1. MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Oct 21, 2013: Cohomology
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

2. triangle-zigzag.py
from dionysus import Simplex, ZigzagPersistence, \
vertex_cmp, data_cmp \
# ,enable_log
complex = {Simplex((0,), ): None, # A
Simplex((1,), ): None, # B
Simplex((2,), ): None, # C
Simplex((0,1), ): None, # AB
Simplex((1,2), ): None, # BC
Simplex((0,2), ): None, # CA
Simplex((0,1,2), 5): None} # ABC

3. print "Complex:"
for s in sorted(complex.keys()): print s
print
triangle idarcy$ python2.7 triangle-zigzag.py
Complex:
<0>
<1>
<2>
<0, 1>
<1, 2>
<0, 2>
<0, 1, 2>

4. #enable_log("topology/persistence")
zz = ZigzagPersistence()
# Add all the simplices
b = 1
for s in sorted(complex.keys(), data_cmp):
print "%d: Adding %s" % (b, s)
1: Adding <0>
2: Adding <1>
3: Adding <2> 1 2

5. # Add all the simplices
b = 1
for s in sorted(complex.keys(), data_cmp):
print "%d: Adding %s" % (b, s)
i,d = zz.add([complex[ss] for ss in s.boundary], b)
complex[s] = i
if d: print "Interval (%d, %d)" % (d, b-1)
b += 1

6. i,d = zz.add([complex[ss] for ss in s.boundary], b) complex[s] = i
# Add all the simplices
b = 1
for s in sorted(complex.keys(), data_cmp):
print "%d: Adding %s" % (b, s)
i,d = zz.add([complex[ss] for ss in s.boundary], b)
complex[s] = i
if d: print "Interval (%d, %d)" % (d, b-1)
b += 1
4: Adding <0, 1>
Interval (2, 3)
5: Adding <1, 2>
Interval (3, 4)
6: Adding <0, 2>
7: Adding <0, 1, 2>
Interval (6, 6) 1 2

7. # Remove all the simplices
for s in sorted(complex.keys(), data_cmp, reverse = True):
print "%d: Removing %s" % (b, s)
d = zz.remove(complex[s], b)
del complex[s]
if d: print "Interval (%d, %d)" % (d, b-1)
b += 1
8: Removing <0, 1, 2>
9: Removing <0, 2>
Interval (8, 8)
10: Removing <1, 2>
11: Removing <0, 1> 1 2

8. 12: Removing <2>
Interval (10, 11)
13: Removing <1>
Interval (11, 12)
14: Removing <0>
Interval (1, 13) 1 2

10. NKI (2002) breast cancer data:
gene expression levels of 24,000 from 272 tumors
Data = 272 points living in R24000

11. NKI (2002) breast cancer data:
gene expression levels of 24,000 from 272 tumors
Data = 272 points living in R24000
In reality one cleans the data first, removing unreliable data, etc.
One may also remove dimensions (genes) that appear irrelevant.

12. NKI (2002) breast cancer data:
gene expression levels of 24,000 from 272 tumors
Data = 272 points living in R24000
Since our dataset consists of only 272 points, it can’t be dimensional.

13. For example, suppose our very fake data is
{ (9, 8, 7, 6, 5, 4, 3, 2, 3) ti | i = 1, 2, 3, … 1000 } =
{ (9, 8, 7, 6, 5, 4, 3, 2, 3), { (18, 16, 14, 12, 10, 8, 6, 4, 6), … }

14. Dimensionality Reduction:
Given dataset D RN
Want: embedding f: D  Rn where n << N
which “preserves” the structure of the data.
Example:
{ (9, 8, 7, 6, 5, 4, 3, 2, 3) ti | i = 1, 2, 3, … 1000 } R9
f: D  R, f((9, 8, 7, 6, 5, 4, 3, 2, 3) ti) = ti U U

15. Example: Principle component analysis (PCA)

16. Many reduction methods:
f1: D  R, f2: D  R, … fn: D  R
(f1, f2, … fn): D  Rn
Many are linear, M: RN  Rn, Mx = y
But there are also non-linear dimensionality reduction algorithms.

