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 {Cop 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

id m ⇤⇥ n :[m] [n] X[n] / [n] X[n] , (4) ! ⇥ 2 ⇥ id 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 dicult 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! manifolds with a tangental ✓-structure ✓.Themaingoalistoidentifytheclassifyingspace B . C 2 C✓

Example 23.8. Let G be a topological group. It gives rise to a topological category 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.

34 2. AspectrumE has well-deﬁned homotopy groups

2 n ⇡n(E):=⇡2(⌦1⌃ E) ,nZ ,k n. 2  We can replace 2 by any other k 2. A morphism between spectra is an equivalence if and only if it induces a morphism in homotopy groups.

3. Asequenceofpointedspacesandmaps

(Xn)n 0 , ⌃Xn Xn+1 ! deﬁnes a spectrum E by

0 1 2 E := colim ⌃ ⌃1X ⌃ ⌃1X ⌃ ⌃1X ... . 0 ! 1 ! 2 ! From this we get the formula for the homotopy groups

⇡n(E) ⇠= colimk⇡n+k(Xk) . We can also start with such a sequence of topological spaces and maps using the maps ⌃ X ⌃X X . h n ! ! n+1 4. Asequenceofmorphismsinacategoryiscalledaﬁbresequenceifateverysegment

A B C ... ···! ! ! ! the morphism A B is represents the fibre of B C.Similarlywedefinethe notion of a cofibre! sequence. In spectra the notions! of fibre and cofibre sequences coincide. AmapbetweenspectraX Y can functorially be extended to a fibre sequence ! Z X Y ⌃Z ... . ···! ! ! ! ! 5. The category of pointed spaces and spectra have symmetric monoidal structures denoted by .Thefunctor⌃1 is a symmetric monoidal functor, i.e. ^

⌃1(X Y ) ⌃1X ⌃1Y. ^ ' ^ For two spectra X, Y we have a mapping spectrum map(X, Y )suchthat

⌦1map(X, Y ) Map(X, Y ) . ' We have the rule map(X Y,Z) map(X, map(Y,Z)) . (5) ^ '

35 6. The (reduced) homology of a pointed space X with values in a spectrum E is deﬁned by H (X, E):=⇡ (⌃1X E) . ⇤ ⇤ ^

The functor ⌃1 (as a left adjoint) preserves coﬁbres. Given a map X Y of spaces we get a coﬁbre sequence !

1 ⌃ ⌃1Cofib(f) ⌃1X ⌃1Y ⌃1Cofib(f) . (6) ! ! ! If we take the product with E (which again yields a (co)fibre sequence as a conse- quence of (5)), then the associated long exact sequence is the long exact homology sequence

H 1(Cofib(f); E) H (X; E) H (Y ; E) H (Cofib(f); E) ... . ···! ⇤ ! ⇤ ! ⇤ ! ⇤ ! Note that if we want to interpret this in pointed topological spaces Top ,thenwe must use homotopy cofibres e.g. represented by the cone of f. ⇤

7. The functor ⌦1 preserves ﬁbre sequences (as a right adjoint) and hence we get a long exact sequence in homotopy groups

⇡n(Z) ⇡n(X) ⇡n(Y ) ⇡n 1(Z) ... . ···! ! ! ! ! The (reduced) cohomology of a space X with coecients in a spectrum E is deﬁned by H⇤(X; E):=⇡ (map(⌃1X, E)) . ⇤ If we insert (6) into Map( ,E)wegetaﬁbresequenceinspectrawhichyieldsthe long exact sequence in cohomology.

