Twisted generalized cohomology and applications
Hisham Sati
New York University Abu Dhabi (NYUAD)
MIMS Summer School, Tunis, Tunisia
9-12 July 2018
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Outline
0. Global overview
I. Generalized cohomology:
1 Axiomatics
2 Spectra
3 Chromatic approach
4 Examples: K-theory, Morava K-theory, Elliptic cohomology
II. Twisted theories:
1 Basic example: Twisted de Rham
2 General approach
3 K-theory
4 Morava K-theory and E-theory
5 Iterated algebraic K-theory
III. Differential refinements
1 Motivation for including geometry
2 General approach
3 Example 1: Twisted differential integral cohomology
4 Example 2: Twisted differential K-theory
IV. Calculations and applications:
1 Calculations via the Atiyah-Hirzebruch spectral sequence (AHSS)
2 Calculations via the universal coefficient sequence (UCT)
3 T-duality as an isomorphism of twisted theories
4 Fields and branes in string theory 2 / 100
0. Global overview
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Theme
Declaration: All twisted generalized cohomology theories we consider (or even all explicitly constructed) are motivated to various degrees by physics.
Mathfromphysics
Q1: What new mathematical structures and constructions can we extract from studying physical models?
Mathin Physics
Q2: What mathematical structures/conditions/tools should we have in place in order to properly define a physical theory?
By phrasing in context of physics, the math becomes more transparent.
Upshot
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Generalities on what physics wants
Nontrivial physical entities, such as fields, charges, etc. generically take values in cohomology.
Cohomology
ww ''
Generalized Twisted Differential
1 Generalized: Algebraically via formal groups or topologically via bundles.
2 Twisting: Symmetries via automorphisms.
3 Differentially refined: Include geometric data, such as connections, Chern character form, smooth structure, smooth representatives of maps ...
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A motivation for generalized cohomology
Modelling of fields in physics, in particular quantum field theory, string theory and M-theory.
Ω•(M)
exact complex
Classical
HdR• (M)
quantization
H•(M;Z)
anomaly cancellation
E•(M) Quantum
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Twists
We would like to introduce automorphisms.
These arise from geometric and physical considerations.
For the homotopy point of view: moduli/family setting; bundles of spectra.
Ω•(M)
twistΩ
exact
complex //HdR• (M)
twistdR
quantization //H•(M;Z)
twistH
anomaly
cancellation //E•(M)
twistE
Relations among various twists?
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Twisted de Rham cohomology
The de Rham complex
(Ω•,d) : . . .→d Ωi(X)→d Ωi+1(X)→d . . .
Twist by a 1-formbuilt out of scalar function: d;dφ:=d+dφ∧with dφ2=0.
Example (Witten’s deformation of Morse theory)
For smoothf :M →R, the Witten differential isds =e−sfdesf =d+sdf∧, wheres∈R. Thends2=0,ds : Ωp→Ωp+1. The term e−sf is a
quasi-isomorphism
. . . //Ωp d //
e−sf
Ωp+1
e−sf
//. . .
. . . //Ωp ds //Ωp+1 //. . . andds yields isomorphic cohomology groups.
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Twist by a closed 3-form: dH3 =d−H3∧, withdH23 =0.
Definition
Twisted de Rham cohomology: Hi(X,H3) := ker(dim(dH3)
H3)
Example (The Ramond-Ramond (RR) fields in string theory)
F =P
i≤5u−iF2i+,=0 or 1 for type IIA or type IIB string theory. These are twisted by a closed 3-form, the NS-fieldH3.
To make periodic: adjoin a generatoruof degree 2 which implements the periodicity & makes total degree uniform:
dH3 =d−u−1H3∧.
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In fact, one can build a differential by adding todH all expressions of the formu−iH2i+1∧, i.e.
dH0 =d+
∞
X
i=0
u−iH2i+1∧ .
There is a twisted graded de Rham complex with differential d+P∞
i=1u−iH2i+1∧, provided the differential formsH2i+1are closed.
Example (S.)
From string theory withF=F+∗F, whereF is the Yang-Mills field and∗F its dual, one gets
(d - H7∧)F=0 .
This gives a twisted differentialdH7 =d−H7∧which in nilpotent, i.e. squares to zero,dH2
7 =0, sinceH7 is closed.
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Twists of integral cohomology
1-d twists of integral cohomology given by alocal systemZ →M.
This is a bundle of groups isomorphic toZ, so is determined by H1(M;Aut(Z))∼=H1(M;Z2)
since the only nontrivial automorphism ofZis multiplication by−1.
Čech description. Let{Ui} be an open covering ofM and
gij :Ui∩Uj→ {±1} a cocycle defining the local systemZ. Then an element ofHq(M;Z)is represented by a collection ofq-cochainsai ∈Zq(Ui)which satisfy
aj =gijai onUij =Ui∩Uj . (1) Use a model as maps to Eilenberg-MacLane spaceK(Z,q).
1 K(Z,0)'Zon which−1 acts by multiplication.
2 K(Z,1)'S1 on which−1 acts by reflection.
Action ofAut(Z)onK(Z,q)and the cocyclegij ⇒ Associated bundleHq→M with fiberK(Z,q).
Eq. (1) says that twisted cohomology classes are represented by sections of Hq→M. The twisted cohomology groupHq(M;Z)is the set of homotopy classes of sections ofHq→M.
