CW Complexes and Cell Attachments

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T = S^1 \times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1 \times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).

2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) coefficients.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute the homology groups of \(\mathbb {R}P^n\) with coefficients in \(\mathbb {Z}/2\).

Compute the homology groups of the cell complex obtained by identifying the sides of the polygon below as indicated:

Figure: a hexagon with boundary edges, in clockwise order starting at the top, labeled \(a^{-1}\), \(c\), \(b\), \(a\), \(c^{-1}\), and \(b^{-1}\), with arrows indicating the side identifications.

Let \(X\) be a CW complex consisting of one vertex p, two edges a and b attached to \(p\) along their boundaries, and two faces \(f_1\) and \(f_2\), attached along \(ab^2\) and \(ba^2\), respectively.

1. Find \(\pi _1(X)\). Is it a finite group?
2. Compute the homology groups \(H_i(X;\mathbb {Z})\) of \(X\) for all \(i\).

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.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). For each cell define its attaching map.

Suppose a space \(Y\) is obtained from a space \(X\) by attaching a single \(n\)-cell

via an attaching map \(f:S^{n-1}\to X\), i.e.

Show that \(H_i(X)\cong H_i(Y)\) for all \(i\) except at most two and give an example where there are two values of \(i\) where the two groups are not isomorphic.

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.

Let \(X\) be a connected cell complex and \(f:\mathbb {R}P^{2n}\to X\) a covering map from the real projective space of dimension \(2n\). Show that \(f\) is a homeo- morphism.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). For each cell define its attaching map.

Let \(X\) be a connected CW complex and \(A\subset X\) a connected subcomplex. Suppose that inclusion \(A\subset X\) is \(\pi _1\)-injective (for some, hence all, basepoints in \(A\)). Let \(p:\widetilde {X}\to X\) be the universal cover of \(X\). Prove that each component of \(p^{-1}(A)\) is simply connected.

Let \(X_n\) be the topological space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to \(S^1\times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).
2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\) and \(\mathbb {Z}_m\)-coefficients.

For \(n\ge 3\) let \(X=S^{n-1}\times S^1\) be the product of an \((n-1)\)-sphere and a circle and let \(Y\) be obtained by attaching an \(n\)-cell to \(X\) where the attaching map is a degree \(k\) from the boundary of the cell to the \((n-1)\)-sphere \(S^{n-1}\times \{p\}\subset X\) via a degree \(k\) map. Calculate the homology of \(Y\).

Let \(X\) be obtained from attaching two tori along a circle in their product structure. In particular each individual torus is \(S^1\times S^1\) and we attach the two tori along circles \(S^1\times \{p\}\) in the first torus and \(S^1\times \{q\}\) in the second torus. Calculate the cohomology and cup product structure on \(X\) with \(\mathbb {Z}\) coefficients. You can use the cup product structure of a torus as a given. (Hint: \(X\) retracts onto each of the individual tori.)

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 \(\chi (X)=0\).

Suppose \(X\) and \(Y\) are cell complexes and that \(A\subseteq X\) is a subcomplex. Let \(f:A\to Y\) be continuous and let \(Y\sqcup _f X\) be \(Y\) with \(X\) attached along \(A\) via \(f\). Prove that \(Y\sqcup _f X\) is homotopy equivalent to a cell complex.

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

Let \(X=S^1\vee S^1\) and \(x_0\in X\) be the attaching point.

1. Carefully state van Kampen’s Theorem and use it to compute \(\pi _1(X,x_0)\).
2. Find two covering spaces of \(X\) with three sheets; one of which is regular (determining a normal subgroup of \(\pi _1(X,x_0)\)) and the other of which is irregular.

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 \(p,q\ge 2\) be integers and denote by \(X_{p,q}\) the join of discrete sets \(\{v_1,\ldots ,v_p\}\) and \(\{w_1,\ldots ,w_q\}\). Thus \(X_{p,q}\) is a graph with \(p+q\) vertices and \(pq\) edges.

1. What is the fundamental group of \(X_{p,q}\)?
2. Denote by \(Y_{p,q}\) the 2-complex obtained from \(X_{p,q}\) by attaching 2-cells with attaching maps \(S^1\to X_{p,q}\) homeomorphisms to all possible cycles of length 4. (Thus there are \(\binom {p}{2}\binom {q}{2}\) 2-cells.) Show that \(Y_{p,q}\) is simply connected.

Describe a cell structure on the real projective space \(\mathbb {R}P^n\). Explicitly describe the attaching maps of all cells. Use this cell structure to compute \(H_i(\mathbb {R}P^n;\mathbb {Z}/2)\) for \(i\ge 0\).