+1 H⇤ (Cofib(f); E) H⇤(Y ; E) H⇤(X; E) H⇤(Cofib(f); E) ... . ···! ! ! ! ! Example 23.10. Using the symmetric monoidal structure of given by the cartesian product we can define the notion of a commutative monoid. AS monoid X in spaces is a group if ⇡0(X)isagroup.LetCGrp( ) CMon( )bethecategoriesofcommutative groups and monoids in spaces. Note thatS commutative✓ S mopnoids in Top represent a very restrictive class of commutative commutative in spaces. In order to model the general case one needs the notion of E -spaces. 1 Since Sp is stable, the forgetful functor CGrp(Sp) Sp is an equivalence. As a right- adjoint of a symmetric monodical functor the infinite loop! space functor ⌦is lax symmetric monodidal. It refines to a functor

⌦1 : Sp CGrp(Sp) CGrp( ) ' ! S which after restriction to the subcategory of connective spectra (i.e. those spectra with trivial homotopy groups in negative degree) induces an equivalence

0 ⌦1 : Sp ' CGrp( ) . ! S 36 Models for the inverse (denoted by sp) which turn a commutative group in into a connective spectrum are usually called -loop space machines. S 1 Let A be a commutative topological abelian group, i.e. an object in CGrp(Top). Then HA := sp(A)iscalledtheEilenberg-MacLanespectrumofA.Wehave An=0 ⇡ (HA) = . n ⇠ 0 n =0 ⇢ 6 The ordinary homology and cohomology of a space X is given in this language by

H (X; Z) = H (X+; HZ) ,H⇤(X; Z) = H⇤(X+; HZ) , ⇤ ⇠ ⇤ ⇠

(X+ is obtained from X by attaching a disjoint base point). For a spectrum E and a space X there is a Atiyah-Hirzebruch spectral sequence con- 2 2 verging to H (X+,E)withnaturalE -term Ep,q = Hp(X+; ⇡q(E)). In this way ordinary ⇤ ⇠ cohomology can be considered as a ﬁrst approximation to the cohomology of a space with coecients in a general spectrum. Assume that X is a CW-complex. Then the associated E1-term of the Atiyah-Hirzebruch spectral sequence is the cellular chain complex E1 = C (X ) ⇡ (E) . p,q ⇠ p + ⌦ q 2

Example 23.11. We consider the topological symmetric monoidal (with respect to ) 1 category of complex vector spaces VectC as a commutative monoid CMon(CatTop [W ]). Since the nerve and the realization preserve products we get BVectC CMon( ). The inlcusion of monoids into groups ﬁts into an adjunction 2 S GrCompl : CMon( ) CGrp( ):incl . S ⌧ S The connective topological K-theory spectrum is deﬁned by

ku := sp(GrCompl(BVectC)) . This is actually a ring space with ring structure induced by the tensor product of vector spaces. Let b ⇡2(ku) ⇠= Z be a generator. Then we get the periodic K-theory spectrum as 2 1 KU := ku[b ] .

It gives rise to K-theory H⇤(X+; KU)andK-homology H (X+; KU)ofaspaceX. ⇤ The algebraic tool to calculate the K-homology is the Atiyah-Hirzebruch spectral sequence whose second term is given by H (X , ) q even E2 = H (X ; ⇡ (KU)) = p + Z . p,q ⇠ p + q ⇠ 0 q odd ⇢ 2 3 3 The ﬁrst non-trivial di↵erential is d3 = Sq : Ep,q Ep 3,q+2. ! 37 Example 23.12. Let X be a topological space and ⇠ X be a real vector bundle. We deﬁne the associated Thom spectrum by !

⇠ X := ⌃1Coﬁb((⇠ X) X) . \ ! h This is meant to a model for a construction in spaces. As an illustration of the usage of the language of spaces we write this construction in that language. We start with the action of GL(n, R)onRn.WeapplythequotientconstructiontotheG-equivariant diagram Rn 0 / Rn \{ }

$ ~ ⇤ and get f n⇤ / n

% y BGL(n, R) the complement zero section of the universal n-dimensional vector bundle. For a map of spaces ⇠ : X BGL(n, R)wedeﬁnetheThomspectrumby ! ⇠ X := ⌃1Coﬁb(X f) . ⇥BGL(n,R)