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First main idea in a nutshell
Rational twisted cohomology arises as image of some Chern character.
Example
Degreethree twistH3:
chH3: K•(X,H3)
| {z }
twisted K−theory
−→ Hev(X,H3)
| {z }
twisted de Rham cohomology
Now if we are presented withhigher degree twists on the left-hand-side, would they be images of somegeneralizedChern character whose domain is somegeneralizedcohomology theory?
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Differential refinement
We would like to introduce geometric data, say via differential forms. That is we would like to retain differential form representatives of cohomology classes.
Ω•(M)
adjoin
uu adjoin
adjoin
((
HdR• (M)
refinement
H•(M;Z)
refinement
E•(M)
refinement
Hc•dR(M) Hc•(M;Z) Ec•(M)
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Why differential cohomology?
Provides a way to refine secondary torsion invariants toR/Z-cohomology classes:
Chern-Simons invariants
Characteristic classes of flat vector bundles
invariants of elements in stable homotopy groups: e-invariant,f-invariant,· · ·
Leads to results in geometry:
e.g. [Chern-Simons] Differential Pontrjagin classespˆi show thatSO(3)∼=RP3 does not admit conformal immersion inR4.
Provides a desirable setting for mathematical physics:
Proper description of actions functionals in (topological) field theories.
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Integral Cohomology - differential refinement
Three approaches to differential integral cohomology:
Integral cohomology
geometric
zz
algebraic approach
axiomatic
$$
Differential characters ks +3 Deligne cohomology ks +3 Homotopy pullback
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(i) Differential characters:
A differential character of deg. k on a smoothM is a homomorphism h:Zk−1(M:Z)→U(1)
from smooth integral-valued singular cocycles of degreek−1.
Such that there exists a differential formcurve(h)∈Ωk(M), uniquely determined byhand is called its curvature, such that
h(∂c) = exp 2πi
Z
c
curv(h) .
Set of all differential characters onM of degreek is Hbk(M;Z). Pointwise multiplication provides an abelian group structure. There is also another multiplicationHbk(M;Z)×Hbl(M;Z)−→Hbk+l(M;Z)which turns this into a ring.
1 k=1 : U(1)-valued functions. Given aU(1)-bundle with connection over M, one can associate a differential character by mapping any 1-cycle to the holonomy of the bundle along this cycle.
2 k>1 : (higher) gerbes.
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(ii) Axiomatic/Diagrammatic approach:
Ωn(M)/Ωn(M)Z d //
a
''
Ωn+1(M)
dR
''
Hn(M;R)
77''
Hconnn+1(X;Z)
F(−)
88
I
&&
Hn+1(M;R)
Hn(M,U(1))
77
β //Hn+1(M;Z)
Ch
88
whered is the de Rham differential,F is the curvature map,I is the forgetful map,Chis the rationalization, and β is the Bockstein associated with the exponential coefficient sequence. 0−→Z−→R
−−→exp R/Z−→0.
connections on
trivial bundles de Rham differential //
regard as
''
curvature forms
de Rham theorem
%%
closed differential
forms
regard as
88
regard as
&&
connections on geometric bundles
curvature
88
topol. class
&&
rationalized bundle
flat connections
regard as
77
comparison map // shape ofbundles
Chern character
99
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(iii) Deligne Cohomology:
Consider the truncated de Rham complex
Ω0=O d //Ω1 d //Ω2 d //· · · d //Ωn
Replace the structure sheafOwith the multiplicative groupO× under the exponential map to get theDeligne complex
O× d //Ω1 d //Ω2 d //· · · d //Ωn
Deligne cohomologyHDn+1(X)in degreen+1 is the hypercohomology for this complex of sheaves of abelian groups, i.e. abelian sheaf cohomology with coefficients in this chain complex.
Note that
U(1) =C∞(−,U(1)).
Twisted + differential by combining Deligne with Axiomatic;stacks: Part of long term project withDan Grady.
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If{Uα}is a good open cover ofM, form the Čech-Deligne double complex Z(Uα0...αn) 2πi //Ω0(Uα0...αn) d //Ω1(Uα0...αn) d //. . . d //Ωn−1(Uα0...αn)
Z(Uα0...αn−1) 2πi //
(−1)n−1δ
OO
Ω0(Uα0...αn−1) d //
(−1)n−1δ
OO
Ω1(Uα0...αn−1) d //
(−1)n−1δ
OO
. . . d //Ωn−1(Uα0...αn−1)
(−1)n−1δ
OO
...
(−1)n−2δ
OO
...
(−1)n−2δ
OO
...
(−1)n−2δ
OO
...
(−1)n−2δ
OO
Z(Uα0α1) 2πi //
−δ
OO
Ω0(Uα0α1) d //
−δ
OO
Ω1(Uα0α1) d //
−δ
OO
. . . d //Ωn−1(Uα0α1)
−δ
OO
Z(Uα0) 2πi //
δ
OO
Ω0(Uα0) d //
δ
OO
Ω1(Uα0) d //
δ
OO
. . . d //Ωn−1(Uα0),
δ
OO
(2) whereUα0α1...αk denotes the k-fold intersection.
The total operator on the double complex is theČech-Deligne operator D:=d+ (−1)pδ, whered andδis the de Rham and Čech differentials, respectively, acting on elements of degreep.
