Algebraic Topology 12: Intro to Singular Homology

[@swaruppaul4417]

You are a messenger. ! God sent you to teach us algebraic topology! ❤

[@ompatel9017]

The magician is back

[@jerrymahajan]

I love the way you explained singular n complexes, we did it in class with the same definition, but I didn't really understand what it meant, the example really helped

[@ZihanKe]

so impressive, Hatcher's book is great but it is really hard to read for an non-native english speaker(it is always full of examples and description instead of formal definition and proof), and your interpretation of the book reveals the magic

[@maynardtrendle820]

I loved the "behatted man" joke!😂

[@evaxu1325]

you have taken the joke to a new level!

[@hyperduality2838]

Points are dual to lines -- the principle of duality in geometry. Singular homology is dual to simplicial homology -- homology is dual. "Always two there are" -- Yoda. Injective is dual to surjective synthesizes bijection or isomorphism (duality). Homology is dual to co-homology. Categories (form, syntax) are dual to sets (substance, semantics) -- category theory.

[@MarioPucci_mamio]

The hat joke, I'm not sure if it's more of a complex joke than a simplicial one 😂😂😂

[@richardchapman1592]

What if you try orthogonality on S¹?

[@davidhand9721]

How can these spaces be Abelian? We are no longer describing a sequence of paths that end where they started; if the cycles can be in any order, then the map from S1 to the space can be discontinuous. You can pick up the pen and draw the cycle piecewise, so to speak. If the idea is to specify which elements constitute the cycle, then why are they Z and not Z mod 2? There's no way to make sense of this as an abelian group.

[@davidhand9721]

Can someone please explain how these delta basis sets are groups? I get the game we are playing with them, i.e. how they are computed, but I have no clue how they are groups. What is the identity? What is the inverse of a point? What does this group act on? What is the group combinator, i.e. the *+* in those expressions? I keep seeing sets of vertices, but what does it _mean?_ I feel like we waltzed right past that and now I have no hope of understanding homology as a consequence. Also what is a kernel and an image, exactly? I can see where he's getting them, but what is their significance?

[@MarioPucci_mamio]

For the torus example, the homology of the quotient of a simplicial complex is computed rather than that of a simplicial complex, correct?

[@MathatAndrews]

It's a Delta complex (see previous lecture for the distinction), but you're right in that the identification of the sides is why there are only three 1-simplicies and one 0-simplex.

[@xanderlewis]

This is me being silly (because we all know what you meant), but it's not quite true that 'any space with n points' will have singular chain groups isomorphic to Z^n. X needs to be discrete, otherwise you can potentially map the standard 1-simplex into X continuously in ways that cover more than one point, and there are as many ways to do this as there are to chop the unit interval into n pieces - a heck of a lot! If X _does_ have the discrete topology, however, what you say is true.

[@xanderlewis]

I guess it's better to say 'colour the unit interval with n colours' than 'chop it into n pieces' since the individual pieces don't have to be connected.

[@rickroesler5247]

The hat joke was good!

[@davidhand9721]

If there's only one map from a simplex to a point, then why is the group Z? What are these _groups?_

[@nonolethug3010]

This a group generated by one element, like Z which is generated by 1

[@hyperduality2838]

Points are dual to lines -- the principle of duality in geometry. Singular homology is dual to simplicial homology -- homology is dual. "Always two there are" -- Yoda. Injective is dual to surjective synthesizes bijection or isomorphism (duality). Homology is dual to co-homology. Categories (form, syntax) are dual to sets (substance, semantics) -- category theory. The integers are self dual as they are their own conjugates. Real is dual to imaginary -- complex numbers are dual. All numbers fall within the complex planes hence all numbers are dual. Positive is dual to negative -- numbers, charge or curvature. Simplicity is dual to complexity. Domains are dual to co-domains -- group theory.

[@davidhand9721]

@@hyperduality2838 Jesus Christ you are everywhere, aren't you? You might even waste as much time on KZbin as I do.

[@hyperduality2838]

@@davidhand9721 Not quite but KZbin has alot of great videos theses days and I can always find something to watch. I have a very simple philosophy in that I try to learn something new everyday, do this for a few years and eventually you may have a profound idea:- Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Your brain/mind has the goal or target to create reality and this is a syntropic process -- teleological. Your mind likes to solve problems and this synthesizes solutions! Problem, reaction, solution -- the Hegelian dialectic. If you do the hard work by continually pushing yourself and asking the right kind of questions you may get lucky and discover something new. KZbin allows you to find answers much more quicker then before KZbin as there are usually people who have solved the problem to your questions. The teacher (Yoda, Socrates) is dual to the pupil (Skywalker, Plato) -- the Hegelian dialectic. The master (lordship) is dual to the slave (bondsman, apprentice) -- the rule of two, Darth Bane, Sith Lord. If you use KZbin correctly then you are not wasting your time. God (thesis) is dual to the Christ consciousness (anti-thesis) leads to the holy spirit or the mind/soul (synthesis) -- the Hegelian dialectic. Your mind/soul synthesizes or creates reality -- syntropic! Mind (syntropy) is dual to matter (entropy) -- Descartes or Plato's divided line.

[@עמיתלרמן]

The groups are not the groups of these maps, they are groups generated by these maps. That is, given that you have 1 map from Delta to a point, your group would be which is a group generated by 1 element, thus Z.