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Thursday, December 22, 2016

Sampling points uniformly on parametrized manifolds

Here I'll describe how to sample points uniformly on a (parametrized) manifold, along with an actual implementation in Python. Let M be a m-dimensional manifold embedded in Rn via f:RmRn. Moreover, assume that f is Lipschitz (true if M is compact), injective (true if M is embedded), and is a parameterization, in the sense that there is an m-rectangle A=[a1,b1]××[am,bm] such that f(A)=M (the intervals need not be closed). Set (˜Jf)2=det(DfDfT) to be the m-dimensional Jacobian, and calculatec=bmamb1a1˜Jf dx1dxm.Recall the brief statistical background presented in a previous blog post ("Reconstructing a manifold from sample data, with noise," 2016-05-26). A uniform or constant probability density function is valued the same at every point on its domain.

Proposition: In the setting above:
  • (completely separable) Let g1,,gm be probability density functions on [a1,b1],,[am,bm], respectively. If g1gm=˜Jf/c, then the joint probability density function of g1,,gm is uniform on M with respect to the metric induced from Rn.
  • (non-separable) Let g be a probability density function on [a1,b1]××[am,bm]. If g=˜Jf/c, then g is uniform on M with respect to the metric induced from Rn.

A much more abstract statement and proof are given in [2], Section 3.2.5, but assuming f is injective and M is in Rn, we evade the worst notation. Section 2.2 of [1] gives a brief explanation of how the given statement follows, while Section 2 of [3] goes into more detail of why the above is true.

Example:
Let M=S2, the sphere of radius r, and f:[0,2π)×[0,π)R3 the natural embedding given by
(θ,φ)(rcos(θ)sin(φ),rsin(θ)sin(φ),rcos(φ)),withDf=[rsin(φ)sin(θ)rcos(θ)sin(φ)0rcos(φ)cos(θ)rcos(φ)sin(θ)rsin(φ)],˜Jf=r2sin(φ),DfDfT=[r2sin2(φ)00r2],c=4πr2.
Let g1(θ)=1/2π be the uniform distribution over [0,2π), meaning that g2(φ)=sin(φ)/2 over [0,π). Sampling points randomly from these two distributions and applying f will give uniformly sampled points on S2.

Example: Let M=T2, the torus of major radius R and minor radius r, and f:[0,2π)×[0,2π)R3 the natural embedding given by
(θ,φ)((R+rcos(θ))cos(φ),(R+rcos(θ))sin(φ),rsin(θ)),withDf=[rcos(φ)sin(θ)rsin(φ)sin(θ)rcos(θ)(R+rcos(θ))sin(φ)cos(φ)(R+rcos(θ))0],˜Jf=r(R+rcos(θ)),DfDfT=[r200(R+rcos(θ))2],c=4π2rR.
Let g2(φ)=1/2π be the uniform distribution over [0,2π), meaning that g1(θ)=(1+rcos(θ)/R)/(2π) over [0,2π). Sampling points randomly from these two distributions and applying f will give uniformly sampled points on T2.

Below I give a simple implementation of how to actually sample points, in Python using the SciPy package. The functions f,g1,,gm are all assumed to be given.

import scipy.stats as st

class var_g1(st.rv_continuous):
    'Uniform variable 1'
    def _pdf(self, x):
        return g1(x)
...
class var_gm(st.rv_continuous):
    'Uniform variable m'
    def _pdf(self, x):
        return gm(x)

dist_g1 = var_g1(a=a1, b=b1, name='Uniform distribution 1')
...
dist_gm = var_gm(a=am, b=bm, name='Uniform distribution m')

def mfld_sample():
    return f(dist_g1.rvs(),...,dist_gm.rvs())
A further application for this would be to understand how to sample points uniformly on projective manifolds, with a leading example the Grassmannian, embedded via Plucker coordinates.

References:
[1] Diaconis, Holmes, and Shahshahani (Sampling from a manifold, Section 2.2)
[2] Federer (Geometric measure theory, Section 3.2.5)
[3] Rhee, Zhou, and Qiu (An iterative algorithm for sampling from manifolds, Section 2)

Thursday, December 8, 2016

The conditioning number of a helix, part 2

Recall the previous attempt to find the conditioning number of a helix (see post "The conditioning number of a helix, part 1," 2016-10-31). Here we complete the approach and although exact solutions are hard to find, we give close approximations.

The setting was a helix C of radius r and stretch c, so given as the zero locus of xrcos(z/c) and yrsin(z/c), and we wanted to find where the normal plane at a point pC intersects C again. It may intersect C several times, but we are only interested in the shortest distances. Without loss of generality, assume that p=(r,0,0). The normal plane at p is then given by
0=det[xrcos(pz/c)yrsin(pz/c)zpz10rsin(pz/c)/c01rcos(pz/c)/c]=det[xryz10001r/c]=rcy+z.Since the cylinder on which the helix C lies is x2+y2=r2, the curve C representing the intersection of the plane with the cylinder is given by the zero locus of ±rx2r2+cz. This allows us to find the intersection with the helix. However, since C is parametrized with z the free variable and C with x free, its is easier to switch to cylindrical coordinates
(r=x2+y2,θ=arctan(y/x),z=z).Doing so gives a nice description of C and C as below.C:(rcos(z/c),rsin(z/c),z)=(r,θ,θc)C:(x,r2x2,rr2x2/c)=(r.θ,r2sin(θ)/c)The switch in coordinates is represented by the diagram below, where we have only used the top half of C.
Finding CC is equivalent to solving c2r2=sin(θ)θ for θ, a task that can not be solved exactly. Instead we take the tangent lines to C on the unrolled cylinder at its base, and see where those intersect the line θc. Inspecting the areas of the tangent lines closer and calculating the euclidean distances in R3 from p to a and b, which is, I can't believe I'm saying this, a great exercise for the reader, we get the distances to be
d(p,a)=2r2(1+cos(πc2r2c2))+(πcr2r2c2)2,d(p,b)=2r2(1cos(2πc2r2+c2))+(2πcr2r2+c2)2.Truthfully, the diagrams are tricky to draw in TikZ and I don't want to simply have a scan of some rough work. More importantly, d(p,a)=d(p,b) implies c=r/3, meaning that when the stretch c is larger than r/3, the normal planes certainly do not intersect the helix again.

Friday, November 18, 2016

Loose ends of smooth manifolds

 Preliminary exam prep

Here we round up some theorems that have escaped previous roundings-up. Let X,Y be smooth manifolds and f:XY a smooth map.

Theorem: (Inverse function theorem) If dfp is invertible for some pM, then there exist Up and Vf(p) connected such that f|U:UV is a diffeomorphism.

Corollary: (Stack of records theorem) If dim(X)=dim(Y), then every regular value yY has a neighborhood Vy such that f1(Y)=U1Uk, where f|Ui:UiV is a diffeomorphism.

Proof: Since yY is a regular value, dfx is surjective for all xf1(y). Since dim(X)=dim(Y) and dfx is linear, dfx is an isomorphism, hence invertible. By the inverse function theorem, there exist Ux and Vy connected such that f|U:UV is a diffeomorphism. Before we actually apply this, we need to show that f1(y) is a finite set.

First we note that by the preimage theorem, since y is a regular value, f1(y) is a submanifold of X of dimension dim(X)dim(Y)=0. Next, if f1(y)={xi} were infinite, since X is compact, there would be some limit point pX of {xi}. But then by continuity,
y=limi[f(xi)]=f(limi[xi])=f(p),so pf1(y). But then either p cannot be separated from other elements of f1(y), meaning f1(y) is not a manifold, or the sequence {xi} is finite in length. Hence f1(y)={x1,,xk}. Let Uixi and Viy be the sets asserted to exist by the inverse function theorem (the Ui may be assumed to be disjoint without loss of generality). Let V=ki=1Vi and Ui=f1(V)Ui, for which we still have f|Ui:UiV a diffeomorphism.

Theorem: (Classification of manifolds) Up to diffeomorphism,
  • the only 0-dimensional manifolds are collections of points,
  • the only 1-dimensional manifolds are S1 and R, and
  • the only 2-dimensional compact manifolds are S2#(T2)#n or S2#(RP2)#n, for any n0.
Compact 2-manifolds are homeomorphic iff they are both (non)-orientable and have the same Euler characteristic. Note that
χ(S2#(T2)#n)=22n,χ(S2#(RP2)#n)=2n.
These surfaces are called orientable (on the left) and non-orientable (on the right) surfaces of genus n.

Theorem: (Stokes' theorem) For X oriented and ωΩn1X, Xdω=Xω.

Proposition: The tangent bundle TX is always orientable.

Proof: Let U,VX with φ:URn and ψ:VRn trivializing maps, and ψφ1:RnRn the transition function. To show that TX is always orientable, we need to show the Jacobian of the induced transition function (determinant of the derivative) on TX is always non-negative. On TU and TV, we have trivializing maps (φ,dφ) and (ψ,dψ), giving a transition function
(ψφ1,dψdφ1)=(ψφ1,d(ψφ1)).
The Jacobian of this is
det(d(ψφ1,d(ψφ1)))=det(d(ψφ1),d(ψφ1)))=det(d(ψφ1))det(d(ψφ1))0,
and since d(ψφ1)0 (as ψφ1 is a diffeomorphism, its derivative is an isomorphism), the result is always positive.

References: Lee (Introduction to smooth manifolds, Chapter 4), Guillemin and Pollack (Differential topology, Chapter 1

Sunday, November 13, 2016

Covering spaces

 Preliminary exam prep

Let X,Y be topological spaces.

Definition: A space ˜X and a map p:˜XX are called a covering space of X if either of two equivalent conditions hold:
  • There is a cover {Uα}αA of X such that p1(Uα)βBαUβ.
  • Every point xX has a neighborhood UX such that p1(U)βBUβ.
We also demand that every Uβ is carried homeomorphically onto Uα (or U) by p, and the Uα (or U) are called evenly covered.

