphism f in sSet is a weak equivalence, if f is one in Top.Weget the desired model of spaces k k 1 sSet sSet[W − ] . ! 'S 1 1 Note that by construction descends to a functor : sSet[W − ] Top[W − ] which happens to be an equivalence.kk kk ! 2 In the following I list some of the basic properties of the category of spaces .Themodel independent notion of isomorphism is equivalence. S 1. Topological spaces or simplical sets represent spaces. 2. There exists a one-point space ,i.e.afinalobject. ⇤ 30 3. AspaceX has a functorially defined set of connected components ⇡0(X). 4. For each map x : X we have functorially associated homotopy groups ⇡n(X, x), n 1. ⇤! ≥ 5. Amorphismf : X Y between spaces is an equivalence if and only if ⇡n(f)isan isomorphism for all!n 0andbasepointsx X (for n 1). ≥ 2 ≥ 6. For two spaces X, Y we have a functorially associated mapping space Map(X, Y ) .Twomorphisms2f ,fS : X Y are equivalent if they represent the same point2 S 0 1 ! in ⇡0(Map(X, Y )). 7. In spaces we can form limits and colimits. Our language and pictures are adapted to Top. 1 Example 23.4. The fibre f − (y)ofamorphismf : X Y in Top at a point y Y can be written as a limit ! 2 Fib (f) / . y ⇤ y ✏ f ✏ X / Y In general, limits in Top do not preserve weak equivalences. In general, the fibre defined as above for a map of topological spaces does not represent the fibre of f considered as a morphism in .Forexample,wehaveaweakequivalenceofdiagrams S ' , ⇤ !⇤ 1 0 ✏ ✏ / [0, 1] / ⇤ ⇤ ⇤ but the map of limits is ,surelynotaweakequivalence. ;!⇤ One needs a modification of the construction of the fibre in order to obtain a model inside Top for the fibre taken in . In this situation one speaks of homotopy fibres. A model for the homotopy fibre is theS limit of Fib (f) / . y h ⇤ y ✏ ev ✏ X Y I 1 / Y ⇥f,Y,ev0 In general, if the morphism X Y in Top is a so-called fibration, then the fibre taken ! in Top represents the fibre in . Note that the map ev1 is the replacement of f by a fibration. S 31 For a map f : X Y of spaces we can control the homotopy groups of the fibre by the long exact sequence! ⇡n(Fiby(f), ) ⇡n(X, x) ⇡n(Y,y) ⇡n 1(Fiby(f), ) . ⇤ ! ! ! − ⇤ 2 Example 23.5. The cofibre of a map f : X Y is defined as the colimit of ! X / . ⇤ f ✏ ✏ Y / Cofib(f) Agoodmodelforthehomotopycofibreisgivenbytheconeoff X / . ⇤ (0,idX ) ✏ ✏ [0, 1] X 1 X,f Y / Cofib(f)h ⇥ t{ }⇥ Here the left vertical map is the replacement of f by a cofibration. It is difficult to control the homotopy groups of the homotopy cofibre. The appropriate invariant is homology. 2 Example 23.6. The functor ∆op colim op : C C, X X ∆ ! 7! | | is called realization. If applied to simplicial spaces one again observes that it does not preserve weak equivalences and therefore does not model the realization in . AweakequivalencepreservingmodelfortherealizationisthegeometricrealizationgivenS by the formula (4) and will be denoted by X . | |h The homotopy groups of X h can be approached via the Bousfield-Kan spectral sequence whose first page is | | 1 Es,t := ⇡s(X[t]) . 2 Example 23.7. The nerve N(C)ofacategoryisthesimplicialsetgivenby N(C)[n]:=Cat([n],C) . We have a functor N : Cat sSet.Acategoricalequivalenceinducesaweakequivalence between nerves. This provides! a natural factorization 1 1 N : Cat[W − ] sSet[W − ] . ! 'S 32 One can define the notion of a category is a diagrammatic way. Interpreting these dia- grams in topological spaces we obtain the notion of a topological category with space of objects Ob(C)andspaceofmorphismsMor(C). In this case the nerve naturally refines to a simplicial space, N(C)[0] := Ob(C) ,N(C)[1] := Mor(C) ,N(C)[2] := Mor(C) Mor(C) ,... ⇥s,Ob(C),r From now one we consider N(C)asasimplicialspace,i.e.asanobjectin ∆op .The classifying space of a topological category is the space defined by S BC := N(C) . | |2S AfunctorbetweentopologicalcategoriesC D is called a weak equivalence if N(C) N(D) is an equivalence. It induces an equivalence! of classifying spaces BC BD. ! In the seminar we will study the topological bordism category of n 1-dimensional! man- − ifolds with a tangental ✓-structure ✓.Themaingoalistoidentifytheclassifyingspace B . C 2 C✓ Example 23.8. Let G be a topological group. It gives rise to a topological cate- gory G with object and morphism space G.Theclassifyingspaceofthiscategory is the usual classifying⇤ space BG of G. A homomorphism between topological groups which is a weak equivalence induces a weak equivalence between topologically enriched categories and therefore a weak equivalence of classifying spaces. For example, the usual inclusion O(n) GL(n, R)isaweakequivalence,hencewegetanequivalence ! BO(n) BGL(n, R). Let G act' on a space X. Taking the quotient X/G in Top does not preserve weak equivalences. A good model for the homotopy quotient is X/ G := EG X, h ⇥G where EG is the geometric realization of the simplicial G-space [n] G[n] with the usual 7! simplicial structure. So the topological space EG G X is a good model for the quotient X/G taken in . ⇥ The algebraic toolS to calculate the homotopy groups of X/G is again a spectral sequence derived from the Bousfield-Kan spectral sequence with first term E1 = ⇡ (G G X) . s,t ⇠ s ⇥···⇥ ⇥ t ⇥ | {z } 2 In topological spaces we can consider: 1. apointinaspace 33 2. ahomeomorphism 3. alocallytrivialfibrebundle 4. a(modelofa)homotopyquotient In spaces we can talk about: 1. aconnectedcomponent 2. an equivalence 3. some space is equivalent to the fibre of a morphism 4. aquotient Example 23.9. There are similar notions of pointed spaces related to Top and sSet . ⇤ ⇤ The suspension ⌃X of an object X in a pointed category isS defined as a colimit⇤ X / . ⇤ ✏ ✏ / ⌃X ⇤ Note again, that the suspension in Top does not model the suspension in .Torepre- ⇤ sent the latter in Top we must use homotopy⇤ push-outs. For example, in TopS we have ⌃S0 = ,whilein we⇤ get ⌃S0 S1. ⇤ 2 ⇠ ⇤ S⇤ ' We consider the suspension in C as an endofunctor ⌃:C C ! provided it is defined on all objects. The category of spectra Sp can be characterized as the universal presentable -category generated by one object on which the suspension acts as an equivalence. More1 concrete constructions (in the world of -categories) are as 1 1 Sp [⌃− ] , 'S⇤ or as the colimit of the diagram Sp colim ⌃ ⌃ ... ' S⇤ !S⇤ ! ⇣ ⌘ These descriptions yield the main features of the category of spectra: 1. There is an adjunction ⌃1 : ⌧ Sp :⌦1 . S⇤ 1 Note that ⌃1X is just the image of X under [⌃− ] Sp.Wecall⌃1X S⇤ !S⇤ ' the suspension spectrum of X,and⌦1(E) the infinite loop space of the spectrum E.
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