23 Introduction to homotopy theory

The objects of homotopy theory are the categories of spaces S and spectra Sp. In thisoverview I want to explain certain features and constructions with these categories whichwill become relevant in the seminar. One needs 1-categories or model categories in orderto capture theses objects on a technical level. In this overview I want to explain howone can work with spaces and spectra properly by simply obeying some rules of correctlanguage.As a preparation I recall the notion of limits and colimits. For categories C and I (thelatter assumed to be small) let CI be the functor category. There is a constant diagramfunctor

c : C ! CI .

The limit (colimit) is a (in general partially defined) right (left) adjoint of c:

c : C ⌧ CI : limI , colimI : CI ⌧ C : c .

We will illuminate this abstract definition by giving examples.

Example 23.1. A typical example of a limit is a pull-back or fibre product:

X ⇥Z Y

✏✏

// X

✏✏Y // Z

.

A typical example of a colimit is a push-out

Z

✏✏

// X

✏✏Y // X tZ Y

.

The quotient by an action a of a group G on X is the colimit of the diagram

G⇥Xa

,,

pr2

22 X // X/G .

2

We start with a discussion of spaces S. One can characterize S as the presentable 1-category (this essentially means that it admits all limits and colimits) generated by apoint. There are many models for S.Example 23.2. One can start from topological spaces Top. A morphism f in Top iscalled a weak equivalence if ⇡0(f) is a bijection and ⇡n(f) : ⇡n(X, x) ! ⇡n(Y, y) are

29

isomorphisms for all n � 1 and x 2 X. If we invert weak equivalences (technically thishappens inside 1-categories), then we get a model for the category of spaces

Top ! Top[W�1] ' S .

2

Example 23.3. One can define a model for S starting from simplicial sets sSet := Set�op

.Recall that �op is the category whose objects are the posets [n] = {0, . . . , n} and whosemorphisms are order preserving maps. For a category C we call C�op

the category ofsimplicial objects in C.In order to explain the notion of a weak equivalence in sSet we need the geometricrealization. The standard simplices �n ⇢ Rn+1 provide a functor

�� : � ! Top , [n] 7! �n .

Considering sets as topological spaces, a simplicial set X 2 sSet provides a diagram oftopological spaces

F�:[m]![n] �

m ⇥X[n]�⇤⇥id //

id⇥�⇤

✏✏

F[n]2� �n ⇥X[n]

✏✏F[m]2� �m ⇥X[m] // kXk

, (4)

whose colimit is the called the geometric realization of X. A more condensed, but equiv-alent, definition uses the notion of coend

kXk :=

Z �op

�⇥X .

By definition, a morphism f in sSet is a weak equivalence, if kfk is one in Top. We getthe desired model of spaces

sSet ! sSet[W�1] ' S .

Note that by construction k � k descends to a functor k � k : sSet[W�1] ! Top[W�1]which happens to be an equivalence. 2

In the following I list some of the basic properties of the category of spaces S. The modelindependent notion of isomorphism is equivalence.

1. Topological spaces or simplical sets represent spaces.

2. There exists a one-point space ⇤, i.e. a final object.

30

3. A space X 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. A morphism f : X ! Y between spaces is an equivalence if and only if ⇡n(f) is anisomorphism for all n � 0 and base points x 2 X (for n � 1).

6. For two spacesX, Y 2 S we have a functorially associated mapping space Map(X, Y ) 2S. Two morphisms f0, f1 : X ! Y are equivalent if they represent the same pointin ⇡0(Map(X, Y )).

7. In spaces we can form limits and colimits.

Our language and pictures are adapted to Top.

Example 23.4. The fibre f�1(y) of a morphism f : X ! Y in Top at a point y 2 Y canbe written as a limit

Fiby(f) //

✏✏

⇤y

✏✏X

f // Y

.

In general, limits in Top do not preserve weak equivalences. In general, the fibre definedas above for a map of topological spaces does not represent the fibre of f considered as amorphism in S. For example, we have a weak equivalence of diagrams

⇤1✏✏

⇤ 0 // [0, 1]

'! ⇤

✏✏⇤ // ⇤

,

but the map of limits is ; ! ⇤, surely not a weak equivalence.

One needs a modification of the construction of the fibre in order to obtain a model insideTop for the fibre taken in S. In this situation one speaks of homotopy fibres. A modelfor the homotopy fibre is the limit of

Fiby(f)h //

✏✏

⇤y

✏✏X ⇥f,Y,ev0 Y

I ev1 // Y

.

