/sci/ - Science & Math

>>10376695

>>10151242
Cox.
Not a pun btw

>>10037994
??

>>9760147
[math]H_\bullet:({\bf Top},\coprod)\rightarrow({\bf Grp},\otimes)[/math] is a graded covariant functor satisfying the Eilenberg-Steenrod axioms. What exactly are you trying to ask here?
>>9766166
>In homology, you identify "n-dimensional loops" in a space that can be written as the boundary of an n+1-dimensional "surface".
In singular homology on CW complexes*
Homology detects holes, not loops. Homotopy groups detects loops.
Otherwise it's an alright explanation.
>>9768228
Assuming you're in flat space, the metric [math]\eta = \operatorname{diag}(1,-1,-1,-1)[/math] moves the indices up and down. For instance if [math]e^\mu\in\Gamma^{0,1}(M)[/math]is a contravariantvariant rank 1-tensor then [math]\eta_{\mu\nu}e^\nu = e_\mu[/math].
Now since [math]\eta^{-1} = \eta[/math], letting [math]e_\mu\in\Gamma^{1,0}(M)[/math] be the covariant rank-1 tensor dual to [math]e^\mu[/math], we can do [math]e^2 = e^\mu e_\mu = e_\nu \eta^{\mu\nu}\eta_{\rho\mu}e^\rho = e_\nu \delta^{\nu}_{\rho}e^\rho = e_\nu e^\nu[/math], so it doesn't really matter whether we define the quadratic form to be [math]\Gamma^{1,0}(M)\times \Gamma^{0,1}(M)\rightarrow \mathbb{R}[/math] or [math]\Gamma^{0,1}(M)\times\Gamma^{1,0}\rightarrow\mathbb{R}[/math], at least for spaces [math]M[/math] with flat metrics.