The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let 
![\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f_1+%3D+%5Cdots+%3D+f_m+%3D+0%2C+f_i+%5Cin+R%5Bx_1%2C+%5Cdots+x_n%5D&bg=ffffff&fg=333333&s=0&c=20201002)
Then it’s not hard to see that this problem is equivalent to the problem of finding ![S = R[x_1, \dots x_n]/(f_1, \dots f_m)](https://s0.wp.com/latex.php?latex=S+%3D+R%5Bx_1%2C+%5Cdots+x_n%5D%2F%28f_1%2C+%5Cdots+f_m%29&bg=ffffff&fg=333333&s=0&c=20201002)
of commutative rings making 
of affine schemes. Allowing 
The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map 
We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.
of 
of 
there exists some 
such that the tensor product 
is isomorphic to the identity for the tensor product, namely 
be a closed orientable surface of genus 
. (Below we will occasionally write 
, omitting the genus.) Then its Euler characteristic 
is even. In this post we will give five proofs of this fact that do not use the fact that we can directly compute the Euler characteristic to be 
, roughly in increasing order of sophistication. Along the way we’ll end up encountering or proving more general results that have other interesting applications. 
. When 
the resulting complex surfaces give nice examples of 
-torus 
in as many ways as I can think of. Below, if no coefficient ring is specified then the coefficient ring is 
, its tangent space 
at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
, its cohomology 
is a priori a graded abelian group, but it ends up having the structure of a graded ring.
over 
is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-
of a space: that is, we will show that, in a suitable sense, each individual homotopy group 
of a group is an interesting construction: it associates to every group 
(unlike the abelianization ![G/[G, G]](https://s0.wp.com/latex.php?latex=G%2F%5BG%2C+G%5D&bg=ffffff&fg=333333&s=0&c=20201002)
). We might wonder, then, exactly what kind of construction the center is.
we may associate to an object 
its automorphism group 
or endomorphism monoid 
), and this is a canonical construction which again doesn’t extend in an obvious way to a functor 
or 
. (It merely reflects some special part of the 
.) 
, where 
Annoying Precision 