In general, if the morphism X ! Y in Top is a so-called fibration, then the fibre takenin Top represents the fibre in S. Note that the map ev1 is the replacement of f by afibration.

31

For a map f : X ! Y of spaces we can control the homotopy groups of the fibre by thelong 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)

.

A good model for the homotopy cofibre is given by the cone of f

X

(0,idX)✏✏

// ⇤

✏✏[0, 1]⇥X t{1}⇥X,f Y // Cofib(f)h

.

Here the left vertical map is the replacement of f by a cofibration. It is di�cult to controlthe homotopy groups of the homotopy cofibre. The appropriate invariant is homology. 2

Example 23.6. The functor

colim�op : C�op ! C , X 7! |X|

is called realization. If applied to simplicial spaces one again observes that it does notpreserve weak equivalences and therefore does not model the realization in S.A weak equivalence preserving model for the realization is the geometric realization givenby the formula (4) and will be denoted by |X|h.The homotopy groups of |X|h can be approached via the Bousfield-Kan spectral sequencewhose first page is

E1s,t := ⇡s(X[t]) .

2

Example 23.7. The nerve N(C) of a category is the simplicial set given by

N(C)[n] := Cat([n], C) .

We have a functor N : Cat ! sSet. A categorical equivalence induces a weak equivalencebetween nerves. This provides a natural factorization

N : Cat[W�1] ! sSet[W�1] ' 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 ofobjects Ob(C) and space of morphisms Mor(C). In this case the nerve naturally refinesto a simplicial space,

N(C)[0] := Ob(C) , N(C)[1] := Mor(C) , N(C)[2] := Mor(C)⇥s,Ob(C),r Mor(C) , . . . .

From now one we consider N(C) as a simplicial space, i.e. as an object in S�op. The

classifying space of a topological category is the space defined by

BC := |N(C)| 2 S .

A functor between topological categories C ! 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 C✓. The main goal is to identify the classifying spaceBC✓. 2

Example 23.8. Let G be a topological group. It gives rise to a topological cate-gory G with object ⇤ and morphism space G. The classifying space of this categoryis the usual classifying space BG of G. A homomorphism between topological groupswhich is a weak equivalence induces a weak equivalence between topologically enrichedcategories and therefore a weak equivalence of classifying spaces. For example, theusual inclusion O(n) ! GL(n,R) is a weak equivalence, hence we get an equivalenceBO(n) ' BGL(n,R).Let G act on a space X. Taking the quotient X/G in Top does not preserve weakequivalences. A good model for the homotopy quotient is

X/hG := EG⇥G X ,

where EG is the geometric realization of the simplicial G-space [n] 7! G[n] with the usualsimplicial structure. So the topological space EG⇥G X is a good model for the quotientX/G taken in S.The algebraic tool to calculate the homotopy groups of X/G is again a spectral sequencederived from the Bousfield-Kan spectral sequence with first term

E1s,t

⇠= ⇡s(G⇥ · · ·⇥G| {z }t⇥

⇥X) .

2

In topological spaces we can consider:

1. a point in a space

33

2. a homeomorphism

3. a locally trivial fibre bundle

4. a (model of a) homotopy quotient

In spaces we can talk about:

1. a connected component

2. an equivalence

3. some space is equivalent to the fibre of a morphism

4. a quotient

Example 23.9. There are similar notions of pointed spaces S⇤ related to Top⇤ and sSet⇤.The suspension ⌃X of an object X in a pointed category is defined as a colimit

X //

✏✏

⇤

✏✏⇤ // ⌃X

.

Note again, that the suspension in Top⇤ does not model the suspension in S⇤. To repre-sent the latter in Top⇤ we must use homotopy push-outs. For example, in Top⇤ we have⌃S0 ⇠= ⇤, while in S⇤ we get ⌃S0 ' S1. 2

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 asthe universal presentable 1-category generated by one object on which the suspensionacts as an equivalence. More concrete constructions (in the world of 1-categories) are as

Sp ' S⇤[⌃�1] ,

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 : S⇤ ⌧ Sp : ⌦1 .

Note that ⌃1X is just the image of X under S⇤ ! S⇤[⌃�1] ' Sp. We call ⌃1Xthe suspension spectrum of X, and ⌦1(E) the infinite loop space of the spectrumE.

34

2. A spectrum E has well-defined homotopy groups

⇡n(E) := ⇡2(⌦1⌃2�nE) , n 2 Z , k n .

