# Differences between Algebraic Topology and Algebraic Geometry

## Main Question or Discussion Point

i don't know if i can post it here, like this man https://www.physicsforums.com/showthread.php?t=397395, there's alot of usefull comment for me.

anyway, i'm still don't really know which one i like, either algebraic topology, or algebraic geometry. but i really do like algebra... so i'm planning to take my next year course. for now i've decided go to algebraic "topology"( because i've heard topology is more like analysis, and i love analysis and also, geometry are motivated by low dimensional(more to practical thing i guess which i not preferred) but topology defined based on nature),

but i was wondering, is it ok if i did'nt take ANY geometry courses.. or rather is it possible for me to study rigorously in topology without any fundamental on geometry? it's not that i don't want to learn them, just sometimes it's really redundant for me to learn something which is not really usefull for me to pursue something i like,
(example: like there's course on magic and latin square, which have nothing to do with algebraic topology, sooner or later, i'll forget about that course)

thanks for the time, and sorry if anything wrong with my english.

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Office_Shredder
Staff Emeritus
Gold Member
Neither of these courses are going to look like a classical geometry course, and wouldn't require any such background.

Algebraic topology starts by taking a topological space and examining all the loops contained in it. For example, in the plane every loop can be contracted to a single point. But on a torus, if you have a loop going around it through the middle, this cannot be contracted to a single point. Algebraic topology makes this rigorous by constructing a group consisting of all "distinct" loops (they can't be wiggled to form another one) I don't see how taking an algebraic topology class before taking a normal topology class makes sense to be honest, so you might want to look into how that would work

Algebraic geometry is the study of the zero sets of polynomials. For example, y-x2=0 just gives the parabola, x2+y2-1=0 just gives the unit circle. Of course you can do this in arbitrary dimensions. You can look at the set of polynomials which are zero on such a set - for example on the parabola, the polynomial y4-x2y3 is always zero as well. These polynomials form an ideal in the ring of all polynomials, and properties about this ring correspond to properties about the set of points where the polynomials are zero.

So to summarize, the two subjects have very little to do with each other, besides the fact that they both involve algebra (not even the same type of algebraic object) and that the words topology and geometry are often assumed to be talking about similar things

i've made my decision, i'm taking topology, algebraic topology and differential topology. without geometry, algebraic geometry and differential geometry. it's ok right?, i hope its ok