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.
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.
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).
Best one for me🥇
Very nice and clear lecture. Do you have cohomology lecture as well ?
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.
Take n+1 points which are not in an n dimensional linear space and we have an n simplex.
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.
Three cups of coffee and he’s good lol
39:15
E≡MC²³
pun intended at 46:00 ?