Some definitions require that p be surjective. A universal cover of X is a covering space that is universal with respect to this property, in that it covers all other covering spaces. Moreover, a cover that is simply connected is immediately a universal cover.

Remark:
Every path connected (pc), locally path connected (lpc), and semi locally simply connected (slsc) space has a universal cover.

Theorem: (Lifting criterion) Let Y be pc and lpc, and ˜X a covering space for X. A map f:YX lifts to a map ˜f:Y˜X iff f(π1(Y))p(π1(˜X)).

Further, if the initial map f0 in a homotopy ft:YX lifts to ˜f0:Y˜X, then ft lifts uniquely to ˜X. This is called the homotopy lifting property. Next, we will see that path connected covers of X may be classified via a correspondence through the fundamental group.

Theorem:
Let X be pc, lpc, and slsc. There is a bijection (up to isomorphism) between pc covers p:˜XX and subgroups of π1(X), described by p(π1(˜X)).

Example:
Let X=T2, the torus, with fundamental group ZZ. Below are some covering spaces of p:˜XX with the corresponding subgroups p(π1(ZZ)).
Definition: Given a covering space p:˜XX, an isomorphism g of ˜X for which idXp=pg, is called a deck transformation, the collection of which form a group G(˜X) under composition. Further, ˜X is called normal (or regular) if for every xX and every ˜x1,˜x2p1(x), there exists gG(˜X) such that g(˜x1)=˜x2.

For path connected covering spaces over path connected and locally path connected bases, being normal is equivalent to p(π1(˜X))π1(X) being normal. In this case, G(˜X)π1(X)/p(π1(˜X)). This simplifies even more for ˜X a universal cover, as π1(˜X)=0 then.

Theorem: Let G be a group, and suppose that every xX has a neighborhood Ux such that g(U)h(U)= whenever ghG. Then:
  • The quotient map q:XX/G describes a normal cover of X/G.
  • If X is pc, then G=G(X).
A group action satisfying the hypothesis of the previous theorem is called a covering space action.

Proposition: For any n-sheeted covering space ˜XX of a finite CW complex, χ(˜X)=nχ(X).

References: Hatcher (Algebraic Topology, Chapter 1)

Thursday, November 10, 2016

Differential 1-forms are closed if and only if they are exact

 Preliminary exam prep

The title refers to 1-forms in Euclidean n-space Rn, for n2. This theorem is instructive to do in the case n=2, but we present it in general. We will use several facts, most importantly that the integral of a function f:XY over a curve γ:[a,b]X is given by
γf dx1dxk=ba(fγ) d(x1γ)d(xnγ),
where x1,,xn is some local frame on X. We will also use the fundamental theorem of calculus and one of its consequences, namely
baft(t) dt=f(b)f(a).

Theorem:
A 1-form on Rn is closed if and only if it is exact, for n2.

Proof: Let ω=a1dx1+andxnΩ1Rn be a 1-form on Rn. If there exists ηΩ0Rn such that dη=ω, then dω=d2η=0, so the reverse direction is clear. For the forward direction, since ω is closed, we have
0=dω=ni=1a1xidxidx1++ni=1anxndxidxn      aixj=ajxi  ij.
Now fix some (x1,,xn)Rn, and define fΩ0Rn by
f(x1,,xn)=γ(x1,,xn)ω,
for γ the composition of the paths
γ1 : [0,x1]Rn,t(t,0,,0),γ2 : [0,x2]Rn,t(x1,t,0,,0),γn : [0,xn]Rn,t(x1,,xn1,t).
By applying the definition of a pullback and the change of variables formula (use s=γi(t) for every i),
γ(x1,,xn)ω=ni=1γia1dx1++ni=1γiandxn=ni=1γia1(x1,,xn) dx1++ni=1γian(x1,,xn) dxn=ni=1xi0a1(γi(t)) d(x1γi)(t)++ni=1xi0an(γi(t)) d(xnγi)(t)=x10a1(γ1(t))γ1(t) dt++xn0an(γn(t))γn(t) dt=(x1,0,,0)(0,,0)a1(s) ds++(x1,,xn)(x1,,xn1,0)an(s) ds=x10a1(s,0,,0) ds++xn0an(x1,,xn1,s) ds.
To take the derivative of this, we consider the partial derivatives first. In the last variable, we have
fxn=xnxn0an(x1,,xn1,s) ds=an(x1,,xn)=an.
In the second-last variable, applying one of the identities from ω being closed, we have
fxn1=xn1xn10an1(x1,,xn2,s,0) ds+xn1xn0an(x1,,xn1,s) ds=an1(x1,,xn1,0)+xn0anxn1(x1,,xn1,s) ds=an1(x1,,xn1,0)+xn0an1s(x1,,xn1,s) ds=an1(x1,,xn1,0)+an1(x1,,xn)an1(x1,,xn1,0)=an1(x1,,xn)=an1.
This pattern continues. For the other variables we have telescoping sums, and we compute the partial derivative in the first variable as an example:
fx1=x1x10a1(s,0,,0) ds+ni=2x1xi0ai(x1,,xi1,s,0,,0) ds=a1(x1,0,,0)+ni=2xi0aix1(x1,,xi1,s,0,,0) ds=a1(x1,0,,0)+ni=2xi0a1s(x1,,xi1,s,0,,0) ds=a1(x1,0,,0)+ni=2(a1(x1,,xi,0,,0)a1(x1,,xi1,0,,0))=a1(x1,,xn)=a1.
Hence we get that
df=fx1dx1++fxndxn=a1dx1++andxn=ω,
so ω is exact.

References: Lee (Introduction to smooth manifolds, Chapter 11)

Tuesday, November 8, 2016

More (co)homological constructions

 Preliminary exam prep

Recall a previous post (2016-09-16, "Complexes and their homology") that focused on constructing topological spaces in different ways and recovering the homology. Here we complete that task, introducing cellular homology. Recall a cell complex (or CW complex) X was a sequence of skeleta Xk for k=0,,dim(X) consisting of k-cells eki and their attaching maps to the (k1)-skeleton.

Cellular homology


Definition: The long exact sequence in relative homology for the pair Xk,Xk1 shares terms with the long exact sequence for the pair Xk+1,Xk, as well as Xk1,Xk2. By letting dk be the composition of maps in different long exact sequences, for k>1, that make the diagram
commute, we get a complex of equivalence classes of chains
Hk+1(Xk+1,Xk)dk+1Hk(Xk,Xk1)dkHk1(Xk1,Xk2)H1(X1,X0)d1H0(X0)d00,
whose homology HCWk(X)=ker(dk)/im(dk1) is called the cellular homology of X. The map d1 is the connecting map in the long exact sequence of the pair X1,X0, and d0=0.

This seems quite a roundabout way of defining homology groups, but it turns out to be very useful. Note that for k=1, the map d1 is the same as for a simplicial complex, hence

Theorem:
In the context above,
  1. for k0, HCWk(X)Hk(X);
  2. for k1, Hk(Xk,Xk1)=Z, where is the number of k-cells in X; and
  3. for k2, dk(eki)=jdeg(ekiSk1fk,iXk1πXk1/Xk1ek1jSk1)ek1j.
Example: Real projective space RPn has a cell decomposition with one cell in each dimension, and 2-to-1 attaching maps (ek)=2Xk1 for k>1. This gives us a construction
X0=e0,X1=e1(e1)=e0X0,X2=e2(e2)=2e1X1,X3=e3(e3)=2e2X2,It is immediate that d0=d1=0, and for higher degrees, we have
dk(ek)=deg(Sk1RPk1Sk1)ek1.
Since this is a map between spheres, we may apply local degree calculations. The first part is the 2-to-1 cover, where every point in RPk1 is covered by two points from Sk1, one in each hemisphere. One covers it via the identity, the other via the antipodal map. As long as we choose a point not in RPk2RPk1, the second step doesn't affect these degree calculations. The antipodal map Sk1Sk1 has degree (1)k, hence for a the antipodal map, the composition has degree
deg(Sk1RPk1Sk1)=deg(idSk1)+deg(aSk1)=1+(1)k={2k even,0k odd.

Products in (co)homology


Recall that an n-chain on X is a map σ:ΔnX, where Δn=[v0,,vn] is an n-simplex. These form the group Cn of n-chains. An n-cochain is an element of Cn=Hom(Cn,Z), though the coefficient group does not need to be Z necessarily.

Definition: The diagonal map XX×X induces a map on cohomology H(X×X)H(X), and by Kunneth, this gives a map H(X)H(X)H(X), and is called the cup product.

For aHp(X) and bHq(X), representatives of the class a are in Hom(Cp,Z) and representatives of the class b are in Hom(Cq,Z), though we will conflate the notation for the class with that of a representative. Hence for a (p+q)-chain σ the cup product of a and b acts as
(ab)σ=a(σ|[v0,,vp])b(σ|[vp,,vp+q]).
Definition: The cap product combines p-cochains with q-chains to give (qp)-chains, by
 : Hp(X)×Hq(X)Hqp(X), (a,σ)a(σ|[v0,,vp])σ|[vp,,vq].
The cap product with the orientation form of an orientable manifold X gives the isomorphism of Poincare duality.

Remark: Given a map f:XY, the cup and cap products satisfy certain identities via the induced map on cohomology groups. Let a,bH(Y) and cH(X) be cochain and chain classes, for which
f(ab)=f(a)f(b),afc=f(fac).
The first identity asserts that f is a ring homomorphism and the second describes the commutativity of an appropriate diagram. The cup and cap products are related by the equation
a(bσ)=(ab)σ,for aHp, bHq and σCp+q.