17. f1: D  R, f2: D  R, … fn: D  R
(f1, f2, … fn): D  Rn

18. Goal
f1: D  S1, f2: D  S1, … fn: D  S1

19. Note S1 is a ring:

20. Cohomology and circular coordinates:
Let X = finite simplicial complex.
X0 = set of 0-simplices = vertices.
X1 = set of 1-simplices = edges.
X2 = set of 2-simplices = faces. . v2 e2 e1 e3 v1 v3

21. Homology
Let X = finite simplicial complex.
C0 = set of 0-chains = linear combinations of vertices
C1 = set of 1-chains = linear combinations of edges.
C2 = set of 2-chains = linear combinations of faces. . v2 e2 e1 e3 v1 v3

22. Cohomology and circular coordinates:
Let X = finite simplicial complex, R a ring.
C0 = set of 0-cochains = { f : C0  R | f homomorphism}
C1 = set of 1-cochains = { f : C1  R | f homomorphism}
C2 = set of 2-cochains = { f : C2  R | f homomorphism} .

23. Cohomology and circular coordinates:
Let X = finite simplicial complex, R a ring.
C0(X, R) = set of 0-cochains = { f : X0  R }
C1(X, R) = set of 1-cochains = { f : X1  R }
C2(X, R) = set of 2-cochains = { f : X2  R } .

24. → Proposition 1: Let α ∈ C1(X;Z) be a cocycle.
Then there exists a continuous function
θ : X → S1
which maps each vertex to 0, and
each edge ab around the entire
circle with winding number α(ab). v2 e2 e1 e3 v1 v3 →

25. Proposition 2: 2 Let α ∈ C1(X;R) be a cocycle
Proposition 2: 2 Let α ∈ C1(X;R) be a cocycle. Suppose there exists α ∈ C1(X;Z) and f ∈ C0(X;R) such that α = α + d0f . Then there exists a continuous function
θ : X → S1
which maps each edge ab linearly to an interval of length α(ab), measured with sign. v2 e2 e1 e3 v1 v3 →

26. θ : X → R/Z = S1

27. (d1α)(abc) = α(bc)− α(ac)+α(ab).
coboundary maps
d0 : C0 = { f : X0  R } → C1 = { f : X1  R }
(d0f )(ab) = f (b)− f (a)
d1 : C1 = { f : X1  R } → C2 = { f : X2  R }
(d1α)(abc) = α(bc)− α(ac)+α(ab).

28. Let α ∈ Ci ,
di : Ci = { f : Xi  R } → Ci+1 = { f : Xi+1  R }
α is a cocycle if diα = 0.
α is a coboundary if there exists f in Ci-1 s.t. di-1f = α
Note d1 d0 f = 0 for any f ∈ C0.
(d0f )(ab) = f (b) − f (a)
(d1d0f )(abc) = d0f(bc) − d0f(ac) + d0f(ab)
= d0f(c) - d0f(b) - d0f(c) + d0f(a) + d0f(b) - d0f(a)

29. d1 d0 f = 0 implies Im (d0 ) ⊆ Ker (d1 ).
H1 (X;A ) = Ker (d1 )/ Im (d0 ).
Two cocycles α,β are cohomologous if α − β is a coboundary.

30. d1 d0 f = 0 implies Im (d0 ) ⊆ Ker (d1 ).
H1 (X;A ) = Ker (d1 )/ Im (d0 ). v2 e2 e1 e3 v1 v3

31. d1 d0 f = 0 implies Im (d0 ) ⊆ Ker (d1 ).
H1 (X;A ) = Ker (d1 )/ Im (d0 ). v2 e2 e1 e3 v1 v3
0  C0  C1  0

32. C0(X, R) = set of 0-cochains = { f : X0  R }v2 e2 e1 e3 v1 v3

33. C0(X, R) = set of 0-cochains = { f : X0  R }
= { f | f(v1) = r1, f(v2) = r2, f(v3) = r3; (r1, r2, r3) in R3 }
C1(X, R) = set of 1-cochains = { f : X1  R }
= { f | f(e1) = r1, f(e2) = r2, f(e3) = r3; (r1, r2, r3) in R3 } v2 e2 e1 e3 v1 v3

34. Hi (X;A ) = Ker (di )/ Im (di-1 ).
C0(X, R) = set of 0-cochains = { f : X0  R }
= { f | f(v1) = r1, f(v2) = r2, f(v3) = r3; (r1, r2, r3) in R3 }
C1(X, R) = set of 1-cochains = { f : X1  R }
= { f | f(e1) = r1, f(e2) = r2, f(e3) = r3; (r1, r2, r3) in R3 }
Hi (X;A ) = Ker (di )/ Im (di-1 ).
0  C0  C1  0 v2 e2 e1 e3 v1 v3