Amapf : Y X induces a map of Thom spectra Y f ⇤⇠ X⇠.Inparticular,the inclusion of a point! in X gives a map ⌃dim(⇠)S X⇠. ! ! For all n N we have equivalences 2 n ⇠ ✏n ⇠ ⌃ X X . ' We can deﬁne KO0(X)asthegroupofstableisomorphismclassesofvectorbundleson dim(⇠) ⇠ X.Therefore⌃ X only depends up to equivalence on the KO-theory class of ⇠. We thus can consider the Thom spectrum X⇠ for a class ⇠ KO0(X). Using this we can deﬁne e.g. the Thom spectrum 2

MSO := BSO⇠ , where ⇠ KO0(BSO)istheclassofthetautologicalbundle.BythePontrjagin-Thom 2 construction the homotopy groups ⇡n(MSO)arethebordismgroupsofn-dimensional oriented manifolds. More generally, we have a cohomology theory X H⇤(X; MSO) and a homology theory X H (X; MSO). 7! 7! ⇤ In general it is dicult to calculate the homotopy groups of a Thom spectrum. Often one considers multiplicative cohomology theories represented by E CAlg(Sp). 2 38 The Adams spectral sequence is a machine which tries to calculate the homotopy groups of a spectrum A starting from its E-homology H (A; E) = ⇡ (A E). In good cases its ⇤ ⇠ ⇤ ^ E2-terms has an algebraic description as

2 E = ExtE E(⇡ (E),H (A; E)) . ⇠ ⇤ ⇤ ⇤ An orientation of a vector bundle ⇠ for E is a class or Hdim(⇠)(X⇠; E)whoserestriction 2 to every point x X aunitin⇡0(E). If the vector bundle ⇠ is oriented for E,thenwe have a Thom isomorphism2

⇠ H +dim(⇠)(X; E) = H (X ; E) . ⇤ ⇠ ⇤ This allows to calculate the ﬁrst input into the Adams spectral sequence.

1. The tautological bundle on BO(n)isorientedforHZ/2Z.

2. The tautological bundle on BSO(n)isorientedforHZ. 3. The tautological bundle on BSpinc(n)isorientedforKU.

4. The tautological bundle on BSpin(n)isorientedforKO.

In the last two cases the orientations are called Atiyah-Bott-Shapiro orientations. For example, in the cases of MSO, MU, MSpin and MSpinc the Adams spectral sequence works successfully with E = HZ/pZ and E = HQ and ﬁnally gives a complete understanding of the homotopy groups. 2

Example 23.13. Amap✓ : X BO(n)representsavectorbundlealsodenotedby✓. The Madsen-Tillman spectrum associated! to this datum is deﬁned as

✓ MT✓ := X .

In simple cases like X = BSpin(n)orX = BSO(n)onecancalculate⇡n(MT✓)using the methods indicated above. The main goal of the seminar is to construct an equivalence

B ⌦1⌃MT✓. (7) C✓ '

Remark 23.14. In this remark we sketch how (7) can be reﬁned to an -loop map. 1 Let Fin+ be the category of ﬁnite pointed sets. We get a functor

Fin+ /BO(n) ,F F X+ . !S⇤ 7! ^

39 The Thom spectrum construction can be considered as a functor /BO(n) Sp.There- ⇤ fore by composition we get the functor S !

MT✓ : Fin Sp . + ! This is actually an object of the full subcategory of Sp SpFin+ of -spectra E SpFin+ characterized by ✓ 2

n E[n] E[1] and a condition of being ”group-like” . + ' + i=1 Y It reﬁnes ⌃MT✓ to a grouplike spectrum, and its -loop space to a grouplike -space. We have an equivalence 1 Sp CGrp(Sp) , ' which under ⌦1 goes to CGrp( ) . S' S This says that the -loop space structure on ⌦1⌃MT✓ is equivalent to the -loop space 1 1 structure on ⌦1⌃MT✓ derived from the -space structure ⌦1⌃ MT✓.

On the other hand, using the functoriality of the bordism category in ✓ we can consider Fin+ the object B ✓ .Sincetheequivalence(7)isnaturalin✓ it reﬁnes to an equiv- Fin+C 2S alence in .Consequently,B ✓ is a group like -space, and (7) is an -loop map, if we equipS B with the -loop spaceC structure coming from the -space structure.1 2 C✓ 1