References: Hatcher (Algebraic topology, Chapter 2.2), Prasolov (Elements of homology theory, Chapter 2)

Monday, November 7, 2016

Images of manifolds and transversality

 Preliminary exam prep

Let X,Y be manifolds embedded in Rn, and f:XY a map, with dfx:TxXTf(x)Y the induced map on tangent spaces.

Definition: The map f is a
  • homeomorphism if it is continuous and has a continuous inverse, 
  • diffeomorphism if it is smooth and has a smooth inverse,
  • injection if f(a)=f(b) implies a=b,
  • immersion if dfx is injective for all xX,
  • embedding if it is an immersion and dfx is a homeomorphism onto its image,
  • submersion if dfx is surjective for all xX.
Transversality is a mathematical relic whose only practical use is, perhaps, in classical algebraic geometry.

Definition: The manifolds X and Y are transverse if TpXTpYRn for every pXY. The map f and Y are transverse if im(f) and Y are transverse.

Note that being transverse (or transversal) is a symmetric, but not a reflexive, nor a transitive relation. Recall that a regular value of f is yY such that dfx:TxXTf(x)Y is surjective for all xf1(y). If y is not in the image of f, then f1(y) is empty, so y is trivially a regular value. Every value that is not a regular value is a critical value.

Theorem: (Preimage theorem) For every regular value y of f, the subset f1(y)X is a submanifold of X of dimension dim(X)dim(Y).

Now let M be a submanifold of Y.

Corollary: If f is transverse to M, then f1(M) is a manifold, with codimY(M)=codimX(f1(M)).

Theorem: (Transversality theorem) Let {gs:XY | sS} be a smooth family of maps. If g:X×SY is transverse to M, then for almost every sS the map gs is transverse to M.

If we replace f with df, and ask that it be transverse to M, then df|s is also transverse to M.

Example: Consider the map gs:XRn given by gs(X)=i(X)+s=X+s, where i is the embedding of X into Rn. Since g(X×Rn)=Rn and g varies smoothly in both variables, we have that g is transverse to X. Hence by the transversality theorem, X is transverse to its translates X+s for almost all sRn.

Theorem: (Sard) For f smooth and Y=, almost every yY is a regular value of f and f|X. Equivalently, the set of critical values of f has measure zero.

Resources: Guillemin and Pollack (Differential topology, Chapters 1, 2), Lee (Introduction to smooth manifolds, Chapter 6)

Friday, November 4, 2016

Tools of homotopy

 Preliminary exam prep

Let X,Y be topological spaces and A a subspace of X. Recall that a path in X is a continuous map γ:IX, and it is closed (or a loop), if γ(0)=γ(1). When X is pointed at x0, we often require γ(0)=x0, and call such paths (and similarly loops) based.

Definitions


Definition:
  • X is connected if it is not the union of two disjoint nonempty open sets.
  • X is path connected if any two points in X have a path connecting them, or equivalently, if π0(X)=0.
  • X is simply connected if every loop is contractible, or equivalently, if π1(X)=0.
  • X is semi-locally simply connected if every point has a neighborhood whose inclusion into X is π1-trivial.
Path connectedness and simply connectedness have local variants. That is, for P either of those properties, a space is locally P if for every point x and every neighborhood Ux, there is a subset VU on which P is satisfied.

Remark: In general, X is n-connected whenever πr(X)=0 for all rn. Note that 0-connected is path connected and 1-connected is simply connected and connected. Also observe that the suspension of path connected space is simply connected.

Definition:
  • A retraction (or retract) from X to A is a map r:XA such that r|A=idA.
  • A deformation retraction (or deformation retract) from X to A is a family of maps ft:XX continuous in t,X such that f0=idX, f1(X)=A, and ft|A=idA for all t.
  • A homotopy from X to Y is a family of maps ft:XY continuous in t,X.
  • A homotopy equivalence from X to Y is a map f:XY and a map g:YX such that gfidX and fgidY.
Definition: A pair (X,A), where AX is a closed subspace, is a good pair, or has the homotopy extension property (HEP), if any of the following equivalent properties hold:
  • there exists a neighborhood UX of A such that U deformation retracts onto A,
  • X×{0}A×I is a retract of X×I, or
  • the inclusion i:AX is a cofibration.
In some texts such a pair (X,A) is called a neighborhood deformation retract pair, and HEP is reserved for any map AX, not necessarily the inclusion, that is a cofibration. For more on cofibrations, see a previous blog post (2016-07-31, "(Co)fibrations, suspensions, and loop spaces").

Definition: There is a functor π1:TopGrp called the fundamental group, that takes a pointed topological space X to the space of all pointed loops on X, modulo path homotopy.

This may be generalized to πn, which takes X to the space of all pointed embeddings of Sn.

Definition: Let G,H be groups. The free product of G and H is the group
GH={a1an : nZ0,aiG or H,aiG(H)ai+1H(G)},
with group operation concatenation, and identity element the empty string . We also assume eGeH=eHeG=eG=eH=, for eG (eH) the identity element of G (H).

The above construction may be generalized to a collection of groups G1Gm, where the index may be uncountable. If every Gα=Z (equivalently, has one generator), then αAGα is called the free group on |A| generators.

Theorems


Theorem: (Borsuk-Ulam) Every continuous map SnRn takes a pair of antipodal points to the same value.

Theorem: (Ham Sandwich theorem) Let U1,,Un be bounded open sets in Rn. There exists a hyperplane in Rn that divides each of the open sets Ui into two sets of equal volume.

Volume is taken to be Lebsegue measure. The Ham sandwich theorem is an application of Borsuk-Ulam (see Terry Tao's blog post for more).

Theorem: If X and Y are path-connected, then π1(X×Y)π1(X)×π1(Y).

Now suppose that X=αAα is based at x0 with x0Aα for all α. There are natural inclusions iα:AαX as well as jα:AαAβAα and jβ:AαAβAβ.
Both iα and jα induce maps on the fundamental group, each (and all) of the iα:π1(Aα)π1(X) extending to a map Φ:απ1(Aα)π1(X).

Theorem: (van Kampen)
  • If AαAβ is path-connected, then Φ is a surjection. 
  • If AαAβAγ is path connected, then ker(Φ)=jα(g)(jβ(g))1 | gπ1(AαAβ,x0).
As a consequence, if triple intersections are path connected, then π1(X)αAα/ker(Φ). Moreover, if all double intersections are contractible, then ker(Φ)=0 and π1(X)αAα.

Proposition: If π1(X)=0 and ˜Hn(X)=0 for all n, then X is contractible.

References: Hatcher (Algebraic topology, Chapter 1), Tao (blog post "The Kakeya conjecture and the Ham Sandwich theorem")

Tuesday, November 1, 2016

Explicit pushforwards and pullbacks

 Preliminary exam prep

Here we consider a map f:MN between manifolds of dimension m and n, respectively, and the maps that it induces. Let pM with x1,,xm a local chart for Up and y1,,yn a local chart for Vf(p). Induced from f are the differential (or pushforward) df and the pullback df, which are duals of each other:
dfp : TpMTf(p)Ndf : TMTNα(βα(βf))dfp : Tf(p)NTpMdf : TNTMωωfkTNkTMω dy1dyk(ωf) d(y1f)d(ykf)
These maps may be described by the diagram below.
Example: For example, consider the map f:R3R3 given by f(x,y,z)=(xy,3z2,xz+yz), with the image having coordinates (u,v,w). With elements
2xx5zyTM,2uv+w5C(N),cos(uv)TN,
we have
dfp(2xx5zy)(2uv+w5)=(2xx5zy)(6(xy)z2+xz+yz5)(p),dfp(cos(uv))=cos((xy)3z2),(2dfp)(cos(uv)dudw)=cos((xy)3z2)d(3z2)d(xz+yz)=cos((xy)3z2)(6z2 dxdz6z2 dydz).

Monday, October 31, 2016

The conditioning number of a helix, part 1

Definition: Let M be a smooth d-manifold embedded in Rn and NϵpM=NpMB(p,ϵ) the natural embedding of the ϵ-normal plane at pM. The pairwise conditioning number of p and q is
τp,q=sup{ϵ : NϵpMNϵqM embeds in Rn}.

The condition on ϵ is the same as saying i(NϵpM)i(NϵqM)=, where i is induced by the embedding of M. It is immediate that τ=infp,q{τp,q}, so we will try to find τp,q first. Recall that a helix of radius r and vertical period 2πc is a 1-dimensional manifold
embedded in R3 as the zero locus of
f(x,y,z)=xrcos(z/c),g(x,y,z)=yrsin(z/c).
We first find the normal plane at two arbitrary points p1,p2 on the helix, then their intersection (which is a line), and then the distance from p1 and p2 to that line. The smallest of these two distances bounds τp1,p2 from below (and the bound is achieved on pairs of points defining the medial axis). Then take the infimum of this value over all points on the helix. However, this excludes the case when the normal planes are parallel (for instance when the two points have the same x- and y-values).

