Chain Complexes and Chain Homotopy

We can regard \(\pi _1(X,x_0)\) as the set of basepoint-preserving homotopy classes of maps \((S^1,s_0) \to \) \((X,x_0)\). Let \([S^1,X]\) be the set of homotopy classes of maps \(S^1 \to X\) with no condition on basepoints. Thus there is a natural map \(\psi : \pi _1(X,x_0) \to [S^1,X]\) obtained by ignoring basepoints. Show that \(\psi \) is onto if \(X\) is path connected.

1. Construct a single chain complex \(C\) such that \(C\) has the following three properties:

   • \(H_i(C)\) is non-zero if and only \(i=1\).
   • There is a coefficient group \(G_0\) such that \(\tilde {H}^i(C;G_0)=0\) for all \(i\).
   • There is a coefficient group \(G_1\) such that \(\tilde {H}^i(C;G_1)\) is non- zero if and only if \(i=1,2\).
2. Find a topological space \(X\) such that \(H_i(C)=\tilde {H}_i(X)\) for all \(i\).

Show that \(S^2\vee S^4\) is not homotopy equivalent to a compact manifold without boundary.

Let \(T^2\) be the real 2-dimensional torus and \(f:S^1\vee S^1\to T^2\) a continuous map. Prove or disprove the following assertions:

1. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(f\circ g:T^2\to T^2\) is homotopy to the identity map.

2. There is a continuous map \(g:T^2\to S^1\vee S^1\) such that \(g\circ f\) is homotopy to the identity map.

   Justify your answers.

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Let \(p : \tilde {X} \to X\) be a covering map between two connected CW-complexes. If \(p\) is null-homotopic, prove that \(\tilde {X}\) is contractible. If you use a theorem make sure you state it carefully and include all the hypotheses.

1. Define the notion of a chain morphism between two chain complexes and prove that it induces a homomorphism in homology.
2. Define the notion of a chain homotopy equivalence.
3. Prove that a chain homotopy equivalence induces an isomorphism in homology.

Give an example of a chain complex whose homology and cohomology are different.

Let \(C_\bullet \) be a chain complex of finitely-generated \(\mathbb {Z}\)-modules such that \(C_i=0\) for all \(|i|\gg 0\) (i.e., the complex \(C_\bullet \) is bounded). Define the Euler characteristic of \(C\) to be \[ \chi (C)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(C_i), \] where \(\operatorname {rank}(C_i)\) denotes the rank of the (finitely-generated) abelian group \(C_i\).

1. Let \(X\) be a finite CW complex, i.e. one having finitely many cells in total. Define \(\chi (X)\) to be \(\chi (C_\bullet ^{\mathrm {CW}}(X))\), where \(C_\bullet ^{\mathrm {CW}}(X)\) denotes the cellular chain complex of \(X\). Show that \[ \chi (X)=\sum _{i\in \mathbb {Z}}(-1)^i\operatorname {rank}(H_i(X)), \] and that this quantity equals \(\sum _{i\in \mathbb {Z}}(-1)^i\dim _F(H_i(X;F))\) for any field \(F\).
2. Let \(X\) be a connected compact manifold of odd dimension; you may assume \(X\) is homeomorphic to a finite CW complex. Show that \(\chi (X)=0\).

Give the chain complex used to define cellular homology of a cell complex \(X\). Prove that it’s a chain complex.

Prove that \(H_i(S^2\vee S^4,\mathbb {Z})\) and \(H_i(\mathbb {C}P^2,\mathbb {Z})\) are isomorphic for all \(i\), but that \(S^2\vee S^4\) is not homotopy equivalent to \(\mathbb {C}P^2\).

Let \(\{B_n\}_{n\in \mathbb {Z}}\) and \(\{C_n\}_{n\in \mathbb {Z}}\) be chain complexes. What is the definition of a chain map from \(\{B_n\}_{n\in \mathbb {Z}}\) to \(\{C_n\}_{n\in \mathbb {Z}}\)?

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2\}\). Define \(\pi :\mathbb {R}^3\backslash \{0\}\to S^2\) by \[ \pi (x,y,z)= \left ( \frac {x}{\sqrt {x^2+y^2+z^2}}, \frac {y}{\sqrt {x^2+y^2+z^2}}, \frac {z}{\sqrt {x^2+y^2+z^2}} \right ) \] and \(T_\epsilon (x,y,z)=(x,y,z+\epsilon )\). For \(\epsilon \in (0,1)\) the map \(f_\epsilon =\pi \circ T_\epsilon \) is a Lefschetz map from \(S^2\) to itself. Calculate its Lefschetz number and conclude every map of \(S^2\) to itself that is homotopic to the identity has a fixed point. (Hint: It will be easier to calculate the derivative of \(\pi \) and \(T_\epsilon \) separately and use the chain rule to find the derivative of \(f_\epsilon \).)

