Topology 2. Problem sheet 2

Problem 2.1. For a simplicial complex $\mathcal{K}$, construct a chain map $C_{\ast }^{\Delta }(\mathcal{K})\to C_{\ast }(|\mathcal{K}|)$.

▷ In a few lectures we will prove that it induces an isomorphism $H_{\ast }^{\Delta }(\mathcal{K})\cong H_{\ast }(|\mathcal{K}|)$.

Problem 2.2. Suppose the inclusion of a subspace $i:A\hookrightarrow X$ admits a retraction (by the way, what is that?). Prove that $i_{\ast }:H_{\ast }(A)\to H_{\ast }(X)$ is a) injective; b) the inclusion of a direct summand.

Problem 2.3. a) Construct a group homomorphism $h:{\pi }_1(X,x_0)\to H_1(X)$. b) Let $X$ be path-connected. Prove that $h$ is surjective and $[{\pi }_1(X),{\pi }_1(X)]\subseteq \mathrm{Ker}(h)$.

▷ In a few lectures we will prove that $[{\pi }_1,{\pi }_1]=\mathrm{Ker}(h)$, that is, ${\pi }_1(X{)}_{ab}\cong H_1(X)$ ("Poincaré's theorem").

Problem 2.4. Let $x\in H_n(X)$ be an arbitrary element. Construct a finite CW complex $Y$, an element $y\in H_n(Y)$ and a map $f:Y\to X$ such that $x=f_{\ast }(y)$.

Problem 2.5. Let $X_0\subset X_1\subset \cdots \subset X={\bigcup }_{n\ge 0}{X}_n$ be a chain of nested topological spaces such that every compact set $K\subset X$ lies in some $X_n$. a) Prove that the natural map ${\mathrm{colim}}_n{H}_{\ast }(X_n)\to H_{\ast }(X)$ is an isomorphism. b) The same for ${\pi }_1(X,x_0)$.

Problem 2.6. a) By constructing a chain homotopy, prove that the simplicial homology of a simplex is trivial. b) Compute the simplicial homology of the $d$-dimensional skeleton of the $n$-dimensional simplex.

Problem. A simplicial map ${\mathcal{K}}_1\to {\mathcal{K}}_2$ is a map $f:V_1\to V_2$ such that $f(I)\in {\mathcal{K}}_2$ for all $I\in {\mathcal{K}}_1$. Construct a chain map $f_{\#}:{\tilde{C}}_{\ast }^{\Delta }({\mathcal{K}}_1)\to {\tilde{C}}_{\ast }^{\Delta }({\mathcal{K}}_2)$ and check that $(f\circ g{)}_{\#}=f_{\#}\circ g_{\#}$. Consider three cases: a) $f$ is monotone; b) $f$ is bijective; c) $f$ is arbitrary.

Problem 2.8. a) Let $C_{\bullet }$ be a chain complex over a field. Construct a chain homotopy equivalence between $C_{\bullet }$ and $(H_{\ast }(C_{\bullet }),0)$. b) Is this true for complexes of free abelian groups?

Problem 2.9. Prove: a) chain homotopy is an equivalence relation (on the set of chain maps $C_{\bullet }\to C_{\bullet }^{\prime }$); b) if $f\sim g:C_{\bullet }\to C_{\bullet }^{\prime }$ and $f^{\prime }\sim g^{\prime }:C_{\bullet }^{\prime }\to C_{\bullet }^{\prime \prime }$, then $f^{\prime }\circ f\sim g^{\prime }\circ g$.

Problem 2.10. Suppose the diagram below commutes and has exact rows. Prove: a) If $l$ is surjective and $m$, $p$ are injective, then $n$ is injective. b) If $m$, $p$ are surjective and $q$ is injective, then $n$ is surjective. c) (The five lemma) If $l,m,p,q$ are isomorphisms, then $n$ is an isomorphism.