Moreover, even just calculating the infimum for points whose normal planes are not parallel yields a result of zero. We describe the process nonetheless. For the first step, we need the equations of the normal planes. Let
Df=[10rsin(z/c)/c],Dg=[01rcos(z/c)/c].
be the Jacobians of f and g. The points p1, p2 are completely described by the z-coordinate, so we have two values z1, z2 for p1, p2, respectively. The normal plane at pi is the zero locus of
det[xrcos(zi/c)yrsin(zi/c)zzi10rsin(zi/c)/c01rcos(zi/c)/c]=zzixrcsin(zi/c)+yrccos(zi/c).
We have two equations and three unknowns, so one independent variable. Solving for x and y gives us
x=(zz1)cos(z2/c)(zz2)cos(z1/c)rsin(z1z2c)/c,y=(zz1)sin(z2/c)(zz2)sin(z1/c)rsin(z1z2c)/c.
These are functions of z, giving us two new functions
hi(z)=(x(z)rcos(zi/c))2+(y(z)rsin(zi/c))2+(zzi)2,
for i=1,2, which, when minimized, give a lower bound for the pairwise conditioning number of p1 and p2. Indeed, by slowly increasing the ϵ until the ϵ-normal planes at p1 and p2 intersect, the first point of intersection will happen on the intersection Np1MNp2M. Hence finding the shortest distance from p1 and p2 to this line gives a definite lower bound. The functions hi are quadratic in z, and we know the function az2+bz+c, for a>0, has minimum at b/2a. The values of h1 and h2 at their minima are the same and equal to
hm:=hi(b2a)=2(c2+r2)cos2(z1z22c)(r2+c(z1z2)csc(z1z2c))22c2r2+r4+r4cos(z1z2c).
A natural limit of hm to consider is z2z1. If any of the factors in the numerator are zero, we also get a minimum, so another limit to look for is z2cz1, which makes the cosine factor zero. These are
limz2z1[hm]=(c2+r2)2r2,limz2z1+cπ[hm]=c2π2(c2+r2)4r2,
which are finite nonzero for positive values of c and r. For the last factor, fix z1=0. Then finding when the factor vanishes is equivalent to finding when sin(z2/c) and z2c/r2 intersect. There are values for which this happens, and the other factors in hm are all finite at these values, so infzR[hi(z)]=0. Visual confirmation is given by the cases below.
Hence this is not the best approach to calculate the conditioning number of a curve. The next attempt will be to calculate the actual pairwise conditioning number, rather than trying to bound it from below.

Tuesday, October 25, 2016

Basic topological constructions

 Preliminary exam prep

Let X,Y be topological spaces based at x0,y0, respectively, and I=[0,1] the unit interval.
coneCX=X×I/X×{0}suspensionΣX=X×I/X×{0},X×{1}reduced suspension˜ΣX=X×I/X×{0},X×{0},{x0}×IwedgeXY=XY/{x0}{y0}smashXY=X×Y/X×{y0},{x0}×YjoinXY=X×Y×I/X×{y}×{0} yY{x}×Y×{1} xXconnected sumX#Y=(XDnX)(YDnY)/DnXDnY
In the last description, X and Y are assumed to be n-manifolds, with DnX a closed n-dimensional disk in X (similarly for Y). The quotient identification may also be made via some non-trivial map. In fact, only the interior of each n-disk is removed from X and Y, so that the quotient makes sense.

Remark: Some of the above constructions may be expressed in terms of others, for example
XY=X×Y/XY,XY=Σ(XY).
The first is clear by viewing X=X×{y0} and Y={x0}×Y as sitting inside X×Y. The second is clear by letting X×{y}×{0} be identified to {x0}×{y}×{0} for every yY, and analogously with Y.

Example:
Here are some of the constructions above applied to some common spaces.
CXptΣSn=Sn+1SnSm=Sn+mΣX=S1XSnSm=Sn+m+1
Remark: We may also calculate the homology of the new spaces in terms of the old ones.
˜Hk(CX)=0via homotopy˜Hk(ΣX)=˜Hk1(X)via Mayer--Vietoris˜Hk(XY)=˜Hk(X)˜Hk(Y)via Mayer--Vietoris˜Hk(XS)=˜Hk(X)via Kunneth˜Hk(X#Y)=˜Hk(X)˜Hk(Y)via Mayer--Vietoris and relative homology
The last equality holds for k<n1, for M and N both n-manifolds, and for k=n1 when at least one of them is orientable.

References: Hatcher (Algebraic Topology, Chapters 0, 2)

Thursday, October 13, 2016

Tools of (co)homology

 Preliminary exam prep

Let X,Y be topological spaces, G a group, and R a unital commutative ring.

Defining homology groups


Theorem: If (X,A) is a good pair (there exists a neighborhood UX of A such that U deformation retracts onto A), then for i:AX the inclusion and q:X the quotient maps, there exists a long exact sequence of reduced homology groups
\cdots \to \widetilde H_n(A) \tov{i_*} \widetilde H_n(X) \tov{q_*} \widetilde H_n(X/A) \to \cdots.

Theorem: For any pair (X,A), there exists a long exact sequence of homology groups
\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to \cdots,
where the last is called a relative homology group. Hence H_n(X,A)\cong \widetilde H_n(X/A) for a good pair (X,A).

Theorem (Excision): For any triple of spaces (Z,A,X) with \text{cl}(Z)\subset \text{int}(A), there is an isomorphism H_n(X-Z,A-Z)\cong H_n(X,A).

For any x\in X, the local homology of X at x is the relative homology groups H_n(X,X-\{x\}). By excision, these are isomorphic to H_n(U,U-\{x\}) for U any neighborhood of x. If X is nice enough around x (that is, if U\cong \R^k), then these groups are isomorphic to H_n(\R^k,\R^k-\{x\})\cong H_n(D^k,\dy D^k) = H_n(S^k).

Theorem (Mayer-Vietoris): For X=A\cup B, there is a long exact sequence of homology groups
\cdots \to H_n(A\cap B) \to H_n(A)\oplus H_n(B) \to H_n(X) \to \cdots,
and if A\cap B is non-empty, there is an analogous sequence for reduced homology groups.

Extending with coefficients


Recall the \Tor and \Ext groups, which were, respectively, the left and right derived functors of, respectively, \otimes and \Hom (see post "Exactness and derived functors," 2016-03-20). Here we only need \Tor_1 and \Ext^1, which are given by, for any groups (that is, \Z-modules) A, B,
\begin{array}{r c c c l} \Tor(A,B) & = & H_1(\text{projres}(A)\otimes B) & = & H_1(A\otimes \text{projres}(B)), \\ \Ext(A,B) & = & H^1(\Hom(A,\text{injres}(B))) & = & H^1(\Hom(\text{projres}(A),B)). \end{array}
Note that \Tor is symmetric in its arguments, while \Ext is not. Recall that \Tor_0(A,B)=A\otimes B and \Ext^0(A,B) = \Hom(A,B).

Theorem (Universal coefficient theorem): There exist isomorphisms
\begin{array}{r c c c l} H_n(X;G) & \cong & \Hom(H^n(X),G)\oplus \Ext(H^{n+1}(X),G) & \cong & H_n(X)\otimes G\ \oplus\ \Tor(H_{n-1}(X),G),  \\ H^n(X;G) & \cong & \Hom(H_n(X),G)\oplus \Ext(H_{n-1}(X),G) & \cong & H^n(X)\otimes G\ \oplus\ \Tor(H^{n+1}(X),G). \end{array}

Here are some common \Hom, \Tor, and \Ext groups:
\begin{align*} \Hom(\Z,G) & = G & \Tor(\Z,G) & = 0 & \Ext(\Z,G) & = 0 \\ \Hom(\Z_m,\Z) & = 0 & \Tor(G,\Z) & = 0 & \Ext(\Z_m,\Z) & = \Z_m \\ \Hom(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Tor(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} & \Ext(\Z_m,\Z_n) & = \Z_{\gcd(m,n)} \\ \Hom(\Q,\Z_n) & = 0 & & & \Ext(\Q,\Z_n) & = 0 \\ \Hom(\Q,\Q) & = \Q & & & \Ext(G,\Q) & = 0 \end{align*}
Theorem (Künneth formula): For X,Y CW-complexes, F a field, and H^k(Y;G) or H^k(X;G) finitely generated for all k, there are isomorphisms, for all k,
H_k(X\times Y;F) \cong \bigoplus_{i+j=k} H_i(X;F)\otimes_FH_j(Y;F), \hspace{1cm} H^k(X\times Y;G) \cong \bigoplus_{i+j=k} H^i(X;G)\otimes_GH^j(Y;G)

Dualities


Theorem (Poincaré duality): For X a closed n-manifold (compact, without boundary) that is R-orientable (consistent choice of R-generator for each local homology group), for k=0,\dots,n there are isomorphisms
H^k(X;R)\cong H_{n-k}(X;R).

Note that a simply orientable manifold means \Z-orientable. A manifold that is not \Z-orientable is always \Z_2-orientable (in fact all manifolds are \Z_2-orientable).

Theorem (Alexander duality): For X\subsetneq S^n a non-empty closed locally contractible subset, for k=0,\dots,n-1 there are isomorphisms
\widetilde H^k(X) \cong \widetilde H_{n-k-1}(S^n-X).

References: Hatcher (Algebraic topology, Chapters 2, 3), Aguilar, Gitler, and Prieto (Algebraic Topology from a Homotopical Viewpoint, Chapter 7)

Monday, October 10, 2016

Vector fields

 Preliminary exam prep

Here we will have an overview of vector fields and all things related to them. Let M be an n-dimensional manifold, and \pi:M\to TM its tangent bundle.

Definition: A vector field is a map X:M\to TM such that \pi\circ X = \id_M.

A vector field may also be viewed as a section of the tangent bundle, and smooth vector fields as the space of smooth sections \Gamma(TM). Given a chart (U,\varphi) of M near p, we have the pushforward \varphi_*:T_pM\to T_{\varphi(p)}(\R^n) = \R^n, where we may assume \varphi(p)=0. Given the standard basis \{e_i\} of \R^n, we get a basis of T_pM given by
\left\{\left.\frac{\dy}{\dy x_i}\right|_p = (\varphi_*)^{-1}(e_i)\right\}_{i=1}^n.
Recall that TM may be viewed as the space of derivations, or maps C^\infty(M)\to \R satisfying the Leibniz rule. Then for p\in M, we have X(p):C^\infty(M)\to \R, so we have X(p)(f) = X_p(f)\in \R for all f\in C^\infty(M). Hence X_p\in T_pM, and X(f)\in C^\infty(M). Briefly,
\begin{array}{r c l} f\ :\ M & \to & \R, \\ X\ :\ M & \to & TM, \end{array} \hspace{2cm} \begin{array}{r c l} Xf\ :\ M & \to & \R, \\ fX\ :\ M & \to & TM. \end{array}

Definition: Given a vector field X\in \Gamma(TM), an integral curve of X is a smooth curve \gamma:\R \to M such that \gamma'(t) = X_{\gamma(t)} for all t\in \R.