The sheaf cohomology groupH0(M;D(n))can be identified with the group of diagonal elementsαk,k in the double complex which are Čech-Deligne closed(d+ (−1)pδ)αk,k =0, modulo those which are Čech-Deligne exact.
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Proposition (Properties of Deligne cohomology)
Deligne cohomology satisfies the following properties:
(i)(Functoriality)For a smooth map between manifoldsM →N, we have an induced map
Hbn(N;Z)→Hbn(M;Z).
(ii)(Additivity)ForM =`Mα a disjoint union of smooth manifolds, we have an isomorphism
Hbn(M;Z)∼=M
α
Hbn(Mα;Z).
(iii)(Mayer-Vietoris)For an open cover ofM by open smooth manifolds U andV, we have a sequence
. . . //H∗−2(U∩V;R/Z) //Hb∗(M;Z) //Hb∗(U;Z)⊕Hb∗(U;Z)
// bH∗(U∩V;Z) //H∗+1(M;Z) //. . . .
Mix between ordinary integral cohomology and cohomology withR/Z-coefficients.
Recall Diamond: Deligne cohomology is really a mixture of three different cohomology theories (integral,R/Z-coefficients, and de Rham) and captures the
interactions between these theories. 20 / 100
Differential generalized cohomology
Start with a generalized cohomology theoryh
Ω(X,h∗) := Ω(X)⊗Zh∗ Smooth differential forms with coefficients inh∗:=h(∗) Ωcl(X,h∗)⊆Ω(X,h∗)closed forms
HdR(X,h∗)cohomology of the complex Ω(X,h∗),d Def. Asmooth extensionofhis a contravariant functor
bh:Compact Smooth Manifolds−→Graded Abelian Grps Ωcl(X,h∗)
bh(X)
R
66
I
((((
HdR(X,h∗)
h(X)
OO
[Chern-Simons, Cheeger-Simons, Simons-Sullivan, Hopkins-Singer, Bunke-Schick]
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Full structure
Twisted∩Differential∩Generalized
Ω•(M)
adjoin
uu
adjoin
((
Geometric
HdR• (M)
twistdR
refinement
H•(M;Z)
twistH
refinement
E•(M)
twistE
refinement
Topological
Hc•dR(M)
twistΩb
WW cH•(M;Z)
twistHb
WW cE•(M)
twistEb
WW Both
Examples
1 With Craig Westerland & John Lind: Morava K-theory/E-theory, K-theories ofn-vector bundles.
2 With Dan Grady: Differential refinements of cohomology theories including above.
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Orientations
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Orientation
hcohom, [M]homi ∈C,R,Z,Z2,· · · . Disk neighborhoodU of a pt minMn.
Mapεm,U :Mn→Sncollapse complement ofU ⊂M.
π0(E) =Ee0(S0)∼=Een(Sn) =En(Sn,∗)
1 //sn, canonical orientation ofSn.
Definition
[M]E ∈En(Mn)is anE-orientationofM if, for allm,U, εm,U∗ [M]E=±sn.
E-orientation⇔existence of aThom classu ofTM (orNM).
Thom isomorphism: E∗(M)−→∼= E∗(Thη),x 7→x∪u η(O(n)metric space-) vector bundle onM.
ThηThom space (‘one-pt compactification’):
Th(V) :=D(V)/S(V).
NoteDn/Sn−1homeom' Sn.
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Classifying spaces
Aclassifying spacefor the groupG is a connected topological spaceBG, together with a principalG-bundleEG →BG such that
PrinG(X)∼= [X,BG]
TheUniversal principal G-bundle P=f∗(EG)
EG
X f //BG
Example: PrinG(Sn)∼=πn(BG) Properties:
1 LetE →B be a principalG-bundle with the property that the total space of E iscontractible. Then(B,E)is a classifying space forG.
2 ForanyLie groupG there exists a classifying spaceBG.
3 Each nonzero class inH∗(BG)isa universal characteristic class for principal G-bundles.
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Obstructions to orientations are given bycharacteristic classes:
Pontrjagin classespi andStiefel-Whitney classeswi in the real case, Chern classesci in the complex case.
The vanishing of one or more specific characteristic classes amounts to the ability to define and erect a desirable geometric and topological structure on the space.
Recall
Z2→Spin→SO Needed to define spinors (fermions), Dirac equation etc.
Perspectives:
1 Double cover of manifolds.
2 Part of short exact sequence (SES) of Lie groups.
3 Central extension of Lie groups.
4 Principal bundle of manifolds.
5 Fibration sequence of topological spaces.
We would like to generalize this. For that, writeZ2=K(Z2,0), U(1) =S1=K(Z,1)Eilenberg-MacLane space.
Consider higher degree: K(Z,n)forn>1.
πiK(Z,n) =
Z if i=n 0 otherwise.
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The homotopy groups of the orthogonal group
The homotopy groups of the orthogonal groupO(n), fornsufficiently large, are
πk(O(n)) =
Z2 fork =0,1mod 8 Z fork =3,7mod 8 0 otherwise
. (3)
The condition onnis best understood by considering thestable orthogonal group, also know as the infinite orthogonal group, which is defined as the direct limit of the sequence of inclusions
O(1)⊂O(2)⊂ · · · ⊂O=
∞
[
k=0
O(k). (4)
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k 0 1 2 3 4 5 6 7
πk(O(n)) Z2 Z2 0 Z 0 0 0 Z
O(n)hki O(n) SO(n) Spin(n) String(n) Fivebrane(n)
s
killπ0
EE{
killπ1
AA
killπ3
;;
killπ7
77
“kill"→“factorize".
