§ An invitation to homology and cohomology, Part 2 --- Cohomology

Once again, we have our humble triangle with vertices V={r,g,b}V = \{r, g, b\}, edges E={o,m,c}E = \{o, m, c \}, faces F={f}F = \{ f \} with boundary maps EV\partial_{EV}, FE\partial_{FE}:
  • FE(f)=o+m+c\partial_{FE}(f)= o + m + c
  • EV(o)=rg\partial_{EV}(o) = r - g
  • EV(m)=br\partial_{EV}(m) = b - r
  • EV(c)=gb\partial_{EV}(c)= g - b
We define a function hv:VRh_v: V \rightarrow \mathbb R on the vertices as:
  • hv(r)=3h_v(r) = 3, hv(g)=4h_v(g) = 4, hv(b)=10h_v(b) = 10.
We now learn how to extend this function to the higher dimensional objects, the edges and the faces of the triangle. To extend this function to the edges, we define a new function:
  • he:ERh_e: E \rightarrow R
  • he(e)iαihv(vi)h_e(e) \equiv \sum_i \alpha_i h_v(v_i) where EVe=iαivi\partial_{EV} e = \sum_i \alpha_i v_i
Expanded out on the example, we evaluate hvh_v as:
  • he(o)dhv(o)=hv(r)hv(g)=34=1h_e(o) \equiv d h_v(o) = h_v(r) - h_v(g) = 3 - 4 = -1
  • he(m)dhv(m)=hv(b)hv(r)=103=+7h_e(m) \equiv d h_v(m) = h_v(b) - h_v(r) = 10 - 3 = +7
  • he(c)dhv(c)=hv(g)hv(b)=410=6h_e(c) \equiv d h_v(c) = h_v(g) - h_v(b) = 4 - 10 = -6
More conceptually, we have created an operator called dd (the coboundary operator) which takes functions defined on vertices to functions defined on edges. This uses the boundary map on the edges to "lift" a function on the vertices to a function on the edges. It does so by assigning the "potential difference" of the vertices to the edges.
  • d:(VR)(ER)d: (V \rightarrow \mathbb R) \rightarrow (E \rightarrow \mathbb R)
  • d(hv)hed(h_v) \equiv h_e, he(e)iαif(vi)h_e(e) \equiv \sum_i \alpha_i f(v_i) where EVe=iαivi\partial_{EV} e = \sum_i \alpha_i v_i
We can repeat the construction we performed above, to construct another operator d:(ER)(FR)d : (E \rightarrow \mathbb R) \rightarrow (F \rightarrow \mathbb R), defined in exactly the same way as we did before. For example, we can evaluate:
  • hfd(he)h_f \equiv d(h_e)
  • hf(f)dhe(f)=he(o)+he(m)+he(c)=1+76=0h_f(f) \equiv d h_e(f) = h_e(o) + h_e(m) + h_e(c) = -1 + 7 -6 = 0
What we have is a chain:
  • hvdhedhfh_v \xrightarrow{d} h_e \xrightarrow{d} h_f
Where we notice that d2=dd=0d^2 = d \circ d = 0, since the function hfh_f that we have gotten evaluates to zero on the face ff. We can prove this will happen in general, for any choice of hvh_v. (it's a good exercise in definition chasing). Introducing some terminology, A differential form ff is said to be a closed differential form iff df=0df = 0. In our case, heh_e is closed, since dhe=hf=0d h_e = h_f = 0. On the other hand hvh_v is not closed, since dhv=he0d h_v = h_e \neq 0. The intuition for why this is called "closed" is that its coboundary vanishes.

§ Exploring the structure of functions defined on the edges

Here, we try to understand what functions defined on the edges can look like, and their relationship with the dd operator. We discover that there are some functions ge:ERg_e: E \rightarrow \mathbb R which can be realised as the differential of another function gv:VRg_v: V \rightarrow \mathbb R. The differential forms such as geg_e which can be generated a gvg_v through the dd operator are called as exact differential forms. That is, ge=dgvg_e = d g_v exactly, such that there is no "remainder term" on applying the dd operator. We take an example of a differential form that is not exact, which has been defined on the edges of the triangle above. Let's call it heh_e. It is defined on the edges as:
  • he(c)=3h_e(c) = 3
  • he(m)=2h_e(m) = 2
  • he(o)=1h_e(o) = 1
We can calcuate hf=dheh_f = d h_e the same way we had before:
  • hf(f)dhe(f)=he(o)+he(m)+he(c)=3+1+2=6h_f(f) \equiv d h_e(f) = h_e(o) + h_e(m) + h_e(c) = 3 + 1 + 2 = 6.
Since dhe0d h_e \neq 0, this form is not exact. Let's also try to generate heh_e from a potential. We arbitrarily fix the potential of bb to 00. That is, we fix hv(b)=0h_v(b) = 0, and we then try to see what values we are forced to values of hvh_v across the rest of the triangle.
  • hvb=0h_v b = 0
  • he(c)=hv(g)hv(b)h_e(c) = h_v(g) - h_v(b). hv(g)=hv(b)+he(c)=0+3=3h_v(g) = h_v(b) + h_e(c) = 0 + 3 = 3.
  • he(o)=hv(r)hv(g)h_e(o) = h_v(r) - h_v(g). hv(r)=hv(g)+he(o)=3+1=4h_v(r) = h_v(g) + h_e(o) = 3 + 1 = 4.
  • he(m)=hv(b)hv(r)h_e(m) = h_v(b) - h_v(r) 2=042 = 0 - 4. This is a contradiction!
  • Ideally, we need hv(b)=6h_v(b) = 6 for the values to work out.
Hence, there can exist no such hvh_v such that hedhvh_e \equiv d h_v. The interesting thing is, when we started out by assigning hv(b)=0h_v(b) = 0, we could make local choices of potentials that seemed like they would fit together, but they failed to fit globally throughout the triangle. This failure of locally consistent choices to be globally consistent is the essence of cohomology.

§ Cohomology of half-filled butterfly

Here, we have vertices Vr,g,b,b,pV \equiv \\{ r, g, b, b, p \\}, edges Erb,gr,bg,m,o,cE \equiv \\{rb, gr, bg, m, o, c \\} and faces FfF \equiv \\{ f \\}. Here, we see a differential form heh_e that is defined on the edges, and also obeys the equation dhe=0dh_e = 0 (Hence is closed). However, it does not have an associated potential energy to derive it from. That is, there cannot exist a certain hvh_v such that dhv=hed h_v = h_e. So, while every exact form is closed, not every closed form is exact. Hence, this gg that we have found is a non-trivial element of Kernel(dFE)/Image(dEV)Kernel(d_{FE}) / Image(d_{EV}), since dhe=0dh_e = 0, hence heKernel(dFE)h_e \in Kernel(d_{FE}), while there does not exist a hvh_v such that dhv=hed h_v = h_e, hence it is not quotiented by the image of dEVd_{EV}. So the failure of the space to be fully filled in (ie, the space has a hole), is measured by the existence of a function heh_e that is closed but not exact! This reveals a deep connection between homology and cohomology, which is made explicit by the Universal Coefficient Theorem