The domain of \gamma need not be all of \R, though any integral curve may be extended to a maximal integral curve, for which the domain can not be made larger. A collection of integral curves for a particular vector field is a flow.


Definition: A flow, or a one paramater group of diffeomorphisms, is a smooth map \psi:\R\times M\to M such that
  1. \psi(t,\cdot) is a diffeomorphism of M, for all t,
  2. \psi(0,\cdot) = \id_M,
  3. \psi(s+t,\cdot) = \psi(s,\cdot)\circ \psi(t,\cdot).
For convenience, we write \psi_t(p) = \psi(t,p), Note that fixing p\in M, the map \psi(\cdot,p) is a integral curve. Moreover, flows and vector fields are related uniquely by
\left.\frac{d f}{d t} \psi_t(p)\right|_{t=0} = X_p(f).
Indeed, if we have a flow \psi and an element f\in \Hom(T^*_pM,\R), this gives us a vector field X\in \Gamma(TM). Conversely, if we have a vector field X, by the existence and uniqueness of solutions to first order ordinary differential equations (with boundary conditions), we can find a \psi that satisfies this equality.

Definition: Let X,Y\in \Gamma(TM) and \psi be the associated flow of X. The Lie derivative of Y in the direction of X, or Lie bracket of X and Y, is an element of \Gamma(TM) given by
\begin{align*} \left(\mathcal L_XY\right)_p(f) & = \left.\frac{df}{dt}\right|_{t=0}\bigg((\psi_t)_*^{-1}(Y_{\psi_t(p)}(f))\bigg) \\ & = [X,Y]_p(f) \\ & = X_p(Y(f)) - Y_p(X(f)) \end{align*}

The Lie derivative has some properties, among them \mathcal L_X(fY) = X(fY) + f(\mathcal L_XY) for any f\in C^\infty(M). If we let Y be the map M\to TM given by
\begin{array}{r c l} Y\ :\ M & \to & \Hom(T^*M,\R),\\ p & \mapsto & \left(\begin{array}{r c l} f_p\ :\ C^\infty(M) & \to & \R, \\ g & \mapsto & g(p), \end{array}\right), \end{array}
then Yf = f, so \mathcal L_XY = X-X = 0, and we have \mathcal L_X f = Xf.

Remark:
Vector fields are 1-forms, or elements of \mathcal A^0_M(TM) = \Gamma(TM\otimes \bigwedge^0T^*M) = \Gamma(TM). We may generalize the definition above to consider the Lie derivative \mathcal L_X\omega of a differential k-form \omega . Note that a differential k-form takes in k vector fields and gives back a smooth function M\to \R. With this in mind, we may define new operations on vector fields:
\begin{align*} (\mathcal L_X\omega)(Y_1,\dots,Y_k) & = \mathcal L_X(\omega(Y_1,\dots,Y_k)) - \sum_{i=1}^k\omega(Y_1,\dots,\mathcal L_XY_i,\dots,Y_k) \\ (d\omega)(Y_1,\dots,Y_{k+1}) & = \sum_{i=1}^{k+1}(-1)^{i-1}Y_i(\omega(Y_1,..,\widehat{Y_i},..,Y_{k+1})) + \sum_{j>i=1}^{k+1}(-1)^{i+j}\omega([Y_i,Y_j],Y_1,..,\widehat{Y_i},..,\widehat{Y_j},..,Y_{k+1}) \\ (i_X\omega)(Y_1,\dots,Y_{k-1}) & = \omega(X,Y_1,\dots,Y_{k-1}) \end{align*}

The last is the interior product. All three are related by Cartan's formula \mathcal L_X\omega = d(i_X\omega)+i_X(d\omega):
\begin{align*} (\mathcal L_{Y_1}\omega)(Y_2,\dots,Y_{k+1}) & = Y_1(\omega(Y_2,\dots,Y_{k+1})) - \sum_{i=2}^{k+1}\omega(Y_2,\dots,[Y_1,Y_i],\dots,Y_k) \\ & = Y_1(\omega(Y_2,\dots,Y_{k+1})) - \sum_{i=2}^{k+1}(-1)^i\omega([Y_1,Y_i],Y_2,\dots,\widehat{Y_i},\dots,Y_k) \\ (d(i_{Y_1}\omega))(Y_2,\dots,Y_{k+1}) & =  \sum_{i=2}^{k+1}(-1)^iY_i(\omega(Y_1,..,\widehat{Y_i},..,Y_{k+1})) - \sum_{j>i=2}^{k+1}(-1)^{i+j}\omega([Y_i,Y_j],Y_1,..,\widehat{Y_i},..,\widehat{Y_j},..,Y_{k+1})\\ (i_{Y_1}(d\omega))(Y_2,\dots,Y_{k+1}) & = (d\omega)(Y_1,\dots,Y_{k+1}) \\ & = \sum_{i=1}^{k+1}(-1)^{i-1}Y_i(\omega(Y_1,..,\widehat{Y_i},..,Y_{k+1})) + \sum_{j>i=1}^{k+1}(-1)^{i+j}\omega([Y_i,Y_j],Y_1,..,\widehat{Y_i},..,\widehat{Y_j},..,Y_{k+1}) \end{align*}

Remark: The action of a k-differential form on a k-vector field is given by \left(dx_1\wedge \cdots \wedge dx_k\right)\left(\frac\dy{\dy y_1},\dots,\frac\dy{\dy y_p}\right) = \det\begin{bmatrix} dx_1\frac\dy{\dy y_1} & dx_1\frac{\dy}{\dy y_2} & \cdots & dx_1\frac\dy{\dy y_p} \\ dx_2\frac\dy{\dy y_1} & dx_2\frac{\dy}{\dy y_2} & \cdots & dx_2\frac\dy{\dy y_p} \\ \vdots & \vdots & \ddots & \vdots \\ dx_p\frac\dy{\dy y_1} & dx_p\frac{\dy}{\dy y_2} & \cdots & dx_p\frac\dy{\dy y_p} \end{bmatrix} = \det\left(dx_i\frac\dy{\dy y_j}\right). This may be generalized to get a map \wedge^k T^*M \oplus \Gamma(TM)^{\oplus \ell} \to \bigwedge^{k-\ell}T^*M, for \ell\leqslant k. For example, given a basis x,y of our space M, (dx\wedge dy)\left(x\frac\dy{\dy x} + y \frac\dy{\dy y}\right) = dx\left(x\frac\dy{\dy x} + y \frac\dy{\dy y}\right)dy - dy\left(x\frac\dy{\dy x} + y \frac\dy{\dy y}\right)dx = x\ dy - y\ dx. When \ell=1, this is just the interior product.

References: Lee (Introduction to smooth manifolds, Chapter 8), Hitchin (Differentiable manifolds, Chapter 3)

Thursday, September 29, 2016

The tangent space and differentials

 Preliminary exam prep

Let M,N be smooth n-manifolds. Here we discuss different definitions of the tangent space and differentials, or pushforwards, of smooth maps f:M\to N.

Derivations (Lee)

Definition: A derivation of M at p\in M is a linear map v:C^\infty(M)\to \R such that for all f,g\in C^\infty(M),
v(fg) = f(p)v(g) + g(p)v(f).
The tangent space T_pM to M at p is the set of all derivations of M at p.

Given a smooth map F:M\to N and p\in M, define the differential dF_p:T_pM\to T_{f(p)}N, which, for v\in T_pM and f\in C^\infty(N) acts as
dF_p(v)(f) = v(f\circ F)\in \R.

Dual of cotangent (Hitchin)

Definition: Let Z_p\subset C^\infty(M) be the functions whose derivative vanishes at p\in M. The cotangent space T_p^*M to M at P is the quotient space C^\infty(M)/Z_p. The tangent space to M at P is the dual of the cotangent space T_pM = (T_p^*M)^* = \Hom(T_p^*M,\R).

Given a smooth map F:M\to N and p\in M, define the differential
\begin{array}{r c l} dF_p\ :\ T_pM & \to & T_{F(p)}N, \\ \left(f:C^\infty(M)/Z_p \to \R\right) & \mapsto & \left(\begin{array}{r c l} g\ :\ C^\infty(N)/Z_{F(p)} & \to & \R, \\ h & \mapsto & f(h\circ F). \end{array}\right) \end{array}
This definition makes clear the relation to the first approach. Since h\not\in Z_{F(p)}, the derivative of h does not vanish at F(p). Hence the derivative of h\circ F at p, which is the derivative of h at F(p) multiplied by the derivative of F at p, does not a priori vanish at p.

Derivative of chart map (Guillemin and Pollack)

Definition: Let f:\R^n\to \R^m be a smooth map. Then the derivative of f at x\in \R^n in the direction y\in \R^n is defined as
df_x(y) = \lim_{h\to 0}\left[\frac{f(x+yh)-f(x)}h\right].
Given x\in M and charts \varphi:\R^n\to M\subset \R^m, the tangent space to M at p is the image T_pM = d\varphi_0(\R^n), where we assume \varphi(0)=p.

Given a smooth map F:M\to N and charts \varphi:\R^n\to M, \psi:\R^n\to N, with \varphi(0)=p and \psi(0)=F(p), define the differential dF_p:T_pM\to T_{F(p)}N via the diagrams below.
Here h = \psi^{-1}\circ F\circ \varphi, so dh_0 is well-defined. Hence dF_p = d\psi_0\circ dh_0\circ d\varphi_0^{-1} is also well-defined.