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 3 attaching map and the other with degree 5 attaching map.

1. Using van Kampen’s theorem show that \(X\) is simply connected.
2. Show that \(X\) is homotopy equivalent to the 2-sphere. Hint: Use repeateadly the fact that the spaces \(Y\cup _f e^2\) and \(Y\cup _g e^2\) obtained from \(Y\) by attaching a single 2-cell with homotopic attaching maps \(f,g\) are homotopy equivalent.

Let \(X\) be the space obtained from the disjoint union of a circle \(S^1\) and a cylinder \(S^1\times [0,1]\) by attaching the cylinder along its boundary to the circle with the attaching map on \(S^1\times \{0\}\) having degree 2 and on \(S^1\times \{1\}\) degree 3. Find a presentation of the fundamental group of \(X\).

Let \(Y\) be a space obtained from a space \(X\) by attaching an \(n\)-cell. What is the relationship between the homology groups of \(X\) and of \(Y\)? Specifically, prove the following, where \(f:X\to Y\) is inclusion.

• \(f_*:H_i(X)\to H_i(Y)\) is an isomorphism for \(i\ne n-1,n\).
• \(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) is surjective.
• \(f_*:H_n(X)\to H_n(Y)\) is injective.
• Either \(f_*:H_{n-1}(X)\to H_{n-1}(Y)\) has finite kernel or \(f_*:H_n(X)\to H_n(Y)\) is an isomorphism (but not both).

Assuming the ranks of \(H_{n-1}(X)\) and \(H_n(X)\) are finite, also prove that \[ \operatorname {rank}(H_n(X))-\operatorname {rank}(H_{n-1}(X)) =\operatorname {rank}(H_n(Y))-\operatorname {rank}(H_{n-1}(Y))-1. \]

Let \(p:\widetilde {X}\to X\) be a covering map between cell complexes. Assume that \(\widetilde {X}\) is connected and that \(p\) is null-homotopic. Prove that \(\widetilde {X}\) is contractible.

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give a proof by induction on \(n\) using (a). State any theorems you are using.
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give an inductive proof. You may use Poincaré duality.

1. Describe the simplest cell structures on \(\mathbb {C}\mathrm {P}^2\) (including attaching maps).
2. Prove that \(H_i(\mathbb {C}\mathrm {P}^2;G)\) is isomorphic to \(H_i(S^2\vee S^4;G)\) for every abelian group \(G\) and all \(i\), and the same is true for cohomology.
3. Prove that \(\mathbb {C}\mathrm {P}^2\) is not homotopy equivalent to \(S^2\vee S^4\) by considering ring structures in cohomology.

Let \(X\) be the space obtained from the circle \(S^1\) by attaching two 2-cells, one with degree 2 attaching map, and the other with degree 4 attaching map. What is the cardinality of the fundamental group of \(X\)? If you use van Kampen’s theorem, state it.

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n)\) (integral coefficients)? Give a proof by induction on \(n\) using (a).
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n)\)? Give an inductive proof. You may use Poincaré duality.

Define a deformation retract. Construct a 2-dimensional cell complex that contains both an annulus \(S^1\times I\) and a Möbius band as deformation retracts.

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.

Let \(X\) be a finite connected cell complex such that \(H^2(X;\mathbb {Z})=\mathbb {Z}/2\mathbb {Z}\) and \(H^i(X;\mathbb {Z})=0\) for \(i=1\) and \(i>2\). Compute \(H_i(X;\mathbb {Z})\) for all \(i\). State all theorems you are using.

Let \(X\) be the cell complex obtained from the circle \(S^1\) by attaching two 2-cells \(e^2_2\) and \(e^2_3\) with attaching maps of degrees 2 and 3 respectively. Compute the fundamental group of \(X\). Carefully state any theorems you are using.

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.

Give an example of a connected cell complex \(X\) such that \(H_k(X;\mathbb {Z})=0\) for all \(k>0\) but \(\pi _1(X)\ne \{1\}\). You do not have to verify the second claim, but you should argue the first.

Prove that a closed orientable surface \(S_g\) of genus \(g\ge 1\) is not homotopy equivalent to the wedge \(X\vee Y\) of two finite cell complexes both of which have nontrivial \(H_1(\cdot ;\mathbb {Z})\).

Let \(X\) be a cell complex obtained from the circle by attaching a 2-cell with a degree 3 attaching map. Prove that \(X\) cannot be embedded into \(\mathbb {R}^3\). You may assume that the embedding is such that there exists a “regular neighborhood”, i.e. a smooth compact manifold \(N\subset \mathbb {R}^3\) such that \(X\hookrightarrow N\) is a homotopy equivalence.

