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Slides for Lectures 3 and 4 Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) H∗ without tears Shapes of Spaces III Attaching cells Axioms for homology Bjørn Ian Dundas Homology – of SO(3) and in general Summer School, Lisbon, July 2017 The orthogonal groups Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) H∗ without tears Let O(n) be the space of all real orthogonal n × n-matrices. Attaching cells I.e., the real n × n-matrices A satisfying Axioms for homology AtA = I Includes reflections: ignoring these we have the special orthogonal group SO(n)={A ∈ O(n) | det A =1} consisting exactly of the rotations. t O(n)={A ∈ Mn(R) | A A = I } n2 subspace of Mn(R)=R At :thetransposeofS2 → S2A, p "→ −p, At A = I ⇔ theis not columns homotopic in A are to orthonormal S2 → S2, p "→ p. The space SO(3) of rotations in R3. Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) H∗ without tears R3 Understanding the space SO(3) of rotations in is all Attaching cells important for Axioms for homology robotics/prosthetics computer visualisation/games navigation 9 It is a 3D-subspace of M3R = R , but curves and folds up on itself in a manner that makes the flat 9D coordinates useless. At this point I hope you all did the hands-on exercise about rotations on the first exercise sheet. The space SO(3) of rotations in R3. Shapes of Spaces Bjørn Ian Dundas R3 A nontrivial rotation in has a unique axis: the eigenspace ∼ SO(3) = RP3. (w/eigenvalue 1) of the rotation matrix. H∗SO(3) Choosing an eigen vector corresponds to choosing an H∗ without tears orientation of the rotation. Attaching cells 3 Axioms for D → SO(3) homology p $→ the rotation around p by |p|π radians A rotation around p by |p|π radians is the same as a rotation around −p by −|p|π radians, which isn’t a problem until |p| =1. SO(3) =∼ D3/(|p| =1⇒ p ∼−p) =∼ RP3 SO(3) may be identified with the quotient of D3 by the equivalence relation that a point on the boundary is identified with its antipodal point, which is a model for the 3 real projective space RP . D3 = {p ∈ R3 ||p|≤1} The shape of SO(3) Shapes of Spaces Bjørn Ian Dundas ∼ Hence, to understand the shape of the space of rotations, it SO(3) = RP3. 3 suffices to understand the projective space RP obtained by H∗SO(3) 3 identifying antipodal points on the boundary of D . H∗ without tears n More generally, the (real) projective n-space RR is obtained Attaching cells by identifying antipodal points on the boundary of Dn. Axioms for homology E.g. RP2 is obtained from the (filled) square, with points on the boundary identified accordingly: 1 B z o a O A A a / z B As a toy example towards understanding the shape of 2 SO(3), we’ll calculate H∗RP . 1Shout if I forget to talk about electrons The shape of RP2 Shapes of Spaces Bjørn Ian Dundas B z o a ∼ RP3. O SO(3) = H∗SO(3) A A H∗ without tears Attaching cells a / z B Axioms for homology This is not a simplicial complex. We could model RP2 by a simplicial complex, but it would be much bigger e.g. requiring six vertices (as opposed to two), but H∗ can be calculated directly for “CW-complexes” of which the above is an example:2 it has two “0-cells”: z, a, two “1-cells”: A, B and one “2-cell”: the (filled) square S itself. The chain complex Homology has an ∂1 ∂2 ∂3 C0 o C1 o C2 o axiomatic description on CW-complexes 2Oh, no! he is repeating Dan’s lecture The shape of RP2 Shapes of Spaces Bjørn Ian Dundas B z o a ∼ RP3. O SO(3) = H∗SO(3) A A H∗ without tears Attaching cells a / z B Axioms for homology two “0-cells”: z, a, two “1-cells”: A, B and one “2-cell”: the (filled) square S itself. ∂1 ∂2 ∂3 C0 o C1 o C2 o ... a−z←A! 2A+2B←S! ⟨z, a⟩⟨o A, B⟩ o ⟨S⟩ z−a←B! Z0 = ⟨z, a⟩, B0 = ⟨z − a⟩, H0 = Z Z1 = ⟨A + B⟩,B1 = ⟨2(A + B)⟩,H1 = Z/2Z Z2 =0, B2 =0, H2 =0. The shape of RP2 Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) H∗ without tears B Attaching cells z o a O Axioms for homology A A a / z B 2 H0RP =0-pathconnected 2 H1RP = Z/2Z - has a hole, but “going around it twice, it vanishes”.3 3This can also be viewed as a manifestation of the fact that S 2 is the “universal covering space” of RP2 from another lecture. Shapes of Spaces H∗SO(3) Bjørn Ian Dundas 3 SO(3) = RP : b z ∼ 3 ⑧ ⑧ SO(3) = RP . ⑧⑧ ⑧⑧ ⑧ ⑧ H∗SO(3) z a H∗ without tears a z Attaching cells ⑧⑧ ⑧⑧ Axioms for ⑧⑧ ⑧⑧ homology c b Zero cells: z, a, b, c,1-cells: za, zb, zc, ab, ac, bc,2-cells: F , S, T (“front, side, top”), 3-cell: C.Cellcomplex za ∂1 za zb zc ∂2 ∂3 ⟨ bc⟩⟨o ab ac bc ⟩ o ⟨F , S, T ⟩ o ⟨C⟩ 1 . −1 −1 −1 −1 ... −11 1 ..