# Torus and period integrals

I'm following a course in Riemann surfaces, and I'd like to solve the exercise below. Let $$L$$ a lattice in $$\mathbb{C}$$, and let $$T:= \mathbb{C}/L$$ the corresponding torus.

i) Prove that $$dx$$ and $$dy$$ span $$H_1^{dR}(T)$$ (where $$x,y$$ are the standard coordinate in $$\mathbb{C}$$ and $$H_1^{dR}(T)$$ the first de Rham cohomology group of $$T$$).

ii) Chose a canonical basis of $$H^1(T)$$ and compute the period integrals of $$dx$$ and $$dy$$ with respect to that basis (where $$H^1(T)$$ is the first homology group of $$T$$).

Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $$H^1$$? For example I know that $$H^1(T)$$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.

Many thanks.

Remark that for every real $$a,b, adx+bdy$$ is closed, suppose that this form is exact, if you have $$df=adx+bdy$$ where $$f$$ is a differentiable function defined on the torus, then $$f$$ has an extremum point $$u$$ and $$df_u=0$$, this implies that $$(adx+dy)_u$$ which is equivalent to $$a=b=0$$. So $$dx$$ and $$dy$$ are linearly independent. Since $$H^1(T)$$ is $$2$$-dimensional you have your basis.