Example (String)
There is a fibrationK(Z,2)→String→Spin. Take the corresponding long exact sequence on homotopy groups
. . . //πi(K(Z,2)) //πi(String) //πi(Spin)
//πi−1(K(Z,2)) //πi−1(String) //. . . . From the known homotopy groups of the fiber and the base:
Fori=3,· · · ,6 andj =7 this givesπi(String) =0 andπ7(String)∼=Z. Effectively means that in going from Spin to String we have “killed"π3(Spin).
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BFivebrane(n)
Fivebrane structure
BString(n)
String structure
BSpin(n)
Spin structure
BSO(n)
Orientation
X 44//99>>BB
BO(n) Riemannian
target space Whitehead tower ofBO(n)
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String structures
Homotopy fibration seq.: K(Z,2)→String(n)→Spin(n).
Geometric description ofK(Z,2)asPU(H), the projective unitary group on an infinite-dimensional Hilbert spaceH. Canonical degree three classDDof PU(H)bundles. In the operator algebra language this is called the
Dixmier-Douady class.
Classifying functor: K(Z,3)→BString(n)→BSpin(n).
String structure onX:
1. The obstruction is: 12p1(X)∈H4(X;Z) .
2. The set of lifts, i.e. the set of String structures for a fixed Spin structure is a torsor for a quotient ofH3(X;Z).
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Fivebrane structures
Homotopy fibration: K(Z,6)→Fivebrane(n)→String(n).
Classifying functor: K(Z,7)→BFivebrane(n)→BString(n).
Proposition (SSS)
1. The obstruction is given by 16p2(X)∈H8(X;Z) .
2. The set of lifts, i.e. the set of Fivebrane structures for a fixed String structure is a torsor for a quotient of the seventh integral cohomology groupH7(X;Z).
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Going higher [S.]
k 7 8 9 10 11 12
πk(O(n)) Z Z2 Z2 0 Z 0
O(n)hki String(n) Fivebrane(n) Oh9i(n) Oh10i(n) Ninebrane(n)
s
killπ7
EE s
killπ8
EE o
killπ9
GG w
killπ11
CC
The mod 8 (Bott) periodicity of the homotopy groups of the orthogonal group motivates the following for the correspondingG-structures:
1 SpaceOh9icorresponds to a ‘shift by 8’ analog of orientation: 2-Orient.
2 SpaceOh10icorresponds to a ‘shift by 8’ analog of Spin structure: 2-Spin.
One usually starts with classifying spaces and then take the loop space to define the above groups.
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Connected covers [S.-Schreiber-Stasheff, S.]
...
K(Z,11) //BOh13i= BNinebrane
K(Z/2,9) //BOh11i= B2-Spin
1 240p3
//K(Z,12)
K(Z/2,8) //BOh10i= B2-Orient
α10 //K(Z/2,10) K(Z,7) //BOh9i= BFivebrane
α9 //K(Z/2,9) K(Z,4) //BOh8i= BString
1 6p2
//K(Z,8)
K(Z/2,1) //BOh4i= BSpin
1 2p1
//K(Z,3)
BOh2i= BSO
w2 //K(Z/2,2) BO w1 //K(Z/2,2)
(5)
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I. Generalized cohomology
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Generalized Cohomology Theories
CW: the category whose objects are CW-complexes with a 0-cell chosen as a base-point and whose maps are basepoint preserving maps.
Ab: the category of abelian groups.
Definition
Acohomology theoryonCWis a sequence{En}n∈Zof functorsEn:CWop→Ab together with natural isomorphismsEn(X)∼=En+1(ΣX)for allX ∈CW, such that theEilenberg-Steenrod axiomsare satisfied:
1 Homotopy: iff,g :X →Y are homotopic (preserving basepoints) then the induced mapsEn(Y)→En(X)are isomorphic.
2 Inclusion: For each inclusionA,→X in CW, the sequence En(X/A)→En(X)→En(A)is exact.
Excision: U is contractible, inclusion X−U,→X induces an iso inE.
3 Additivity: For a wedge sumX =W
αXαwith inclusionsια:Xα,→X, the product mapQ
αιαEn(X)−∼=→Q
αEn(Xα).
4 Long exact sequence: in cohomology for pairs of topological spaces (X, A):
· · · →En(X,A)→En(X)→En(A)→En+1(X,A)→ · · ·
5 Dimension: E∗(pt) =⊕nEn(pt)is a graded abelian group,coefficient group.
TheGrothendieck construction K associates an abelian group to any semigroup by formally adjoining inverses.
K(N) =Z: Identify(n,m)∼(n+k,m+k)which we think of asn−m. 35 / 100
Example (K-theory K or KU)
Vect(X): set of isomorphism classes of complex vector bundles overX. Abelian semigroup with op. +coming from Whitney sum of vector bundles.
K0(X): the universal group associated toVect(X). Elements are formal differences of isomorphism classes of vector bundles.
Given a mapf :X →Y and vector bundles ξoverY, we have pullback bundlef∗ξoverX. Passing to isomorphism classes and formal differences, getf∗:K0(Y)→K0(X).