We can replace 2 by any other k � 2. A morphism between spectra is an equivalenceif and only if it induces a morphism in homotopy groups.

3. A sequence of pointed spaces and maps

(Xn)n�0 , ⌃Xn ! Xn+1

defines a spectrum E by

E := colim�⌃0⌃1X0 ! ⌃�1⌃1X1 ! ⌃�2⌃1X2 ! . . .

�.

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 themaps ⌃hXn ! ⌃X ! Xn+1.

4. A sequence of morphisms in a category is called a fibre sequence if at every segment

· · · ! A ! B ! C ! . . .

the morphism A ! B is represents the fibre of B ! C. Similarly we define thenotion of a cofibre sequence. In spectra the notions of fibre and cofibre sequencescoincide.

A map between spectra X ! Y can functorially be extended to a fibre sequence

· · · ! Z ! X ! Y ! ⌃Z ! . . . .

5. The category of pointed spaces and spectra have symmetric monoidal structuresdenoted by ^. The functor ⌃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 ) such that

⌦1map(X, Y ) ' Map(X, Y ) .

We have the rulemap(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 definedby

H⇤(X,E) := ⇡⇤(⌃1X ^ E) .

The functor ⌃1 (as a left adjoint) preserves cofibres. Given a map X ! Y of spaceswe 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 homologysequence

· · · ! 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⇤, then wemust 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 along exact sequence in homotopy groups

· · · ! ⇡n(Z) ! ⇡n(X) ! ⇡n(Y ) ! ⇡n�1(Z) ! . . . .

The (reduced) cohomology of a space X with coe�cients in a spectrum E is definedby

H⇤(X;E) := ⇡�⇤(map(⌃1X,E)) .

If we insert (6) into Map(�, E) we get a fibre sequence in spectra which yields thelong exact sequence in cohomology.

· · · ! H⇤+1(Cofib(f);E) ! H⇤(Y ;E) ! H⇤(X;E) ! H⇤(Cofib(f);E) ! . . . .

Example 23.10. Using the symmetric monoidal structure of S given by the cartesianproduct we can define the notion of a commutative monoid. A monoid X in spaces is agroup if ⇡0(X) is a group. Let CGrp(S) ✓ CMon(S) be the categories of commutativegroups and monoids in spaces. Note that commutative mopnoids in Top represent a veryrestrictive class of commutative commutative in spaces. In order to model the generalcase one needs the notion of E1-spaces.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 symmetricmonodidal. It refines to a functor

⌦1 : Sp ' CGrp(Sp) ! CGrp(S)

which after restriction to the subcategory of connective spectra (i.e. those spectra withtrivial homotopy groups in negative degree) induces an equivalence

⌦1 : Sp�0 '! CGrp(S) .

36

Models for the inverse (denoted by sp) which turn a commutative group in S into aconnective spectrum are usually called 1-loop space machines.

Let A be a commutative topological abelian group, i.e. an object in CGrp(Top). ThenHA := sp(A) is called the Eilenberg-MacLane spectrum of A. We have

⇡n(HA) ⇠=⇢

A n = 00 n 6= 0

�.

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-verging to H⇤(X+, E) with natural E2-term E2

p,q⇠= Hp(X+; ⇡q(E)). In this way ordinary

cohomology can be considered as a first approximation to the cohomology of a space withcoe�cients in a general spectrum.Assume that X is a CW -complex. Then the associated E1-term of the Atiyah-Hirzebruchspectral sequence is the cellular chain complex

E1p,q

⇠= Cp(X+)⌦ ⇡q(E) .

2

Example 23.11. We consider the topological symmetric monoidal (with respect to �)category of complex vector spacesVectC as a commutative monoidCMon(Cat

Top

[W�1]).Since the nerve and the realization preserve products we get BVectC 2 CMon(S).The inlcusion of monoids into groups fits into an adjunction

GrCompl : CMon(S) ⌧ CGrp(S) : incl .

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 vectorspaces. Let b 2 ⇡2(ku) ⇠= Z be a generator. Then we get the periodic K-theory spectrumas

KU := ku[b�1] .

It gives rise to K-theory H⇤(X+;KU) and K-homology H⇤(X+;KU) of a space X.The algebraic tool to calculate theK-homology is the Atiyah-Hirzebruch spectral sequencewhose second term is given by

E2p,q

⇠= Hp(X+; ⇡q(KU)) ⇠=⇢

Hp(X+,Z) q even0 q odd

.

