Lecture 2 Addition and free abelian groups of
Lecture 2: Addition (and free abelian groups) of a series of preparatory lectures for the Fall 2013 online course MATH: 7450 (22 M: 305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Target Audience: Anyone interested in topological data analysis including graduate students, faculty, industrial researchers in bioinformatics, biology, computer science, cosmology, engineering, imaging, mathematics, neurology, physics, statistics, etc. Isabel K. Darcy Mathematics Department/Applied Mathematical & Computational Sciences University of Iowa http: //www. math. uiowa. edu/~idarcy/Applied. Topology. html
A free abelian group generated by the elements x 1, x 2, …, xk consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k xk where ni are integers. Z = The set of integers = { …, -2, -1, 0, 1, 2, …} = the set of all whole numbers (positive, negative, 0) Addition: (n 1 x 1 + n 2 x 2 + … + nkxk) + (m 1 x 1 + m 2 x 2 + … + mkxk) = (n 1 + m 1) x 1 + (n 2 + m 2)x 2 + … + (nk + mk)xk
Will add video clips when video becomes available. Formal sum: 4 cone flower + 2 rose + 3 cone flower + 1 rose = 7 cone flower + 3 rose
A free abelian group generated by the elements x 1, x 2, …, xk consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k xk where ni are integers. Z = The set of integers = { …, -2, -1, 0, 1, 2, …} = the set of all whole numbers (positive, negative, 0) Addition: (n 1 x 1 + n 2 x 2 + … + nkxk) + (m 1 x 1 + m 2 x 2 + … + mkxk) = (n 1 + m 1) x 1 + (n 2 + m 2)x 2 + … + (nk + mk)xk
A free abelian group generated by the elements x 1, x 2, …, xk consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k xk where ni are integers. Example: Z[x 1, x 2] 4 x 1 + 2 x 2 x 1 - 2 x 2 -3 x 1 kx 1 + nx 2 Z = The set of integers = { …, -2, -1, 0, 1, 2, …}
A free abelian group generated by the elements x 1, x 2, …, xk consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k xk where ni are integers. Example: Z[x , x ] 4 ix + 2 I x – 2 i -3 x k + n iii Z = The set of integers = { …, -2, -1, 0, 1, 2, …}
A free abelian group generated by the elements x 1, x 2, …, xk consists of all elements of the form n 1 x 1 + n 2 x 2 + … + n k xk where ni are integers. Example: Z[x 1, x 2] 4 x 1 + 2 x 2 x 1 - 2 x 2 -3 x 1 kx 1 + nx 2 Z = The set of integers = { …, -2, -1, 0, 1, 2, …}
Addition: (n 1 x 1 + n 2 x 2 + … + nkxk) + (m 1 x 1 + m 2 x 2 + … + mkxk) = (n 1 + m 1) x 1 + (n 2 + m 2)x 2 + … + (nk + mk)xk Example: Z[x 1, x 2] (4 x 1 + 2 x 2) + (3 x 1 + x 2) = 7 x 1 + 3 x 2 (4 x 1 + 2 x 2) + (x 1 - 2 x 2) = 5 x 1
Addition: (n 1 x 1 + n 2 x 2 + … + nkxk) + (m 1 x 1 + m 2 x 2 + … + mkxk) = (n 1 + m 1) x 1 + (n 2 + m 2)x 2 + … + (nk + mk)xk Example: Z[x , x 2] (4 x + 2 x 2) + (3 x + x 2) = 7 x + 3 x 2 (4 x 1 + 2 x 2) + (x 1 - 2 x 2) = 5 x 1
Addition: (n 1 x 1 + n 2 x 2 + … + nkxk) + (m 1 x 1 + m 2 x 2 + … + mkxk) = (n 1 + m 1) x 1 + (n 2 + m 2)x 2 + … + (nk + mk)xk Example: Z[x 1, x 2] (4 x 1 + 2 x 2) + (3 x 1 + x 2) = 7 x 1 + 3 x 2 (4 x 1 + 2 x 2) + (x 1 - 2 x 2) = 5 x 1
Example: 4 vertices + 5 edges + 1 faces 4 v + 5 e + f. v = vertex e = edge f = face
Example 2: 4 vertices + 5 edges 4 v + 5 e v = vertex e = edge
v 2 e 1 v 1 e 4 e 2 e 3 v 3 e 5 v 4 v 1 + v 2 + v 3 + v 4 + e 1 + e 2 + e 3 + e 4 + e 5
v 2 e 1 v 1 e 4 e 2 e 3 v 3 e 5 v 4 v 1 + v 2 + v 3 + v 4 in Z[v 1, v 2, , v 3, v 4] e 1 + e 2 + e 3 + e 4 + e 5 in Z[e 1, e 2, , e 3, e 4, e 5]
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 3 e 5 v 4 Technical note: In graph theory, the cycle also includes vertices. I. e, this cycle in graph theory is the path v 1, e 1, v 2, e 2, v 3, e 3, v 1, . Since we are interested in simplicial complexes (see later lecture), we only need the edges, so e 1 + e 2 + e 3 is a cycle.