MakesK0(−)into a contravariant functorfrom spaces to abelian groups.
Reduced theory: Ke:= kerrank, rank: K→Z∼=K(x0)
[E] - [F]7→[Ex0]−[Fx0] =rankx0. DefineKe−n(X)asKe0(ΣnX)forn∈N.
Kn(X)∼=Kn−2(X),n∈Z, 2-periodic cohomology theory (Bott).
K0(pt) =
Z neven 0 nodd
Graded ring: K∗(pt) =Z[u,u−1],|u|=−2.
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Spectra
A contravariant functorH:Space→Sets is called ahomotopy functorif H(f) =H(g)holds wheneverf 'g.
H is calledrepresentable if it is naturally isomorphic to the hom functor Hom(−,X)for some objectX of Spaces.
Brown representability theorem: H is representable iffH satisfies the additivity and Mayer-Vietoris axioms.
Definition
Aspectrumis an object representing a generalized cohomology theory.
It is a collection of spaceshn, one for eachn∈Z, such that En(X) = [X,Ω∞hn]
together with connecting maps (inclusions)Σhn→hn+1(note: En(X)∼=En+1(ΣX)).
Example (Cohomology)
Singular cohomologyHn(−;A)is representable sinceHn(X;G)∼= [X,K(A,n)].
Eilenberg-MacLane spectrumHAwithHAn=K(A,n), Eilenberg-MacLane space πi(K(A,n))∼=
A, i =n 0, i 6=n
Note: C∼=K(C,0)forC a discrete group, andU(1)∼=K(Z,1). 37 / 100
Example (K-theory)
K-theoryK(X)is representableK0(X)∼= [X,BU×Z]
BU =classifying space of the stable unitary group as colimitU = lim
−→
n
U(n) The spectrum isU withU2n=BU×ZandU2n+1=U (Bott periodicity).
A spectrum may be constructed out of a space. Thesuspension spectrum of a space X is a spectrumXn=Sn∧X, with the structure maps being the identity.
Example (Sphere spectrum)
The sphere spectrumS= Σ∞S0is the ‘smallest nontrivial’ spectrum.
It is the suspension spectrum ofS0=set of two points.
thenth space is the spectrum isΣnS0=Sn.
the structure mapsΣSn→Sn+1are the canonical homeomorphisms.
Sis a unit for the smash product (suspension).
It is the spectrum forcohomotopy
πn(X) = [X,Σ∞S0]n= [X,Sn]. Note that forhomotopy
πn(X) = [Σ∞S0,X]n= [Sn,X].
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Ring spectra
Definition
Aring spectrumis a spectrumE such that the diagrams that describe the ring axioms in terms of smash product (smash=Cartesian/wedge) commute up to homotopy. There is a multiplication map
µ:E∧E →E
and a unit mapη:S→E, whereSis the sphere spectrum withSn=Sn∼= ΣnS0, such that
Associativeup to homotopy: µ(id∧µ)∼µ(µ∧id).
Unitalup to homotopy: µ(id∧η)∼µ(η∧id)∼id.
(Highly) structured:
A∞-ring spectrum(an algebra over an A∞-operad) or
E∞-ring spectrum(an algebra over an E∞-operad) by taking them as suitable (higher) loop spaces.
Examples
1 BothHR (R=ring) and U areE∞.
2 “Higher" examples: (later) Morava E-theoryE(n): E∞
Morava K-theoryK(n): A∞. 39 / 100
Omega-spectra
A topological space is a loop space if it has a delooping. It is an infinite loop space if this delooping has itself a delooping, and so on.
Infinite loop spaces are the grouplikeE∞algebras in Top (grouplike E∞
spaces).
Consider the following category of spectraR: sequences of spacesRnwith homeomorphismsRn→ΩRn+1, whose zeroth spaces are infinite loop spaces:
E0'ΩE1'Ω2E2' · · · 'Ω∞R∞.
This defines a functorΩ∞from spectra to spaces. If the spectrum has a structure (E∞ etc.) then that will be remembered by the space (E∞-space etc.).
The suspension spectrum functorΣ∞is the left adjoint.
Adjunctions:
Top
(−)+ //Top+
oo Σ
∞ //Spectra
Ω∞
oo
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II. Twisted spectra
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Units of ring spectra
Naively: We want to classify twists. Classifying functorB:Monoids→Top∗, adjoint to loop functorΩ :Top∗→Monoids. What to do if we have rings?
Example (Algebra)
LetGL1R denote the group of units of a commutative ringR.
The free abelian group functorZ:Sets−→AbGrp induces a functor Z:Grpoo //Ring:GL1
whose right adjoint isGL1.
In particular, there is a natural map of rings Z[GL1R]−→R.
Units of ring spectra[May-Quinn-Ray]: FunctorGL1:Rings→Group-like Spaces,R7→GL1(R) =R×. IfR is structured then so will beGL1(R).
UnitsGL1(R)to be the union of the components inΩ∞R that correspond to group of units(π0Ω∞R)× ⊆π0(Ω∞R)in the discrete ring π0Ω∞R.
Definition
Theunit is defined as the homotopy pullback diagram GL1(R) //
Ω∞R=R0
π0
(π0Ω∞R)× ⊆ //π0(Ω∞R) where(π0R)×⊆π0(R)is the group of units of the ring π0(R).