Sometimes the differential is referred to as the pushforward, in which case it is denoted by (F_*)_p.

References: Lee (Introduction to Smooth Manifolds, Chapter 3), Hitchin (Differentiable manifolds, Chapter 3.2), Guillemin and Pollack (Differential topology, Chapter 1.2)

Wednesday, September 28, 2016

Degree and orientation

 Preliminary exam prep

Topology

 Recall that a topological manifold is a Hausdroff space in which every point has a neighborhood homeomorphic to \R^n for some n. An orientation on M is a choice of basis of \R^n in each neighborhood such that every path in M keeps the same orientation in each neighborhood. Every manifold M\owns x appears in a long exact sequence (via relative homology) with three terms
H_n(M-\{x\}) \tov{f} H_n(M) \tov{g} H_n(M,M-\{x\}).
The first term is 0, because removing a point from an n-dimensional space leaves only its (n-1)-skeleton, which is at most (n-1)-dimensional. For U a neighborhood of x in M, the last term (via excision) is
H_n(M-U^c, M-\{x\}-U^c) = H_n(U, U-\{x\})\cong H_n(\R^n, \R^n-\{x\}) \cong H_n(\R^n, S^{n-1}),
which in turn fits into a long exact sequence whose interesting part is
H_n(\R^n)\to H_n(\R^n,S^{n-1}) \to H_{n-1}(S^{n-1})\to H_{n-1}(\R^n),
and since the first and last terms are zero, H_n(M,M-\{x\})=\Z. Since f is zero, g into \Z must be injective, meaning that H_n(M)=\Z or 0.

Theorem: Let M be a connected compact (without boundary) n-manifold. Then
  1. if M is orientable, g is an isomorphism for all x\in M, and
  2. if M is not orientable, g=0.

Definition: Let f:M\to N be a map of connected, oriented n-manifolds. Since H_n(M)=H_n(N) is infinite cyclic, the induced homomorphism f_*:H_n(M)\to H_n(N) must be of the form x\mapsto dx. The number d is called the degree of f.

In the special case when we are computing the degree for a map f:S^n\to S^n, by excision we get
\deg(f) = \sum_{x_i\in f^{-1}(y)} \deg\left(H_n(U_i,U_i-x_i)\tov{f_*} H_n(V,V-y)\right), for any y\in Y, some neighborhood V of y, and preimages U_i of V. This is called the local degree of f.

Geometry

Let M be a smooth n-manifold. Recall \Omega_M^r is the space of r-forms on M and d^r:\Omega^r_M\to \Omega^{r+1}_M is the differential map. Also recall the de Rham cohomology groups H^r(M) = \text{ker}(d^r)/\text{im}(d^{r-1}).

Definition: An n-manifold M is orientable if it has a nowhere-zero n-form \omega\in \Omega^n_M. A choice of \omega is called an orientation of M.

We also have a map H^n(M)\to \R, given by \alpha\mapsto \int_M\alpha, where the integral is normalized by the volume of M, so that integrating 1 across M gives back 1. It is immediate that this doesn't make sense when M is not compact, but when M is compact and orientable, we get that H^n(M)\neq0. Indeed, if \eta\in \Omega^{n-1}_M with d\eta =\omega, by Stokes' theorem we have
\int_M\omega = \int_Md\eta = \int_{\dy M}\eta = \int_\emptyset\eta = 0,
as M has no boundary (since it is a manifold). But \omega is nowhere-zero, meaning the first expression on the left cannot be zero. Hence \omega is not exact and is a non-trivial element of H^n(M).

Theorem: Let M be a smooth, compact, orientable manifold of dimension n. Then H^n(M) is one-dimensional.

Proof: The above discussion demonstrates that \dim(H^n(M))\>1. We can get an upper bound on the dimension by noting that the space of n-forms on M, given by \Omega^n_M = \bigwedge^n(TM)^*, has elements described by dx_{i_1}\wedge \cdots \wedge dx_{i_n}, with \{i_1,\dots,i_n\}\subset \{1,\dots,n\}. By rearranging the order of the dx_{i_j}, every element looks like \alpha dx_1\wedge \cdots\wedge dx_n for some real number \alpha. Hence \dim(\Omega^n_M) \leqslant 1, so \dim(H^n(M)) is either 0 or 1. Therefore \dim(H^n(M))=1. \square

Definition: Let f:M\to N be a map of smooth, compact, oriented manifolds of dimension n. Since H^n(M) and H^n(N) are 1-dimensional, the induced map f^*:H^n(N)\to H^n(M) must be of the form x\mapsto dx. The number d is called the degree of f. Equivalently, for any \omega\in \Omega^n_N,
\int_M f^*\omega = d\int_N\omega

References: Hatcher (Algebraic Topology, Chapters 2, 3.3), Lee (Introduction to Smooth Manifolds, Chapter 17)

Thursday, September 22, 2016

The Grassmannian is a complex manifold

 Lecture topic

Let Gr(k,\C^n) be the space of k-dimensional complex subspaces of \C^n, also known as the complex Grassmannian. We will show that it is a complex manifold of dimension k(n-k). Thanks to Jinhua Xu and professor Mihai Păun for explaining the details.

To begin, take P\in Gr(k,\C^n) and an n-k subspace Q of \C^n, such that P\cap Q = \{0\}. Then P\oplus Q = \C^n, so we have natural projections
A neighborhood of P, depending on Q may be described as U_Q = \{S\in Gr(k,\C^n)\ :\ S\cap Q = \{0\}\}. We claim that U_Q \cong \Hom(P,Q). The isomorphism is described by
\begin{array}{r c c c l} \Hom(P,Q) & \to & U_Q & \to & \Hom(P,Q), \\ A & \mapsto & \{v+Av\ :\ v\in P\}, \\ & & S & \mapsto & \left(\pi_Q|_S\right) \circ \left(\pi_P|_S\right)^{-1}. \end{array}
The map on the right, call it \varphi_Q, is also the chart for the manifold structure. The idea of decomposing \C^n into P and Q and constructing a homomorphism from P to Q may be visualized in the following diagram.
Then \Hom(P,Q) \cong \Hom(\C^k,\C^{n-k})\cong \C^{k(n-k)}, so Gr(k,\C^n) is locally of complex dimension k(n-k). To show that there is a complex manifold structure, we need to show that the transition functions are holomorphic. Let P,P'\in Gr(k,\C^n) and Q,Q'\in Gr(n-k,\C^n) such that P\cap Q = P'\cap Q' = \{0\}. Let X\in \Hom(P,Q) such that X\in \varphi_Q(U_Q\cap U_{Q'}), with \varphi_Q(S)=X and \varphi_{Q'}(S)=X' for some S\in U_Q\cap U_{Q'}. Define I_X(v) = v+Xv, and note the transition map takes X to
\begin{align*} X' & = \varphi_{Q'}\circ \varphi_Q^{-1}(X) & (\text{definition}) \\ & = \varphi_{Q'}(S) & (\text{assumption})\\ & = \left(\pi_{Q'}|_S\right)\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{definition}) \\ & = \left(\pi_{Q'}|_S\right)\circ I_X\circ I_X^{-1}\circ \left(\pi_{P'}|_S\right)^{-1} & (\text{creative identity}) \\ & = \left(\pi_{Q'}|_S\circ I_X\right)\circ \left(\pi_{P'}|_S\circ I_X\right)^{-1} & (\text{redistribution}) \\ & = \left(\pi_{Q'}|_P +\pi_{Q'}|_Q\circ X\right) \circ \left(\pi_{P'}|_P +\pi_{P'}|_Q\circ X\right). & (\text{definition}) \end{align*}
At this last step we have compositions and sums of homomorphisms and linear maps, which are all holomorphic. Hence the transition functions of Gr(k,\C^n) are holomorphic, so it is a complex manifold.

Friday, September 16, 2016

Complexes and their homology

 Preliminary exam prep

Here I'll present complexes from the most restrictive to the most general. Recall the standard n-simplex is
\Delta^n = \{x\in \R^{n+1}\ :\ \textstyle\sum x_i = 1, x_i\>0\}.

Definition: Let V be a finite set. A simplicial complex X on V is a set of distinct subsets of V such that if \sigma\in X, then all the subsets of \sigma are in X.

Every n-simplex in a simplicial complex is uniquely determined by its vertices, hence no pair of lower dimensional faces of a simplex may be identified with each other.

Definition: Let A,B be two indexing sets. A \Delta-complex (or delta complex) X is
X = \left.\bigsqcup_{\alpha\in A} \Delta^{n_\alpha}_\alpha \right/\left\{\mathcal F_{\beta}^{k_\beta}\right\}_{\beta\in B}\ , \hspace{1cm} \mathcal F_\beta^{k_\beta} = \{\Delta_1^{k_\beta},\dots,\Delta_{m_\beta}^{k_\beta}\},
such that if \sigma appears in the disjoint union, all of its lower dimensional faces also appear. The identification of the k-simplices in \mathcal F^k is done in the natural (linear) way, and restricting to identified faces gives the identification of the \mathcal F^{k-1} where the faces appear.

To define simplicial homology of a simplicial or \Delta-complex X, fix an ordering of the set of 0-simplices (which gives an ordering of every \sigma\in X), define C_k to be the free abelian group generated by all \sigma\in X of dimension k (defined by k+1 0-simplices), and define a boundary map
\begin{array}{r c l} \partial_k\ :\ C_k & \to & C_{k-1}, \\\ [v_0,\dots,v_k] & \mapsto & \sum_{i=0}^k(-1)^i[v_0,\dots,\widehat{v_i},\dots,v_k]. \end{array}
Then H_k(X):= \text{ker}(\partial_k)/\text{im}(\partial_{k+1}).

Recall the standard n-cell is e^n = \{x\in \R^n\ :\ | x| \leqslant 1\}, also known as the n-disk or n-ball.