Let \(p:\widetilde {X}\to X\) be an \(n\)-sheeted covering map between two connected, locally path-connected spaces. Let \(C_*(X)\) and \(C_*(\widetilde {X})\) denote the singular chain complexes of \(X\) and \(\widetilde {X}\) respectively.

1. Give a definition of a chain morphism and prove that \(\tau :C_*(X)\to C_*(\widetilde {X})\) given by \[ \tau (\sigma )=\sum _{i=1}^n \widehat {\sigma }_i \] is a chain morphism, where \(\widehat {\sigma }_1,\ldots ,\widehat {\sigma }_n\) are the \(n\) lifts of \(\sigma \) to \(\widetilde {X}\), and \(\sigma \) is a singular simplex in \(X\).
2. Prove that \[ p_*:H_i(\widetilde {X};\mathbb {Q})\to H_i(X;\mathbb {Q}) \] is always surjective. Hint: Consider the composition \(p_*\tau :C_*(X)\to C_*(X)\).
3. Find an example where \[ p_*:H_i(\widetilde {X};\mathbb {Z})\to H_i(X;\mathbb {Z}) \] is not surjective.

Define the concepts of chain maps and chain homotopies, and prove that chain homotopy is an equivalence relation.

Let \(f\) be the homeomorphism of the annulus \(S^1\times [0,1]\) given by \[ f(z,s)=(ze^{2\pi is},s) \] where we view \(S^1\) as the set of unit norm complex numbers.

1. Construct an explicit isotopy (i.e. homotopy through homeomorphisms) between \(f\) and the identity that does not move the points of \(S^1\times \{0\}\).
2. Prove that there is no homotopy between \(f\) and the identity that does not move points on both \(S^1\times \{0\}\) and \(S^1\times \{1\}\).

Give a definition of the chain homotopy between chain maps and prove that chain homotopic chain maps induce the same homomorphism in homology.

Suppose that \(X\) and \(Y\) are two connected manifolds or cell complexes homotopy equivalent to each other. Prove that their universal covers \(\widetilde {X}\) and \(\widetilde {Y}\) are homotopy equivalent to each other. Carefully state all theorems you are using.

An algebraic fact, often used in homology theory, is that a short exact sequence of chain complexes induces a long exact sequence in homology. State this fact carefully and give a definition of the “connecting homomorphism” (the one that lowers the degree of the homology group).

1. Define the concepts of chain morphisms and chain homotopies.
2. Show that every map \(S^2\to T^2\) from the 2-sphere to the 2-torus is null-homotopic.
3. Write down explicit cellular chain complexes \(C(S^2)\) and \(C(T^2)\) for \(S^2\) and \(T^2\) and a chain morphism \(\Phi :C(S^2)\to C(T^2)\) which is not chain homotopic to a chain morphism representing a constant map.

Define the concepts of chain maps and chain homotopies, and prove that chain homotopy is an equivalence relation.

Let \(S^2\) be the 2-sphere and \(T\) the torus.

1. Show that every map from \(S^2\) to \(T\) is null-homotopic.
2. Show that there are infinitely many homotopy classes of maps from \(T\) to \(S^2\).

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.

Let \(p:\widetilde {X}\to X\) be a 2-sheeted covering map between cell complexes such that the cell structure on \(\widetilde {X}\) is obtained by lifting the cell structure on \(X\).

1. Construct a short exact sequence of cellular chain complexes with coefficients in \(\mathbb {Z}/2\mathbb {Z}\) \[ 0\to C_n(X;\mathbb {Z}/2\mathbb {Z})\to C_n(\widetilde {X},\mathbb {Z}/2\mathbb {Z}) \to C_n(X,\mathbb {Z}/2\mathbb {Z})\to 0 \] (i.e. give the maps explicitly and prove that the sequence is exact).
2. Write down the induced long exact sequence of homology groups (the transfer sequence).
3. Prove that if \(\widetilde {X}\) is acyclic over \(\mathbb {Z}/2\mathbb {Z}\) (i.e. \(H_i(\widetilde {X};\mathbb {Z}/2\mathbb {Z})=0\) for \(i>0\) and \(H_0(\widetilde {X};\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\)) then \(H_i(X;\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\) for all \(i\ge 0\).

Prove that \(\mathbb {C}P^3\) and \(S^2\times S^4\) have cell complex structures with isomorphic cellular chain complexes, and therefore \(H^i(\mathbb {C}P^3;\mathbb {Z})\cong H^i(S^2\times S^4;\mathbb {Z})\) for all \(i\). Also prove that cohomology rings \(H^*(\mathbb {C}P^3;\mathbb {Z})\) and \(H^*(S^2\times S^4;\mathbb {Z})\) are not isomorphic. Are \(\mathbb {C}P^3\) and \(S^2\times S^4\) homotopy equivalent? You are allowed to use the standard cell complex structure of spheres and complex projective spaces.