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Twisted Spectra
[Ando-Blumberg-Gepner(-Hopkins-Rezk)]
Can deloop asGL1(R)is ∞-loop space: GL1(R)'Ω∞(gl1(R))ifR E∞ [Heregl1(R)is the spectrum of units of the spectrumR].
Adjoint functor toGL1 is theinfinite suspension functorΣ∞+. Adjunction π0MapE∞/A∞(Σ∞Z+,R)∼=π0MapE∞/A∞(Z,GL1R)∼= [BZ,BGL1R]
whereZ is anE∞ (resp. A∞) space andMapE∞/A∞ denotes the space of E∞(resp. A∞) maps (of spectra on the left and of spaces on the right).
Amodule spectrumis a spectrum with an action of a ring spectrum (it
generalizes a module in abstract algebra). For R anE∞ orA∞ ring spectrum, anR-module spectrum is a spectrum equipped with anR-action.
To a spaceX and a mapsτ :X →BGL1R, one associates an R-module spectrumThτ representing twistedR-theory.
The spaceBGL1R classifies the twists.
Definition
Theτ-twistedR-homologyofX:
Rk(X)τ:=π0HomR(ΣkR,Thτ)∼=πkXτ . Theτ-twistedR-cohomologyofX:
Rk(X)τ:=π0HomR(Thτ,ΣkR).
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Examples
There exist continuous maps
K-theory K(Z,3)→BGL1(K) Periodic de Rham cohom. K(R,2n+1)→BGL1(HR[u,u−1]) Topological modular forms K(Z,4)→BGL1(TMF)
whereu is the periodicity element andK(G,n)is an Eilenberg-MacLanespace (whose sole homotopy group isG in dimensionn).
Since these spaces represent the cohomology functorHn(−,G), such cohomology classes give rise to twistings of the indicated generalized cohomology theory.
Bundles of spectra view: (schematic) Automorphism=symmetries
We have a universalGL1-bundle P=τ∗(E)
E =EGL1(R)
GL1(R)
oo
M τ //BGL1(R)
;Parametrized spectra 44 / 100
Ex: Twisted K-theory (Rosenberg, Atiyah-Segal, ...)
R=U, GL1(Z×BU)∼=Z2×BU =Z××BU.
Pullback GL1(U)∼=Z2×BU //
Z×BU
π0(Z×BU)×∼=Z2
⊆ //π0(Z×BU)∼=Z We have a universalGL1-bundle
P
EGL1(U)
GL1(U)
oo
M τ //BGL1(U)
Twist: τ :M →B(Z2×BU⊗) ∼=B(Z2×BU(1)×BSU⊗)
∼=K(Z2,1)×K(Z,3)×BBSU⊗
(⊗of virtual line bundles).
Usually adegree three determinantal twistis isolated by postcomposing with the inclusioni:K(Z,3)→BGL1(U).
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Twisted Chern character
For a finite CW-complexX there is a ring isomorphism ch:K0/1⊗ZQ−→∼ Hev/odd(X;Q)
This is obtained by usingch(L) = exp(c1(L))for line bundlesL, the splitting principle, and additivity.
At the level of spectra: There is a homotopy equivalence between the rationalized ring spectra(Z×BU)⊗ZQand the Eilenberg-MacLane spectrumQ
n≥0K(Q,2n).
For a finite complexX, the groupsK0/1(X;τ)⊗ZQare the homotopy groups of a bundle of spectra overX with fiberQ
n≥0K(Q,2n), which defines twisted rational cohomology.
The splitting principle cannot be used for twisted K-theory classes, as they are usually represented by infinite-dimensional bundles.
Nevertheless, one can construct a twisted version of the Chern isomorphism chτ:K(X;τ)⊗ZQ−→∼ Hτev/odd(X;Q)
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Twisted orientations
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Twisted structures
Definition
Given a cocycleα4:M →K(Z,4), anα-twisted String structure (or a String structure relative toα) on a Spin manifoldM with classifying map
f :M →BSpin(n)is a homotopyη:
M f //
α4
''
BSpin(n)
1 2p1
K(Z,4)
~ η
. (6)
Condition: 12p1(TX) + [α4] =0 . Ifαis trivial (i.e. factors through a point) then this reduces to ordinary String structure.
For the Fivebrane: replace 12p1with 16p2 andα4withβ8.
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Theorem (Kriz-S.)
1 A manifoldX is orientable with respect toK˜(2)if W7(X) =0.
2 A Spin manifoldX is orientable with respect toEO(2)ifw4=0. spaces.
– Similar results hold also for MoravaE(2)-theory whenX is Spin.
– Proof involves the AHSS and study of the Milnor primitives in the Steenrod algebra.
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Twisted TMF (and M-theory)
From various physical and mathematical considerations:
Conjecture (S.)
A twisted form of TMF describes the C-field in M-theory.
Ando-Blumberg-Gepner indeed construct the spectrumtwisted TMF.
Theorem (ABG)
(i) The spectrum twisted TMF exists.
(ii) Its orientation is a twisted String structure.
(iii) It admits a push forward and a Thom isomorphism.
Application:[S.]The charges of branes in M-theory take values in twisted TMF.
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Generalized cohomology with degree seven twist
Conjecture (S.)
(i) Twisted Morava K-theory and MoravaE-theory exit.
