数学代写|Math8306 Algebraic topology

Suppose we have a $\Delta$-set $X$ with
$$
\begin{aligned}
X_0 & ={p} \
X_1 & ={a, b, c} \
X_2 & ={u, v}
\end{aligned}
$$
and face maps
$$
\begin{aligned}
\partial^i(a) & =p & \partial^i(b) & =p & \partial^i(c) & =p \
\partial^0(u) & =a & \partial^1(u) & =c & \partial^2(u) & =b \
\partial^0(v) & =b & \partial^1(v) & =c & \partial^2(v) & =a
\end{aligned}
$$
What is the resulting space?

Construct a $\Delta$-set whose geometric realization is the 2 -sphere $S^2$.
Compute the simplicial homology groups $H_n\left(\mathbb{R P}^2 ; \mathbb{Z}\right)$ using the $\Delta$ complex structure given in class.
Suppose $f: A \rightarrow B$ and $g: B \rightarrow C$ are homomorphisms of abelian groups. Show that there is an exact sequence
$$
0 \rightarrow \operatorname{ker}(f) \rightarrow \operatorname{ker}(g f) \rightarrow \operatorname{ker}(g) \rightarrow \operatorname{coker}(f) \rightarrow \operatorname{coker}(g f) \rightarrow \operatorname{coker}(g) \rightarrow 0
$$

Draw the image of a system of parallel lines under the inversion $z \mapsto$ $1 / z$. Prove it.

Let
$$
P(z)=y^n+b_1(z) y^{n-1}+b_2(z) y^{n-2} \cdots++b_{n-1}(z) y+b_0(z)
$$
be a monic polynomial of degree $n$ on $y$, such that the coefficients $b_i(z)$ are polynomial functions of $z$. Prove that if $P(z)$ has $n$ distinct roots for some value of $z$ then there are also $n$ distinct roots for all but finitely many values of $z$. (Hint: Use the discriminant. Example: If $P(z)=y^2+b(z) y+c(z)$ then the discriminant equals $\left.(b(z))^2-4 c(z)\right)$.

Make a picture of the three-sheeted surface cover of the completed $z$ plane associated to the equation $y^3=(z-a)(z-b)(z-c)$, where $a, b$ and $c$ are distinct complex numbers remember that the completion is done by adding into the surface the appropriate number of points over the points of infinity of the $z$-plane

What is the multiple connectivity of a surface of genus two? Draw a sequence of pictures like the ones made in class going from the torus to a disk for the surface of genus two going to a disk.