An excellent teacher. The lecture is crystal clear. Interesting points: An excellent way to show the Klein bottle has no orientation. I wonder if Witney's embedding theorem requires 2n embedding dimensions for an n dimensional closed and compact manifold only if the embedded manifold has no orientation and 2n-1 embedding dimensions otherwise or maybe there is a counter example. In RP2 the image of the boundary Phi2 is z1*c + z2*(a-b).
I'm confused, I keep trying to compute the boundary of a 2-chain, but it seems like the answer should always be zero, because a 2-chain is a linear combination of 2-complexes, and the boundary of any 2-complex is 0. But then, if the boundary map is a homomorphism, it follows that any linear combination of 2-complexes has a boundary of 0, i.e. any 2-chain has a boundary of 0.
[too late but might be useful for others] The boundary of a 2-complex is not always zero, but it has zero boundary (see: ∂^2 = 0). That might be the source of the confusion.
The more abstract the math the easier it is to understand, enjoy, and work with.
An excellent teacher. The lecture is crystal clear. Interesting points: An excellent way to show the Klein bottle has no orientation. I wonder if Witney's embedding theorem requires 2n embedding dimensions for an n dimensional closed and compact manifold only if the embedded manifold has no orientation and 2n-1 embedding dimensions otherwise or maybe there is a counter example. In RP2 the image of the boundary Phi2 is z1*c + z2*(a-b).
A small mistake in computing H1 of RP2, should be FAb(a-b+c, 2c)/FAb(a-b+c, c), instead of FAb(a-b, 2c)/FAb(a-b, c). This is example 2.4 in Hatcher.
Very nice and clear lecture. Do you have cohomology lecture as well ?
Best one for me🥇
why do we need the edge "c" when representing the torus T^2 with delta complex?
Read the definition of delta complex structure on X.
Because we’re trying to use triangles as building blocks.
I'm confused, I keep trying to compute the boundary of a 2-chain, but it seems like the answer should always be zero, because a 2-chain is a linear combination of 2-complexes, and the boundary of any 2-complex is 0. But then, if the boundary map is a homomorphism, it follows that any linear combination of 2-complexes has a boundary of 0, i.e. any 2-chain has a boundary of 0.
Boundary of a two complex is not zero always.
[too late but might be useful for others] The boundary of a 2-complex is not always zero, but it has zero boundary (see: ∂^2 = 0). That might be the source of the confusion.
Take n+1 points which are not in an n dimensional linear space and we have an n simplex.
more accurate, an n-dimensional *affine* space.
Three cups of coffee and he’s good lol
39:15
pun intended at 46:00 ?
E≡MC²³