funnily enough, before krispy kreme abominations, donuts were better approximations to mathematical torus
they didn't really half a filling
bagels on the other hand, were never good approximations to mathematical torus
unless you painstakingly kept saying, "like the surface of a bagel, is what I mean by 'torus'!"
anyway.. even if the torus was just the boundary of what you thought it was, i don't see how you wouldn't end up with a filled in sphere at the end
unless by 'identify longitudnal/meridonial circles' to a a point, you meant something very different..
the interior regions of the hemispheres bounded them aren't touching anyway
to lose an entire dimension, you will need to collapse something of comdimension 1
your circles have codimension 2 in the filled in torus
im not able to think of a way to collapse something of dimension 3 to something of dimension 2 in the realm of nice spaces, lets stick to CW complexes or smooth manifolds
without just collapsing it all to a point
like if we just think about things in terms of hausdorff dimension, it would look like to do that you need to collapse some codimension 0 subset... but really what you would have to do is find a way to hollow out via collapsing.. is that possible?
my intuition says no, because if you collapse a codimension 0 subset that is not of full measure.. it seems like you would never lose a dimension
by the way, if you're taking an AT course that follows a book like hatcher, and come from something like a hardcore analysis background, you can justify all this cut and pasting business formally , look into adjunction spaces and see that you can basically view gluing as quotienting by an ideal
in the sense that, when quotienting by $(a,b)$ you can always quotient by $(b)$ and then quotient out $(\overline{a})$ posthoc
thats essentially what you need to justify to make all this business concrete, but the proofs are basically routine
(on a set-theoretic level at least, it should be clear everything checks out)
and the quotient construction is not much more than the set theoretic quotient, theres no other topology you can put on quotients than the final one, basically by definition of colimits
hatchers book imo is not great for these types of minds, and to make do with only using it for a first course you'd have to basically justify by drawing a lot of pictures, and hope that your manipulations are legitimate (but in the context of the spaces he works with they always are)
that being said, i haven't read other AT books, funnily enough, I learned most of my AT using books about riemann surfaces / smooth manifolds
(except a bit of May's , but that is basically antipodal to hatcher in style)
ahlfors-sario is a classic
the first section gives a ground up proof of the classification of surfaces
using triangulations etc, and its written by ahlfors, so if you're used to his style from say his complex analysis classic, then it might be worth looking into
okay.. ive procastinated enough, ciao
What will be the quotient space obtained from torus by identifying the longitudinal and the meridian circles of it?
The problem is that I can't visualize the space but in Hatcher's book it is claimed that the quotient space is homeomorphic to the $2$-sphere $\Bbb S^2.$ Is there any easier way ...