−1 −1 . −11 . [∂1]= . 1 . 1 . −1 , [∂2]= 11 . , [∂3]=0 ..1 . 11 ⎡ . −1 −1 ⎤ ! " 1 . 1 ⎣ ⎦ H∗(SO(3)) is found by finding the null space and column spaces of [∂∗]. The shape of the space of rotations Shapes of Spaces Bjørn Ian Dundas 3 SO(3) = RP : b z SO(3) ∼ RP3. ⑧⑧ ⑧⑧ = ⑧⑧ ⑧⑧ H∗SO(3) z a H∗ without tears Attaching cells a z ⑧⑧ ⑧⑧ Axioms for ⑧⑧ ⑧⑧ homology c b Zero cells: z, a, b, c,1-cells: za, zb, zc, ab, ac, bc,2-cells: F , S, T (“front, side, top”), 3-cell: C. za z−az−b Z Z0 = ⟨ bc⟩, B0 = ⟨ z−c ⟩, H0 = za+ab+bc−zc za+ab+bc−zc Z1 = ⟨ zc−ac+ab−zb ⟩, B1 = ⟨ zc−ac+ab−zb ⟩, H1 = Z/2Z ab+bc−ac 2(ab+bc−ac) Z2 =0, B2 =0, H2 =0 Z3 = ⟨C⟩ B3 =0 H3 = Z. The shape of the space of rotations Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. Z if i =0, 3 H∗SO(3) Z Z H∗ without tears Hi SO(3) = ⎧ /2 if i =1 Attaching cells ⎨⎪0 otherwise. Axioms for homology ⎩⎪ H0 -pathconnected H1 - reflects the technical complications with giving good coordinates for computer graphics, the irritating and surprising behavior of the cables of a computer, robotics, aviation. 4 H2 = 0 - no problem, so nobody talks about it. H3 - reflects that SO(3) is a “closed 3D manifold” - a space that “locally looks like R3,compact(likeS3), and no loops take you to your mirror image”. 4Again, S 3 –thespaceof“unitquaternions”–istheuniversal covering space of SO(3) = RP3 Homology Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) H∗ without tears 1 As we have seen, given a good description of your space Attaching cells it is possible to calculate homology, giving us a powerful Axioms for homology tool in understanding the shape and properties of the space. 2 However, we have still not exhausted the good algebraic properties of homology. 3 Homology is characterized by a list of properties, and in practice, this list is exactly what you want to use for calculations - the construction is immaterial. Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. 5 Kernel, image... H∗SO(3) H∗ without tears Exact sequence = a sequence Attaching cells Axioms for ∂n+2 ∂n+1 ∂n homology ... / Mn+1 / Mn / ..., s.t. ker ∂n = im∂n+1 (exact sequence = “complex whose homology vanishes”) Abelian groups 5ya,ya - Dan did all of this, didn’t he? Except, that he said he didn’t want to teach you how to do linear algebra over the integers... and I won’t either Cell attachments Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) A cell attachment A ⊆ X is the inclusion of spaces you get H∗ without tears from a continuous φ: Sn → A,setting6 Attaching cells Axioms for homology X = A Dn+1 = A Dn+1/φ(p) ∼ p φ + + A CW-complex is a space obtained by repeatedly attaching cells (of increasing dimension, starting with the empty set). Disjoint union: A1 ! A2 = {(i, a) | i =1, 2, a ∈ Ai } 6 n for p ∈ S .AsubsetofX is open if S−1 = ∅, D0 = {0}. “it’s preimage in A and D n+1 are”. CW-complexes Shapes of Spaces Bjørn Ian Dundas ∼ SO(3) = RP3. H∗SO(3) CW-complex: H∗ without tears a space obtained by repeatedly attaching cells (of increasing Attaching cells Axioms for dimension, starting with the empty set). homology A CW-pair (X , A)isaCWcomplexX and a closed subspace A ⊆ X consisting of some of the cells of X . A pointed CW-complex is a CW-pair (X , {point}). Maps between pointed CW-complexes preserve the “base point”. Example: if (X , A) is a CW-pair, the quotient space X /A = X /a ≃ a′ for a, a′ ∈ A is a pointed CW-space (A/A ⊆ X /A is the “base point”) Axioms for homology Shapes of Spaces Bjørn Ian Dundas ˜ ∼ 3 A (reduced) homology theory is a sequence {hn}n of SO(3) = RP . 7 homotopy functors such that for each CW-pair (X , A) H∗SO(3) 8 there is a natural exact sequence H∗ without tears Attaching cells ∂ ˜ in ˜ qn ˜ ∂ ˜ in−1 Axioms for ..
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