Definition: Let X_0 be a finite set. A cell complex (or CW complex) is a collection X_0,X_1,\dots where
X_k := \left.X_{k-1}\bigsqcup_{\alpha\in A_k} e^k_\alpha \right/\left\{\partial e^k_\alpha\sim f_{k,\alpha}(\dy e^k_\alpha)\right\}_{\alpha\in A_k},
where the f_{k,\alpha} describe how to attach k-cells to the (k-1)-skeleton X_{k-1}, for k\>1. X_k may also be described by pushing out e^k\sqcup_{\dy e^k}X_{k-1}. Note that \dy e^k = S^{k-1}, the (k-1)-sphere.

To define cellular homology, we need more tools (relative homology and excision) that require a blog post of their own.

References: Hatcher (Algebraic topology, Chapter 2.1)

Monday, September 5, 2016

Classical Lie groups

 Lecture topic

A Lie group G is both a group and a manifold, with a smooth map G\times G\to G, given by (g,h)\mapsto gh^{-1}. The Lie algebra \mathfrak g of G is the tangent space T_eG of G at the identity.

We distinguish between real and complex Lie groups by saying that the base manifold is either real or complex analytic, respectively.

Example:
Here are some examples of classical Lie groups and their dimension:
\begin{array}{r c r c l} \text{general linear group} & n^2 & GL(n) & = & \{n\times n\ \text{matrices with non-zero determinant}\} \\ \text{special linear group} & n^2-1 & SL(n) & = & \{M\in GL(n)\ :\ \det(M)=1\} \\ \text{orthogonal group} & n(n-1)/2 & O(n) & = & \{M\in GL(n)\ :\ MM^t = I\} \\ \text{special orthogonal group} & n(n-1)/2 & SO(n) & = & \{M\in O(n)\ :\ \det(M)=1\} \\ \text{unitary group} & n^2 & U(n) & = & \{M\in GL(n,\textbf{C})\ :\ MM^* = I\} \\ \text{special unitary group} & n^2-1 & SU(n) & = & \{M\in U(n)\ :\ \det(M)=1\} \\ \text{symplectic group} & n(2n+1) & Sp(n) & = & \{n\times n\ \text{matrices}:\ \omega(Mx,My)=\omega(x,y)\} \end{array}
For the symplectic group, the skew-symmetric bilinear form \omega is defined as
\omega(x,y) = \sum_{i=1}^n x_iy_{i+n} - y_ix_{i+n} = \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} x\cdot y,
where \cdot is the regular dot product (a symmetric bilinear form). Also note that the unitary group is a real Lie group - real because there is no holomorphic map G\times G\to G as would be necessary, so we view the entries of a matrix in U(n) in terms of its real and imaginary parts. Hence the dimension indicated above is real dimension.

References: Kirillov Jr (An introduction to Lie groups and Lie algebras, Chapter 2)

Saturday, September 3, 2016

The real and complex Jacobian

 Lecture topic

Let f:\C^n\to \C^n be a holomorphic function. We will show that the the real Jacobian is the square of the complex Jacobian. Write f = (f_1,\dots,f_n) with f_i = u_i+\img v_i, where the u_i are functions of the z_j = x_j+\img y_j. By the Cauchy-Riemann equations
\frac{\dy u_i}{\dy x_j} = \frac{\dy v_i}{\dy y_j} \hspace{1cm} \text{and} \hspace{1cm} \frac{\dy u_i}{\dy y_j} = -\frac{\dy v_i}{\dy x_j}
and expanding, we have that
\begin{align*} \frac{\dy f_i}{\dy z_j} & = \frac 12\left(\frac{\dy f_i}{\dy x_j} - \img \frac{\dy f_i}{\dy y_j}\right) \\ & = \frac12\left(\frac{\dy u_i}{\dy x_j} + \img \frac{\dy v_i}{\dy x_j} - \img\left(\frac{\dy u_i}{\dy y_j} + \img \frac{\dy v_i}{\dy y_j}\right)\right) \\ & = \frac12 \left(\frac{\dy u_i}{\dy x_j}+ \frac{\dy v_i}{\dy y_j} + \img \left(\frac{\dy v_i}{\dy x_j} - \frac{\dy u_i}{\dy y_j}\right)\right) \\ & = \frac{\dy u_i}{\dy x_j} + \img \frac{\dy v_i}{\dy x_j}. \end{align*}
The complex Jacobian of f is J_\C f (or its determinant), with entries
(J_\C f)_{i,j} = \frac{\dy f_i}{\dy z_j},
and the real Jacobian of f is J_\R f (or its determinant), with entries
\begin{align*} \begin{bmatrix} (J_\R f)_{2i-1,2j-1} & (J_\R f)_{2i-1,2j} \\ (J_\R f)_{2i,2j-1} & (J_\R f)_{2i,2j} \end{bmatrix} & = \begin{bmatrix} \displaystyle \frac{\dy u_i}{\dy x_j} & \displaystyle \frac{\dy u_i}{\dy y_j} \\ \displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \frac{\dy v_i}{\dy y_j} \end{bmatrix} \\ & \tov{R_{2i-1} + \img R_{2i} \to R_{2i-1}} \begin{bmatrix} \displaystyle \frac{\dy f_i}{\dy z_j} & \displaystyle \img\frac{\dy f}{\dy z} \\ \displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \frac{\dy v_i}{\dy y_j} \end{bmatrix} \\ & \tov{C_{2j} - \img C_{2j-i} \to C_{2j}} \begin{bmatrix} \displaystyle \frac{\dy f_i}{\dy z_j} & 0 \\ \displaystyle \frac{\dy v_i}{\dy x_j} & \displaystyle \overline{\frac{\dy f_i}{\dy z_j}} \end{bmatrix}, \end{align*}
where the row and column operations have been performed for all rows 2i and all columns 2j. Moving all the odd-indexed columns to the left and all odd-indexed rows to the top, we get that
J_\R f \simeq \begin{bmatrix} A & 0 \\ * & B \end{bmatrix} \hspace{1cm} \text{with} \hspace{1cm} A_{i,j} = \frac{\dy f_i}{\dy z_j},\ \ \ B_{i,j} = \overline{\frac{\dy f_i}{\dy z_j}}.
Since the number of operations to switch the columns is the same as the number of operations to switch the rows, the sign of the determinant of J_\R f will not change. That is,
\det(J_\R f) = \det(A)\det(B) = \det(J_\C f) \overline{\det( J_\C f)} = |\det(J_\C f)|^2.

Thursday, August 25, 2016

Equations on Riemann surfaces

Recall that a Riemann surface is a complex 1-manifold M with a complex structure \Sigma (a class of analytically equivalent atlases on X). Here we consider equations that relate connections and Higgs fields with solutions on Riemann surfaces. Let G=SU(2) (complex 2-matrices with determinant 1) or SO(3) (real orthogonal 3-matrices with determinant 1), \theta a Higgs field over M, and  P a principal G-bundle over M.

Definition:
The curvature of a principal G-bundle P is the map
\begin{array}{r c l} F_\nabla\ :\ \mathcal A^0_M(P) & \to & \mathcal A^2_M(P), \\ \omega s & \mapsto & (d_\nabla \circ \nabla)(\omega s), \end{array}
where the extension d_\nabla:\mathcal A^k_M(P)\to \mathcal A^{k+1}_M(P) is defined by the Leibniz rule, that is d_\nabla (\omega\otimes s) = (d\omega)\otimes s +(-1)^k\omega \wedge \nabla s, for \omega a k-form and s a smooth section of P.

Since we may write \mathcal A^1 = \mathcal A^{1,0}\oplus\mathcal  A^{0,1} as the sum of its holomorphic and anti-holomorphic parts, respectively (see post "Smooth projective varieties as Kähler manifolds," 2016-06-16), we may consider the restriction of d_\nabla to either of these summands.

Definition: For a vector space V, define the Hodge star * by
\begin{array}{r c l} *\ :\ \bigwedge^k(V^*) & \to & \bigwedge^{n-k}(V^*), \\ e^{i_1}\wedge \cdots \wedge e^{i_k} & \mapsto & e^{j_1}\wedge\cdots \wedge e^{j_{n-k}}, \end{array}
so that e^{i_1}\wedge\cdots \wedge e^{i_k}\wedge e^{j_1}\wedge\cdots \wedge e^{j_{n-k}} = e^1\wedge \cdots \wedge e^n. Extend by linearity from the chosen basis.

The dual of the generalized connection d_\nabla is written d_\nabla^* = (-1)^{m+mk+1}*d_\nabla *, where \dim(M)=m and the argument of d_\nabla^* is in \mathcal A^k_M (this holds for manifolds M that are not necessarily Riemann surfaces as well).