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 3 e 5 Note that e 3 + e 4 + e 5 is a cycle. v 4 Technical note: In graph theory, the cycle also includes vertices. I. e, the cycle in graph theory is the path v 1, e 1, v 2, e 2, v 3, e 3, v 1, . Since we are interested in simplicial complexes (see later lecture), we only need the edges, so e 1 + e 2 + e 3 is a cycle.
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 4 v 3 e 5 Note that – e 3 + e 5 + e 4 is a cycle.
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 4 v 3 e 5 Note that – e 3 + e 4 + e 5 is a cycle. Objects: oriented edges ei
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 4 v 3 e 5 Note that – e 3 + e 5 + e 4 is a cycle. Objects: oriented edges in Z[e 1, e 2, e 3, e 4, e 5] ei
v 2 e 1 v 1 e 4 Note that e 1 + e 2 + e 3 is a cycle. e 2 e 3 v 4 v 3 e 5 Note that – e 3 + e 5 + e 4 is a cycle. Objects: oriented edges in Z[e 1, e 2, e 3, e 4, e 5] ei – ei
v 2 e 1 v 1 e 4 e 2 e 3 e 1 e 2 e 4 e 5 v 3 e 5 v 4 (e 1 + e 2 + e 3) + (–e 3 + e 5 + e 4) = e 1 + e 2 + e 5 + e 4
v 2 e 1 v 1 e 4 e 1 e 2 e 3 v 4 v 3 e 5 v 1 e 4 e 2 e 3 v 3 e 5 v 4 e 1 + e 2 + e 3 + e 1 + e 2 + e 5 + e 4 = 2 e 1 + 2 e 2 + e 3 + e 4 + e 5 e 1 + e 2 + e 5 + e 4 + e 1 + e 2 + e 3 = 2 e 1 + 2 e 2 + e 3 + e 4 + e 5
v 2 e 1 v 1 e 4 The boundary of e 1 = v 2 – v 1 e 2 e 3 v 4 v 3 e 5
v 2 e 1 v 1 e 4 The boundary of e 1 = v 2 – v 1 e 2 e 3 v 3 e 5 The boundary of e 2 = v 3 – v 2 The boundary of e 3 = v 1 – v 3 v 4 The boundary of e 1 + e 2 + e 3 = v 2 – v 1 + v 3 – v 2 + v 1 – v 3 = 0
Add a face v 2 e 1 v 1 e 4 e 2 e 3 v 4 v 3 e 5
Add an oriented face v 2 e 1 v 1 e 4 e 2 e 3 v 4 v 3 e 5
Add an oriented face v 2 e 1 v 1 e 4 v 2 e 3 v 4 e 1 v 3 e 5 v 1 e 4 e 2 e 3 v 4 v 3 e 5
Add an oriented face v 2 e 1 v 1 e 4 e 2 e 3 v 4 v 3 e 5 Note that the boundary of this face is the cycle e 1 + e 2 + e 3
Simplicial complex v 2 e 1 v 1 e 4 e 2 e 3 v 4 v 3 e 5
0 -simplex = vertex = v 1 -simplex = oriented edge = (vj, vk) Note that the boundary vj vk of this edge is v – v k j e 2 -simplex = oriented face = (vi, vj, vk) v 2 Note that the boundary e 1 e 2 of this face is the cycle e 1 + e 2 + e 3 v 1 v 3 e 3
3 -simplex = (v 1, v 2, v 3, v 4) = tetrahedron v 2 v 4 v 1 v 3 4 -simplex = (v 1, v 2, v 3, v 4, v 5)
- Slides: 31