Homology in Mathematics
Explore the concept of homology through mathematical structures such as cycles, boundaries, and kernel/image spaces. Learn how to count connected components and analyze 1-dimensional cycles using linear extensions. Visual aids and examples help in grasping the intricate relationships in this mathematical field.
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Presentation Transcript
on+1 on o2 o1 o0 Cn+1 Cn Cn-1 . . . C2 C1 C0 0 Hn = Zn/Bn = (kernel of )/ (image of ) on on+1 null space of Mn column space of Mn+1 boundaries = cycles = Rank Hn = Rank Zn Rank Bn
Counting number of connected components using homology v2 v5 o0 o1 C1 C0 0 e1 e2 e3 e4 v1 v3 v4 v6 e5 Z0/B0 = <v1, v2, v3, v4, v5, v6 : v1 + v2 = 0, v2 + v3 = 0, v4 + v5 = 0, v5 + v6 = 0, v4 + v6 = 0> Z0/B0 = <[v1], [v4]> where [v1] = {v1, v2, v3} and [v4] = {v4, v5, v6}
v2 v5 Counting 1-dimensional cycles: e1 e2 e3 e4 v1 v3 v4 v6 e5 o2 o1 C2 C1 C0 H1 = Z1/B1 = (kernel of )/ (image of ) o1 o2 null space of M1 column space of M2 = Rank H1 = Rank Z1 Rank B1
v2 v5 o1 e1 e2 e3 e4 C1 C0 v1 v3 v4 v6 e5 Z1 = kernel of = null space of M1 o1
Let e1 = {v1, v2} = Let e2 = {v2, v3} = Let e3 = {v4, v5} = Let e4 = {v5, v6} = Let e5 = {v4, v6} =
v1 = , v2 = , v3 = , v4 = , v5 = v6 =
v2 v5 o(e1) = v1 + v2 e1 e2 e3 e4 v1 v3 v4 v6 e5
v2 v5 o(e2) = v2 + v3 e1 e2 e3 e4 v1 v3 v4 v6 e5
v2 v5 o(e3) = v4 + v5 e1 e2 e3 e4 v1 v3 v4 v6 e5
v2 v5 o(e4) = v5 + v6 e1 e2 e3 e4 v1 v3 v4 v6 e5
v2 v5 o(e5) = v4 + v6 e1 e2 e3 e4 v1 v3 v4 v6 e5
Counting number of connected components using homology v2 v5 e1 e2 e3 e4 v1 v3 v4 v6 e5 : C1 C0 o o(e1) = v1 + v2 o(e2) = v2 + v3 Extend linearly: ( niei) = ni (ei) o o o(e3) = v4 + v5 o(e4) = v5 + v6 o(e5) = v4 + v6
o2 o1 C2 C1 C0 Z1 = kernel of = null space of M1 o1
o2 o1 C2 C1 C0 Z1 = kernel of = null space of M1 o1
New basis Let e1 = {v1, v2} = Let e2 = {v2, v3} = Let e3 = {v4, v5} = Let e4 = {v5, v6} = Let b5 = e3 + e4 + e5
o2 o1 C2 C1 C0 Z1 = kernel of = null space of M1 = <e3 + e4 + e5> o1
o2 o1 v2 C2 C1 C0 e1 e2 v1 v3 v5 e3 e4 v4 v6 e5 B1 = image of = column space of M2 o2
v2 o2 o1 C2 C1 C0 e1 e2 v1 v3 v5 e3 e4 v4 v6 e5 B1 = image of = column space of M2 o2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>
v2 v5 o2 o1 e1 e2 e3 e4 C2 C1 C0 v1 v3 v4 v6 e5 H1 = Z1/B1 = (kernel of )/ (image of ) o1 o2 null space of M1 column space of M2 = <e3 + e4 + e5> <e3 + e4 + e5> = Rank H1 = Rank Z1 Rank B1 = 1 1 = 0
v2 o2 o1 C2 C1 C0 e1 e2 v1 v3 v5 e3 e4 v4 v6 e5 B1 = image of = column space of M2 o2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>
v2 o2 o1 C2 C1 C0 e1 e2 v1 v3 v5 e3 e4 v4 v6 e5 B1 = image of = column space of M2 o2 = <{v4, v5} + {v5, v6} + {v4, v6}> = <e3 + e4 + e5>
v2 v5 o2 o1 e1 e2 e3 e4 C2 C1 C0 v1 v3 v4 v6 e5 H1 = Z1/B1 = (kernel of )/ (image of ) o1 o2 null space of M1 column space of M2 = <e3 + e4 + e5> <e3 + e4 + e5> = Rank H1 = Rank Z1 Rank B1 = 1 1 = 0
