Topology II -- final exam

1. Evaluate the 1-homology of the simplicial complex K given below: $$0-cells: v_0,v_1,...,v_n;$$ $$1-cells: (v_0,v_1),(v_1,v_2),...,(v_n,v_0).$$

2.* Prove $\pi_1(K^2)=\pi_1(K)$, where $K^2$ is the $2$-skeleton of complex $K$.

3. Prove that a simplicial map preserves the equivalence relation of edge paths (on the edge group).

4. Suppose $K$ is the simplicial complex of the tetrahedron. Prove that $∂∂(θ)=0$ for any 2-chain θ in $K$.

5. Suppose $K$ is the unit disk in the plane: $$K = \{(x,y)∈R²: x² + y² ≤ 1\}.$$ Suppose an equivalence relation on K is given by: $(x,y)\sim (-x,y)$ for all $x,y$, $(0,1)\sim (0,-1)$. Sketch $K/\sim $. Is it a surface, a surface with boundary, or neither?

6. For two finite 2-dimensional simplicial complexes $K$ and $L$, prove that $$χ(K×L) = χ(K)χ(L),$$ where $K×L$ is a triangulation of the $|K×L|$.

7. Compute the cycle group $Z₁(K)$, by solving a system of linear equations, of the square frame $K$ with a diagonal (5 edges).

8. Compute the fundamental group of the sphere with the segment connecting the north pole to the south pole attached.

9. The boundary of the Mobius band $M$ is a circle. Let $f:\mathbf{S}^{1}\rightarrow M$ be the function that wraps the circle onto this boundary. Compute $f_*$, the homomorphism on the fundamental group.

10. Suppose $f:K\rightarrow L$ is a deformation retraction. Prove that $f_{\ast}:\pi_1(K)\rightarrow \pi_1(L)$ is an isomorphism.