Computing homology groups | Algebraic Topology | NJ Wildberger
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- čas přidán 9. 07. 2024
- The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2-dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each stage, or dimension.
To make this more understandable, we give in this lecture an in-depth look at some examples. Here we start with the simplest ones: the circle and the disk. For each space it is necessary to look at each dimension separately. The 0-th homology group H_0(X) measures the connectivity of the space X, for a connected space it is the infinite cyclic group Z of the integers. The first homology group H_1 measures the number of independent non-trivial loops in the space (roughly). The second homology group H_2 measures the number of independent non-trivial 2-dim holes in the space, and so on.
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Excellent video. Thank you!!!
MASSIVE thank you for uploading this. It's so much clearer to me now.
I love it. Wildberger saves us from the dreary task of grinding through the usual tomes on this subject, where the authors seem to believe that drawing a single picture would make the subject somehow less glorious :-)
Thanks Reiner!
Idk, Rudin has his appeal
Absolutely impressive! I am wondering if it is possible to post the videos for like: PL Gauss-Bonnet theorem, and about chain derivation results. Thank you for your generosity and your support!!!
I do have a series on Differential Geometry from a course I taught at UNSW. Thanks for the kind words, and remember that I have a Patreon site at www.patreon.com/njwildberger. You can become a patron of my channel and support it directly!
The del-zero map is always zero, since there is no chain group of dimension -1.
"Setting all vertices or elements equal to each other, x1 = x2 = x3 etc." for H(zero) is equivalent to making the mathematics democratic. Mod zero or B(zero) = 0 is like dividing by 1 in physics. You are imposing a condition of objective democracy on the mathematics here or treating all elements equally. The laws of physics are the same and equal for all observers they conform to a principle of objective or 100% democracy! The velocity of light is the same and equal for all observers, hence light or photons conform to a principle of objective democracy.
You are therefore imposing a condition of unbiased, 100% or proper objective democracy on calculating homology groups! Topology is therefore inherently using a principle of objective democracy or duality.
Objective is dual to subjective, absolute is dual relative, independence is dual to dependence.
Homology is dual to co-homology.
Elliptic curves are dual to modular forms -- Fermat's last theorem.
Union is dual to intersection.
Integration is dual to differentiation.
Points (objects) are dual to lines (representations) -- the principle of duality in geometry, category theory.
The initial value theorem (IVT) is dual to the final value theorem (FVT).
"Perpendicularity in hyperbolic geometry is measured in terms of duality".
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas.
Mind is dual to matter -- Descartes.
Matter duality: Bosons are dual to Fermions, waves are dual to particles -- quantum duality. Symmetric waves functions (Bosons) are dual to anti-symmetric wave functions (Fermions). Active matter (life) is dual to passive matter (atoms, forces).
Mind duality is dual to matter duality.
"Always two there are" -- Yoda.
Mod zero = 1 = mod infinity.
Poles (eigenvalues) are dual to zeroes -- optimized control theory.
You can treat poles or infinities as equivalent or dual to zero in control theory -- target tracking, teleology.
The real projective plane contains a mobius loop:-
If you connect positive infinity to negative infinity you get a mobius loop -- atomic spin.
Particles are dual to anti-particles -- Dirac equation.
Spin up is dual to spin down -- Dirac equation.
The Klein bottle is composed of two mobius loops.
The left handed mobius loop (spinor) is dual to the right handed mobius loop (spinor).
The Klein bottle is a surface that intersects itself -- two perspectives, faces = duality!
The big bang is a Janus hole (point) two faces = duality -- Julian Barbour.
Topological holes cannot be shrunk to zero.
Entropy is dual to evolution (syntropy) -- Janna Levin, astrophysicist.
Very clear explanation - thanks!
Great lecture. Thanks!
njwildberger I am in an algebraic topology course that essentially starts from your 35th video (intro to homology) and we are reading hatcher which is good but I find it too verbose, are there any (modern & friendly) books you would recommend on the subject? By the way, I really have enjoyed watching these videos the subject has really come alive and I'm able to appreciate the abstract setting with your great examples.
i think you have probably finished the course now. one of the books that seriously makes me think about Topology rather than just learn it is William Fulton's book on Springer. it appears to have been written in a most non traditional way, promoting thinking.
Just perfect, thanks so much professor
Awesome video! Thanks!
thanks, this video helped a lot!
Thanks for the video. I would actually like to know why you considered (x z) to be equal to - c. Clearly it is the difference in orientation, but is there an argument which would work for higher dimensions?
Thank you
Thanks!
Thanks
Highly useful lectures, thanks! As I understood you used the argument that a map is injective if it is non-zero on generators. Can this always be applied in the setting of homology? I was thinking of the projection from the integers onto Z/2Z as a counterexample...
The nth Homology group is Hn(X) = ker(d(n)) / im(d(n+1)). Remember d maps chains, namely d(n) : C(n) -> C(n-1). The group Hn corresponds to a particular chain C(n). The map (group homomorphism, ie. boundary map on chains) is injective if it is into, unique namely in the sense that for all unique vi, vj in C(n), d(vi) and d(vj) in C(n-1) are also unique.