(ii) The first differential in their Atiyah-Hirzebruch spectral sequence (AHSS) corresponds to the cohomology classW7+ [H7], where[H7]acts as the twist.
Theorem (S.-Westerland)
For mod 2 Morava K-theory, the set of homotopy classes of maps
K(Z,n+2)→BGL1K(n)is a group isomorphic to the 2-adic integersZ2. Furthermore, each component in the space of such maps is contractible.
This allows us to define, for any spaceX and classH∈Hn+2(X), thetwisted Morava K-theory K(n)∗(X;H).
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Structure Condition Generalized cohom. Reference(s)
Spin w2=0 KO-theory Atiyah-Bott-Shapiro
Spinc W3=0 K-theory Atiyah-Bott-Shapiro
Twisted Spin w2+ [B2] =0 Twisted KO-theory Mathai-Murray-Stevenson Twisted Spinc W3+ [H3] =0 Twisted K-theory Freed-Witten, Wang
String 12p1=0 tmf Ando-Hopkins-Strickland
Twisted String 12p1+ [G4] =0 twisted tmf Ando-Blumberg-Gepner
StringK(Z,3) W7=0 MoravaK(2)e Kriz-S.
Twisted StringK(Z,3) W7+ [H7] =0 Twisted MoravaK(2)e H S.-Westerland
Membrane w4=0 Real MoravaEO(2) Kriz-S.
Twisted Membrane w4+ [α4] =0 ?
For the last row:
Conjecture (S.)
Such a theory is a twisted form of real MoravaEO(2)atp=2.
Work in progress with C. Westerland ...
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Twisting chromatic theories
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Formal Group Laws (FGLs)
Whitney sum: c1(L ⊕ L0) =c1(L) +c1(L0) via splitting[(1+c1(L))(1+c1(L0))](2)
Tensor product: c1(L ⊗ L0) =c1(L) +c1(L0) via cup product formula withr=1 cr(L ⊗E) =Pr
i=0c1(L)icr−i(E)
Example (Complex line bundles in K-theory)
IfLis a complex line bundle over a spaceX ;an element [L]of K-theoryK(X).
Definec1(L) = [L]−1(normalization so thatc1(trivial`. b.) =0). Then
c1(L ⊗ L0)= [L ⊗ L0]−1
= [L][L0]−1 (K(X)is a commutative ring under⊗with unit 1=C)
=c1(L) +c1(L0) +c1(L)c1(L0)
=:F(c1(L),c1(L0)) Conditions onF:
1 Sincec1(trivial`. b.) =0, thenF(x,0) =F(0,x) =x.
2 ⊗operation on complex`.b.’s commutative up to isomorphism, so F(x,y) =F(y,x).
3 ⊗operation on complex`.b.’s associative up to isomorphism, so F(x,F(y,z)) =F(F(x,y),z).
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Characterizing generalized cohomology I: Formal groups
c1(L) :X −→CP∞=BU(1), classifying space.
Definition
Acomplex orientationon E is an elementc1E ∈Ee2(CP∞), the universal Chern class inE-theory, whose canonical restriction toS2is a unit±1∈Ee2(S2)via S2=CP1,→CP∞.
Multiplication map classifying tensor product of line bundles µ:CP∞×CP∞−→CP∞ Pulling backµ∗(z)get power seriesF(x,y)∈E∗[[x,y]].
Definition
Aformal group law(FGL) over a commutative ring R is a power seriesF(x,y) with coefficients inR, such that
1 F(x,0) =x,F(0,y) =y;
2 F(x,y) =x+y+H.O.T.:=x+Fy;
3 F(x(F(y,z)) =F(F(x,y),z).
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c1E(L1⊗ L2) =c1E(L1) +F c1E(L2)
Examples
Additive FGL:F(x,y) =x+y;
c1H(L1⊗ L2) =c1H(L1) +c1H(L2).
Multiplicative FGL: F(x,y) =x+y+uxy,
c1K(L) =±1±[L] so c1K(L1⊗ L2) =c1K(L1) +c1K(L2)±c1K(L1)c1K(L2).
Classification: Over an algebraically closed field, every 1-dimensional algebraic group is isomorphic to either:
1 Ga: the additive group.
2 Gm: the multiplicative group.
3 C: an elliptic curve.
FG(L)s are obtained by formal completion of the above groups (at the identity).
Definition
Anelliptic spectrumis a spectrum which represents elliptic cohomology. A triple
1 Even periodic ring spectrumE;
2 an elliptic curveC overE0;
3 an isomorphism of formal groups betweenFGLC andC.b
Examples
1 Topological modular formsTMF∗∼=Z[E4,E6] ring of modular forms.
2 MoravaE-theoryE(n)∗∼=Z[v1,· · · ,vn,vn−1].
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(Co)bordism
Mi,i =1,2, smooth closedn-dimensional manifolds, fi :Mi →X continuous maps.
These maps arebordantif there is a mapf :W →X, with
∂W =M1`(−M2), such thatf|Mi =fi.
M1
M2
W
f1
f
f2
X
Definition
The mapf is called abordismbetweenf1andf2.
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Complex cobordism
Astable complex structureon a real vector bundleEis a fiberwise complex structure on the Whitney sumE⊕Rk.
Definition
The set of bordism classes of stably complex manifoldsMUn(X)is a group under disjoint union, thenth complex cobordism group ofX.