Describe the procedure for computing (i.e. writing down a presentation of) the fundamental group of a cell complex with one 0-cell. Using this, show that \(\mathbb {C}P^n\) is simply connected and \(\pi _1(\mathbb {R}P^n)=\mathbb {Z}/2\mathbb {Z}\) for \(n\ge 2\).

Let \(X_n\) be the space obtained by attaching a disk \(D\) to the torus \(T=S^1\times S^1\) where the attaching map is a degree \(n\) map from \(\partial D\) to the circle \(S^1\times \{p\}\) in \(T\).

1. Calculate \(\pi _1(X_n)\).
2. Calculate the homology and cohomology of \(X_n\) with \(\mathbb {Z}\)-coefficients and with \(\mathbb {Z}_m\)-coefficients where \(\mathbb {Z}_m\) is the cyclic group of order \(m\).

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 \(i\ge 0\).

Suppose \(X\) is a connected 2-dimensional CW complex such that \(H_1(X;\mathbb {Z})\cong \mathbb {Z}/2\) and \(H_2(X;\mathbb {Z})\cong \mathbb {Z}\). Compute \(H_i(X;\mathbb {Z}_2)\), \(H^i(X;\mathbb {Z})\) and \(H^i(X;\mathbb {Z}_2)\) for all \(i\in \mathbb {Z}\).

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.

Let \(X\) be the quotient space of \(S^2\) obtained by identifying the north and south poles to a single point. Carefully define a cell complex structure on \(X\) (including a description of attaching maps) and use this to compute \(\pi _1(X)\).

For a connected CW complex \(X\), call a connected covering space \(\widetilde {X}\to X\) abelian if it is normal and has abelian deck transformation group. Show that \(X\) has an abelian covering space that is a covering space of every other abelian covering space of \(X\), and that such ‘universal’ abelian covering space is unique up to isomorphism. Describe this covering space explicitly for \(X=S^1\vee S^1\). Carefully state theorems you are using.

Let \(X_d\) be the CW-complex obtained by attaching a disk with a degree \(d\) map to the circle \(S^1\times \{x_0\}\) on the torus \(S^1\times S^1\) (and \(x_0\in S^1\) is a basepoint). Calculate the homology and cohomology of \(X_d\) with \(\mathbb {Z}\), \(\mathbb {R}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

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 \(H_n(X;\mathbb {Z}_5)=0\).

Let \(X\) be the space obtained by attaching a disk \(D\) to \(S^1\times S^1\) where the attaching map is a degree \(d\) map from \(\partial D\) to \(S^1\times \{pt\}\). Calculate the homology and cohomology groups for \(X\) with \(\mathbb {Z}\) and \(\mathbb {Z}_p\) coefficients where \(p\) is prime.

Let \(X\) be a connected CW-complex obtained by attaching an \((n+2)\)-dimensional cell to an \(n\)-dimensional CW-complex \(A\). What is the smallest possible dimension \(k\) where \(\pi _k(X)\) is not isomorphic to \(\pi _k(A)\)?

Compute the homology groups of the space obtained from a torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus.

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 \(i\ge 0\).

The 4-dimensional cube \([0,1]\times [0,1]\times [0,1]\times [0,1]\) has 16 vertices, 32 edges and 24 2-dimensional faces. How many 3-dimensional faces does it have?

Describe all possible ways in which the homology of a space can change after attaching an \(n\)-cell.

Let \(X\) be the space obtained from the torus \(S^1\times S^1\) by attaching a Möbius band via a homeomorphism from the boundary circle in the Möbius band to the circle \(S^1\times \{x_0\}\) in the torus. Compute \(\pi _1(X)\) and describe the universal cover of \(X\).

A. Find an explicit CW complex structure for the complex projective space \(\mathbb {C}P^3\), and compute the singular homology groups (over \(\mathbb {Z}\)) using cellular homology.

B. Find an explicit CW complex structure for \(S^2\times S^4\), and compute the singular homology groups using cellular homology.

Describe all possible ways in which the homology of a space can change after attaching an \(n\)-cell.

Let \(X_p=D^4\cup _f S^3\) be the space obtained by attaching a 4-cell to the 3-sphere by an attaching map \(f:\partial D^4\to S^3\) of degree \(p\).

1. Compute the singular homology of \(H_*(X_p;\mathbb {Z})\) and \(H_*(X_p\times X_q;\mathbb {Z})\) for any \((p,q)\) using the CW-complex cell structure.
2. Find \(H_*(X_p\times X_q;\mathbb {Z})\) using the Künneth Theorem.