The first non-trivial di↵erential is d3 = �Sq2 : E3p,q ! E3

p�3,q+2.

37

Example 23.12. Let X be a topological space and ⇠ ! X be a real vector bundle. Wedefine 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 usageof the language of spaces we write this construction in that language. We start withthe action of GL(n,R) on Rn. We apply the quotient construction to the G-equivariantdiagram

Rn \ {0} //

$$

Rn

~~⇤and get

�⇤n

f //

%%

�n

yyBGL(n,R)

the complement zero section of the universal n-dimensional vector bundle. For a map ofspaces ⇠ : X ! BGL(n,R) we define the Thom spectrum by

X⇠ := ⌃1Cofib(X ⇥BGL(n,R) f) .

A map f : Y ! X induces a map of Thom spectra Y f⇤⇠ ! X⇠. In particular, theinclusion of a point in X gives a map ⌃dim(⇠)S ! X⇠.

For all n 2 N we have equivalences

⌃�nX⇠�✏n ' X⇠ .

We can define KO0(X) as the group of stable isomorphism classes of vector bundles onX. Therefore ⌃� dim(⇠)X⇠ only depends up to equivalence on the KO-theory class of ⇠.We thus can consider the Thom spectrum X⇠ for a class ⇠ 2 KO0(X).Using this we can define e.g. the Thom spectrum

MSO := BSO⇠ ,

where ⇠ 2 KO0(BSO) is the class of the tautological bundle. By the Pontrjagin-Thomconstruction the homotopy groups ⇡n(MSO) are the bordism groups of n-dimensionaloriented manifolds. More generally, we have a cohomology theory X 7! H⇤(X;MSO)and a homology theory X 7! H⇤(X;MSO).

In general it is di�cult to calculate the homotopy groups of a Thom spectrum. Often oneconsiders multiplicative cohomology theories represented by E 2 CAlg(Sp).

38

The Adams spectral sequence is a machine which tries to calculate the homotopy groupsof a spectrum A starting from its E-homology H⇤(A;E) ⇠= ⇡⇤(A ^ E). In good cases itsE2-terms has an algebraic description as

E2 ⇠= ExtE⇤E(⇡⇤(E), H⇤(A;E)) .

An orientation of a vector bundle ⇠ for E is a class or 2 Hdim(⇠)(X⇠;E) whose restrictionto every point x 2 X a unit in ⇡0(E). If the vector bundle ⇠ is oriented for E, then wehave a Thom isomorphism

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) is oriented for HZ/2Z.

2. The tautological bundle on BSO(n) is oriented for HZ.

3. The tautological bundle on BSpinc(n) is oriented for KU.

4. The tautological bundle on BSpin(n) is oriented for KO.

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 spectralsequence works successfully with E = HZ/pZ and E = HQ and finally gives a completeunderstanding of the homotopy groups.

2

Example 23.13. A map ✓ : X ! BO(n) represents a vector bundle also denoted by ✓.The Madsen-Tillman spectrum associated to this datum is defined as

MT✓ := X�✓ .

In simple cases like X = BSpin(n) or X = BSO(n) one can calculate ⇡n(MT✓) usingthe methods indicated above.The main goal of the seminar is to construct an equivalence

BC✓ ' ⌦1⌃MT✓ . (7)

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

Fin+ ! S⇤/BO(n) , F 7! F ^X+ .

39

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

�MT✓ : Fin+ ! Sp .

This is actually an object of the full subcategory of �Sp ✓ SpFin+ of �-spectra E 2 SpFin+

characterized by

E[n]+ 'nY

i=1

E[1]+ and a condition of being ”group-like” .

It refines ⌃MT✓ to a grouplike � spectrum, and its 1-loop space to a grouplike �-space.We have an equivalence

� Sp� ' CGrp(Sp�) ,

which under ⌦1 goes to� S ' CGrp(S) .

This says that the 1-loop space structure on ⌦1⌃MT✓ is equivalent to the 1-loop spacestructure on ⌦1⌃MT✓ derived from the �-space structure ⌦1�⌃MT✓.

On the other hand, using the functoriality of the bordism category in ✓ we can considerthe object �BC✓ 2 SFin+ . Since the equivalence (7) is natural in ✓ it refines to an equiv-alence in SFin+ . Consequently, � BC✓ is a group like �-space, and (7) is an 1-loop map,if we equip BC✓ with the 1-loop space structure coming from the �-space structure. 2

40