Now we may understand some equations on Riemann surfaces. They all deal with the connection \nabla, its generalization d_\nabla, its curvature F_\nabla, and the Higgs field \theta. Below we indicate their names and where they are mentioned (and described in further detail).
\begin{align*} \text{Hitchin equations} && \left.d_\nabla\right|_{A^{0,1}}\theta & = 0 && [2],\ \text{Introduction}\\ && F_\nabla + [\theta,\theta^*] & = 0\\[10pt] \text{Yang-Mills equations} && d^*_\nabla d_\nabla \theta + *[*F_\nabla,\theta] & = 0 && [1],\ \text{Section 4} \\ && d_\nabla^*\theta & = 0 \\[10pt] \text{self-dual Yang-Mills equation} && F_\nabla  - *F_\nabla & = 0 && [2],\ \text{Section 1}\\[10pt] \text{Yang-Mills-Higgs equations} && d_\nabla *F_\nabla + [\theta,d_\nabla \theta] & = 0 && [4],\ \text{equation (1)} \\ && d_\nabla * d_\nabla \theta & = 0 \end{align*}

Recall the definitions of \theta and \theta* from a previous post ("Higgs fields of principal bundles," 2016-08-24). Now we look at these equations in more detail. The first of the Hitchin equations says that \theta has no anti-holomorphic component, or in other words, that \theta is holomorphic. In the second equation, the Lie bracket [\cdot,\cdot] of the two 1-forms is
\begin{align*} [\theta,\theta^*] & = \left[\textstyle\frac12f(dz+i\ dy), \frac12\bar f(dz - i\ dy) \right] \\ & = \textstyle -\frac i4f\bar f\ dx\wedge dy + \frac i4 f\bar f\ dy \wedge dx -\frac i4 f\bar f\ dx\wedge dy +\frac i4 f\bar f\ dy\wedge dx \\ & = -i|f|^2\ dx\wedge dy. \end{align*}
In the Yang-Mills and Yang-Mills-Higgs equations, we can simplify some parts by noting that, for a section s of the complexification of P\times_\ad \mathfrak g,
\begin{align*} d_\nabla (\theta\otimes s) & = \textstyle \frac12d_\nabla (fdx\otimes s) + \frac i2 d_\nabla (fdy \otimes s) \\ & = \textstyle \frac12 (df\wedge dx \otimes s - fdx \wedge \nabla s) +\frac i2 (df\wedge dy - fdy \wedge \nabla s) \\ & = \left(\frac i2\frac{\dy f}{\dy x} - \frac 12 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s - \underbrace{\textstyle \frac12f(dx+idy)}_{\theta}\wedge \nabla s. \end{align*}
The Hodge star of \theta is *\theta = \frac 12f(dy -idx), so
\begin{align*} d_\nabla *(\theta\otimes s) & = \textstyle \frac12d_\nabla (fdy\otimes s) - \frac i2 d_\nabla (fdx \otimes s) \\ & = \textstyle \frac12 (df\wedge dy \otimes s - fdy \wedge \nabla s) -\frac i2 (df\wedge dx - fdx \wedge \nabla s) \\ & = \left(\frac 12\frac{\dy f}{\dy x} + \frac i2 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s + \underbrace{\textstyle \frac12f(idx-dy)}_{i\theta}\wedge \nabla s. \end{align*}
We could express \nabla s = (s_1dx + s_2dy)\otimes s^1, but that would not be too enlightening. Next, note the self-dual Yang-Mills equation only makes sense over a (real) 4-dimensional space, since the degrees of the forms have to match up. In that case, with a basis dz_1=dx_1+idy_1, dz_2 = dx_2+idy_2 of \mathcal A^1, we have
\begin{align*} F_\nabla & = F_{12} dx_1\wedge dy_1 + F_{13} dx_1 \wedge dx_2 + F_{14} dx_1\wedge dy_2 + F_{23} dy_1\wedge dx_2 + F_{24} dy_1\wedge dy_2 + F_{34} dx_2\wedge dy_2, \\ *F_\nabla & = F_{12} dx_2\wedge dy_2 - F_{13} dy_1 \wedge dy_2 + F_{14} dy_1\wedge dx_2 + F_{23} dx_1\wedge dy_2 - F_{24} dx_1\wedge dx_2 + F_{34} dx_1\wedge dy_1. \end{align*}
Then the self-dual equation simply claims that
F_{12} = F_{34} \hspace{1cm},\hspace{1cm} F_{13} = -F_{24} \hspace{1cm},\hspace{1cm} F_{14} = F_{23}.

Remark:
This title of this post promises to talk about equations on Riemann surfaces, yet all the differential forms are valued in a principal G-bundle over \R^2 (or \R^4). However, since the given equations are conformally invariant (this is not obvious), and as a Riemann surface locally looks like \R^2, we may consider the solutions to the equations as living on a Riemann surface.

References:
[1] Atiyah and Bott (The Yang-Mills equations over Riemann surfaces)
[2] Hitchin (Self-duality equations on a Riemann surface)
[3] Huybrechts (Complex Geometry, Chapter 4.3)
[4] Taubes (On the Yang-Mills-Higgs equations)

Wednesday, August 24, 2016

Higgs fields of principal bundles

The goal here is to understand the setting of Higgs fields on Riemannian manifolds, in the manner of Hitchin. First we consider general topological spaces X and groups G.

Definition: Let X be a topological space and G a group. A principal bundle (or principal G-bundle) P over X is a fiber bundle \pi:P\to X together with a continuous, free, and transitive right action P\times G\to P that preserves the fibers. That is, if p\in \pi^{-1}(x), then pg\in \pi^{-1}(x) for all g\in G and x\in X.

Now suppose we have a principal bundle \pi:P\to X, a representation \rho of G, and another space Y on which G acts on the left. Define an equivalence relation (p,y)\sim (p',y') on P\times Y iff there is some g\in G for which p'=pg and y'=\rho(g^{-1})y. This is an equivalence relation. We will be interested in the adjoint representation (induced by conjugation).

Proposition: The projection map \pi':P\times_\rho Y := (P\times Y)/\sim\ \to X, where \pi'([p,y]) = \pi(p), defines a vector bundle over X, called the associated bundle of P.

Recall a Lie group G is a group that is also a topological space, in the sense that there is a continuous map G\times G\to G, given by (g,h)\mapsto gh^{-1}. The Lie algebra \mathfrak g of the Lie group G is the tangent space T_eG of G at the identity e. We will be interested in principal G-bundles P\to \R^2 and associated bundles P\times_\ad \mathfrak g\to \R^2, where \ad is the adjoint representation of G.

Next, recall we had the space \mathcal A^k_M of k-differential forms on M (see post "Smooth projective varieties as Kähler manifiolds," 2016-06-16), defined in terms of wedge products of elements in the cotangent bundle (TM)^* = T^*M of M. Now we generalize this to get differential forms over arbitrary vector bundles.

Definition: Let E\to M be a vector bundle. Let
\begin{align*} \mathcal A^k_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^k T^*M) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^k_M, \\ \mathcal A^{p,q}_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^p (T^{1,0}M)^*\otimes \bigwedge^q (T^{0,1}M)^*) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^{p,q}_M \end{align*}
be the spaces of k- and (p,q)-differential forms, respectively, over M with values in E.

Equality above follows by functoriality. Now we are close to understanding where exactly the Higgs field lives, in Hitchin's context.

Definition: Given a function f:\C\to \C, the conjugate of f is \bar f, defined by \bar f(z) = \overline{f(\bar z)}.

Hitchin denotes this as f^*, but we will stick to \bar f. Finally, let P be a G-principal bundle over \R^2 and P\times_\ad \mathfrak g the associated bundle of P. Given f\in \mathcal A^0_{\R^2}( (P\times_\ad \mathfrak g)\otimes \C), set
\begin{align*} \theta & = \textstyle \frac12 f(dx+i\ dy) \in \mathcal A^{1,0}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C) ,\\ \theta^* & = \textstyle \frac12 \bar f(dx-i\ dy) \in \mathcal A^{0,1}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C), \end{align*}
called a Higgs field over \R^2 and (presumably) a dual (or conjugate) Higgs field over \R^2. Note this agrees with the definition in a previous post ("Connections, curvature, and Higgs bundles," 2016-07-25).

References: Hitchin (Self-duality equations on a Riemann surface), Wikipedia (article on associated bundles, article on vector-valued differential forms)

Saturday, August 13, 2016

What is a stack?

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski for explaining the material.

Definition: A groupoid is a category where all the morphisms are invertible. Alternatively, a groupoid is a set of objects A, a set of morphisms \Gamma, and a collection of maps as described by the diagram below.
To describe stacks, we compare them with sheaves. Both start out with a space X and a topology on it, so that we may consider open sets U.
In addition to these conditions, there is a triple intersection condition for stacks that does not have an analogous one in sheaves. It is given by:

for every U_i,U_j,U_k and s_i,s_j,s_k\in \widehat{\mathcal F}(U_i), \widehat{\mathcal F}(U_j), \widehat{\mathcal F}(U_k), respectively, such that there exist isomorphisms \varphi_{ij}:s_i|_{U_i\cap U_j}\to s_j|_{U_i\cap U_j}, \varphi_{jk}:s_j|_{U_j\cap U_k}\to s_k|_{U_j\cap U_k}, and \varphi_{ik}:s_i|_{U_i\cap U_k}\to s_k|_{U_i\cap U_k}, the diagram below commutes:
Example: A Hopf algebroid may be viewed as a functor into groupoids, so that with the appropriate topology, it becomes a stack. Indeed, by definition a Hopf algebroid is a pair of k-algebras (A,\Gamma) such that (\Spec(A),\Spec(\Gamma)) is a groupoid object in affine schemes, or in other words, is a functor from affine schemes into groupoids.

References: nLab (article on groupoids)

Morphisms of schemes

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Zijian Yao for explaining the material.

Consider a morphism of schemes \varphi:S'\to S and coherent sheaves \mathcal F,\mathcal G over S. Consider also a map of sheaves f:\mathcal F\to \mathcal G and a map f' between the pullbacks of \mathcal F and \mathcal G, as described by the diagram below.
There are two natural questions to ask.
  1. When is f' = \varphi^*f?
  2. If we start with \mathcal G' over S', when is \mathcal G' = \varphi^*\mathcal G?
To answer these questions, consider fiber products of schemes and projections from them, as given below.
Remark: If 1. is true, then p_1^*(f') = p_2^*(f'). If the previous statement is an equivalence, then \varphi is a morphism of descent.

Remark:
If 2. is true, then there exists \alpha:p_1^*(\mathcal G') \to p_2^*(\mathcal G') such that \pi_{32}^*(\alpha)\pi_{21}^*(\alpha) = \pi_{31}^*(\alpha) and \pi^*(\Delta) = \alpha. If the previous statement is an equivalence, then \varphi is effective.