Topology 2. Problem sheet 1

Problem 1.1. Prove that $H_0^{\Delta }(\mathcal{K})\cong {\mathbb{Z}}^{\oplus c}$, where $c$ is the number of connected components of the space $|\mathcal{K}|$.

Problem 1.2. Describe the homology of a) a disjoint union; b) a wedge of simplicial complexes.

Problem 1.3. Compute the homology of some triangulation¹ of a) the circle; b) the sphere; c*) the Möbius band.

Problem 1.4. Let ${\mathcal{K}}_1$, ${\mathcal{K}}_2$ be simplicial complexes on vertex sets $V_1$, $V_2$. Their join is

a) Check that this is a simplicial complex and that $|{\mathcal{K}}_1\ast {\mathcal{K}}_2|\cong |{\mathcal{K}}_1|\ast |{\mathcal{K}}_2|$. b) How are the chain complexes ${\tilde{C}}_{\bullet }^{\Delta }({\mathcal{K}}_1)$, ${\tilde{C}}_{\bullet }^{\Delta }({\mathcal{K}}_2)$, ${\tilde{C}}_{\bullet }^{\Delta }({\mathcal{K}}_1\ast {\mathcal{K}}_2)$ related?

Problem 1.5. Let $V_1$, $V_2$ be finite linearly ordered sets. Consider

a) Check that this is a simplicial complex and that $|\mathcal{P}|\cong {\Delta }^{|V_1|-1}\times {\Delta }^{|V_2|-1}$. b) Construct a similar triangulation of a product of simplicial complexes.

Problem 1.6. Let $C_{\bullet }=(0\to C_n\to \cdots \to C_0\to 0)$ be a chain complex of finite-dimensional vector spaces. Its Euler characteristic is the number $\chi (C_{\bullet }):={\sum }_{i\ge 0}(-1{)}^i\dim C_i$. a) Prove that $\chi (C_{\bullet })=\chi (H_{\ast }(C_{\bullet }))$. b) Prove the "strong Morse inequalities":

c*) Generalise to complexes of finitely generated abelian groups ($\dim$ is replaced by $\mathrm{rk}$).

Problem 1.7. How many vertices can a triangulation of a) $S^2$; b) $\mathbb{R}{P}^2$; c) ${\mathbb{T}}^2$ have?

Problem 1.8. Let $C_{\bullet }$, $C_{\bullet }^{\prime }$ be chain complexes and $A$ an abelian group. Define the structure of a chain complex on a) $C_{\bullet }\oplus C_{\bullet }^{\prime }$; b) $C_{\bullet }{\otimes }_{\mathbb{Z}}A$; c) $C_{\bullet }/C_{\bullet }^{\prime }$, if $C_{\bullet }^{\prime }\subset C_{\bullet }$ is a subcomplex.

Problem 1.9. A sequence of homomorphisms is exact if "everywhere the image equals the kernel". Prove: a) The sequence $A\to B\stackrel{f}{\to }C\to D$ is exact $⟺$ it splits into exact sequences $A\to B\to K\to 0$, $0\to K\to C\to D$. b) $0\to A\stackrel{\mathrm{incl}}{\to }A\oplus C\stackrel{\mathrm{pr}}{\to }C\to 0$ is exact, but not every exact sequence $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$ is of this form. c) Nevertheless, it is, if $p$ has a section or $i$ admits a retraction.

Problem 1.10. a) Let $C_{\bullet }$ be a chain complex of finitely generated free abelian groups. Prove that it is isomorphic to a direct sum of complexes of the form $(0\to \mathbb{Z}\to 0)$ and $(0\to \mathbb{Z}\stackrel{\times n}{\to }\mathbb{Z}\to 0)$. b) Let $C_{\bullet }$ be a chain complex of vector spaces over a field $\mathbb{F}$. Prove that $C_{\bullet }$ is isomorphic to a direct sum of complexes of the form $(0\to \mathbb{F}\to 0)$ and $(0\to \mathbb{F}\stackrel{\mathrm{id}}{\to }\mathbb{F}\to 0)$.