M∗ is a graded ring underCartesian products:
MU∗⊗MU∗−→MU∗
[M]⊗[N]7−→[M×N]
From [Milnor, Novikov]: MU∗=Z[x1,x2,x3,· · ·],|xi|=2i.
The product⊗makesMU∗(X)into a gradedMU∗-algebra:
MU∗(CP∞) =MU∗[x],|x|=2.
MU∗(CP∞ ×CP∞)∼=MU∗[x⊗1+1⊗x].
The two skeleton ofCP∞ ×CP∞is just two spheres glued at a point. The two line bundles are tautological on one sphere and trivial on the other sphere
AsCP∞is a topological group, we have a product mapµ:CP∞ ×CP∞ →CP∞inducing MU∗
(CP∞ )−→MU∗
(CP∞×CP∞ ) x7−→µ∗(x) =F(x⊗1,1⊗x)
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The chromatic viewpoint
Systematic approach to information provided by different generalized cohomology theories is given by thechromatic filtration.
Stable homotopy is the most desirable yet most complicated cohomology theory.
Stable homotopy theory (localized at a prime) is naturally filtered by chromatic layers.
layer 0 1 2 n−1
n
.. .
E(p,n)∗(−) Morava E-theory Elliptic cohomology
K-theory Ordinary cohomology
Morava K-theoryK(p,n)∗(−)
measures the difference between level n-1 and level n
nth layer (vn periodic phenomena)↔FGLs of height n
;n-dimensional varieties or higher genus curves.
To a FGG of heightnover an algebraically closed fieldk of charp associate//
Morava K-theory.
Closely related theory: Morava E-theory (ellipticforn=2).
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Introducing coefficients
Workone prime at a time⇒introduce coefficients such as:
1 the prime fieldsFp.
2 thep-local integersZ(p)={ab ∈Q|(p,b) =1}.
3 thep-adic integersZp (inverse limit ofZ/pnZ).
Given a cohomology theoryE∗(−)and a ringR, there is another cohomology theoryER(−) =E∗(−;R),E-cohomology with coefficients.
Examples
1 Homology: Replace chainsC∗(X)byC∗(X)⊗ R.
2 Cohomology: Replace the cochain complexHom(C∗(X),Z)by Hom(C∗(X),R).
3 Alternatively, replace dim. axiom byH∗(pt;R) =
R ifn=0, 0 ifn6=0.
In general: Get universal coefficient spectral sequence relatingER∗(−)and E∗(−).
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Characterizing Generalized cohomology II: Coefficients
There is auniversal FGLF over a ringL(Lazard ring), such that for any commutative ringRand any FGLG overR, there is a unique homomorphism of ringsL→R which carriesF toG.
Quillen: L∼=MU∗ ring of coefficients of complex cobordism.
MU is a complex-oriented cohomology theory.
Localizing at aprimepbreaks MU into a direct sum ofBrown-Peterson theories
BP∗=Fp[v1,v2,v3,· · ·], |vn|=2pn−2.
One inverts and/or kills regular sequences to get:
Examples
Johnson-WilsontheoryBPhni∗=Fp[v1,· · · ,vn] Morava E-theoriesE(n)∗=Fp[v1,· · ·,vn,vn−1].
Morava K-theoriesK(n)∗=Fp[vn,vn−1].
These cohomology theories exist for eachprimepandchromatic leveln∈N.
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Twists of Morava K (n) and E (n)
Recall: K(n)∗=Fp[vn,vn−1],|vn|=2pn−2.
Study homotopy of space of mapsK(Z,m)→BGL1K(n).
there are no nontrivial twistings byHm(X,Z)when m6=n+2.
Form=n+2 at any prime there are many such twistings:
Theorem (S.-Westerland)
For modpMorava K-theory, the set of homotopy classes of maps
K(Z,n+2)→BGL1K(n)is a group isomorphic to thep-adic integersZp.
A representativeu:K(Z,n+2)→BGL1K(n)of a particular topological generator of this group will be called theuniversal twisting. This allows us to define, for any spaceX and classH∈Hn+2(X), thetwisted Morava K-theory K(n)∗(X;H).
Morava E-theory: There is an A∞-mapπ:E(n)→K(n)which gives K(Z,n+1)−→ϕn GL1E(n)−→π GL1K(n)as the twist.
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Properties of twisted Morava K-theory
Twisted Morava K-theory satisfies (generalized) Eilenberg-Steenrod axioms plus the following basic properties, analogous to those of twisted K-theory:
Theorem (Properties of twisted Morava K-theory (SW))
LetK(n)∗(X;H)be twisted Morava K-theory of a spaceX with twisting class H.
Then:
1. (Normalization) IfH=0 thenK(n)∗(X;H) =K(n)∗(X).
2. (Module property)K(n)∗(X;H)is a module overK0(n)(X).
3. (Cup product) There is a cup product homomorphism
K(n)p(X;H)⊗K(n)q(X;H0)−→K(n)p+q(X;H+H0),
which makes⊕HK(n)∗(X;H)into an associative ring (whereH ranges over all of Hn+2(X;Z)).
4. (Naturality) Iff :Y →X is a continuous map, then there is a homomorphism f∗:K(n)∗(X;H)→K(n)∗(Y;f∗H).
This also holds for thetwisted Morava E-theory.
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