Prove that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

Show that \(S^2\vee S^4\) and \(\mathbb {C}P^2\) are not homotopy equivalent.

Define the higher homotopy groups, \(\pi _i\), for \(i\ge 2\) and prove that they are abelian.

1. Describe the (simplest) cell structure of \(\mathbb {R}P^n\) (give the attaching maps of all the cells – you don’t have to give a proof here).
2. Write down the cellular chain complex associated with this cell decomposition.
3. Use this chain complex to compute the homology and cohomology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}\) and \(\mathbb {Z}_2\).

Construct examples of CW-complexes \(X\) with each of the following properties.

1. For some \(n\), \(H_n(X)\not \cong H^n(X;\mathbb {Z})\).
2. For some \(n\), \(H_n(X;\mathbb {Z}_2)\ne 0\) and \(H_n(X;\mathbb {Z}_3)\ne 0\) but \(H_n(X;\mathbb {Z}_5)=0\).

Show that \(S^2\vee S^4\) is not homotopy equivalent to \(\mathbb {C}P^4\).

Show that \(\mathbb {R}P^3\) is not homotopy equivalent to \(\mathbb {R}P^2\vee S^3\).

1. Describe the (simplest) cell structure of \(\mathbb {R}P^n\) (give the attaching maps of all cells - you don’t have to give a proof here).
2. Write down the cellular chain complex associated with this cell decomposition.
3. Use this chain complex to compute homology and cohomology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}\) and \(\mathbb {Z}/2\mathbb {Z}\).

Let \(p:\widetilde {X}\to X\) be a 2-sheeted covering map between cell complexes such that the cell structure on \(\widetilde {X}\) is obtained by lifting the cell structure on \(X\).

1. Construct a short exact sequence of cellular chain complexes with coefficients in \(\mathbb {Z}/2\mathbb {Z}\) \[ 0\to C_n(X;\mathbb {Z}/2\mathbb {Z})\to C_n(\widetilde {X};\mathbb {Z}/2\mathbb {Z}) \to C_n(X;\mathbb {Z}/2\mathbb {Z})\to 0 \] (i.e. give the maps explicitly and prove that the sequence is exact).
2. Write down the induced long exact sequence of homology groups (the transfer sequence).
3. Prove that if \(\widetilde {X}\) is acyclic over \(\mathbb {Z}/2\mathbb {Z}\) (i.e. \(H_i(\widetilde {X};\mathbb {Z}/2\mathbb {Z})=0\) for \(i>0\) and \(H_0(\widetilde {X};\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\)) then \(H_i(X;\mathbb {Z}/2\mathbb {Z})\cong \mathbb {Z}/2\mathbb {Z}\) for all \(i\ge 0\).

Prove that \(\mathbb {C}P^2\) and the wedge sum (one point union) \(S^2\vee S^4\) are not homotopy equivalent.

Explicitly describe a nontrivial homotopy class in \(\pi _3(S^2)\). Show that it is nonzero.

Suppose \((C_k,\partial _k)\) is a free chain complex. Suppose that it is acyclic, i.e. the homology groups \(H_k(C)=0\) for all \(k\). Show that \(C\) is contractible, that is, the identity map and the zero map \(\iota _k,\zeta _k:C_k\to C_k\) are chain homotopic.

Let \(f,g:K\to L\) be two chain maps (\(f_n,g_n:K_n\to L_n\)). Define what it means for \(f\) and \(g\) to be chain homotopic. Prove that if \(f\) and \(g\) are chain homotopic, then they induce the same maps in homology, \(f_*=g_*:H_n(K)\to H_n(L)\).

Let \(n>2\). By \(\mathbb {Z}/n\) denote the cyclic group of order \(n\) with discrete topology and by \(S^2\) the 2-sphere with standard topology. Let \(p\) and \(q\) be two distinct points of \(S^2\). Compute the homology groups of the \(n\)-link sausage space \(X\) which is the space that can be represented as the identification space \((S^2\times \mathbb {Z}/n)/{\sim }\) with the equivalence relation induced by \((p,i)\sim (q,i+1)\).

Diagram description: the space \(X\) is drawn as a cyclic chain of lobe-shaped copies of \(S^2\). Each copy is attached to the next one around the cycle, with the \(p\)-point of one copy identified with the \(q\)-point of the next.

Denote by \([X,Y]\) the set of free homotopy classes of maps \(X\to Y\). Show that \([S^2,\mathbb {R}P^2]\) is infinite. In other words, construct a sequence of maps \(f_i:S^2\to \mathbb {R}P^2\), \(i=1,2,3,\ldots \), such that no two are homotopic to each other.

Let \(\{C_n,d_n\}\) and \(\{C'_n,d'_n\}\) be chain complexes of abelian groups and \(f_n:C_n\to C'_n\) be a chain homotopy equivalence. Then the induced map \(f_{n*}:H_n(C)\to H_n(C')\) is isomorphism in homology.