Cellular Homology

Prove or give a (justified) counterexample. If the first homology group \(H_1(X)\) of a connected CW-complex \(X\) is trivial then \(X\) is contractible.

Compute the relative homology groups \(H_n(X,A)\) where \(X\) is \(S^2\) and \(A\) is a finite set of points in \(X\).

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.

1. Explain how the set of lines in \(\mathbb {C}^{n+1}\) is topologized to yield the space \(\mathbb {C}\mathrm {P}^n\). Describe how to equip \(\mathbb {C}\mathrm {P}^n\) with a CW structure with \(m\)-skeleton isomorphic to \(\mathbb {C}\mathrm {P}^m\) for all \(m\le n\).
2. Compute the singular homology groups \(H_*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\). Be sure you indicate which properties of homology you are using.

Fix integers \(n\ge 1\) and \(m\ge 1\).

1. Construct a connected space \(X\) such that \[ \widetilde {H}_r(X;\mathbb {Z})= \begin {cases} 0 & \text {if } r\ne n,\\ \mathbb {Z}/m & \text {if } r=n. \end {cases} \] (Here \(\widetilde {H}\) denotes reduced homology.)
2. For any connected space \(X\) satisfying the conclusion of part (a), compute the groups \(H_r(X;\mathbb {Z}/2)\) for all \(r\ge 0\).

For \(k\in \N \) and a topological space \(X\), let \(C_kX\) be a collection of \(k\) cones on \(X\), with their bases – each homeomorphic to \(X\) – identified. So for example, \(C_2X\) is the suspension of \(X\). Express the homology groups (with integer coefficients) of \(C_kX\) in terms of the homology groups of \(X\).

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

Define the singular homology groups \(H_i(X;\mathbb {Z})\) of a topological space \(X\) and compute the singular homology groups of the \(n\)-sphere \(S^n\), justifying all your steps.

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 \[ \operatorname {rank}(H_n(X))-\operatorname {rank}(H_{n-1}(X)) =\operatorname {rank}(H_n(Y))-\operatorname {rank}(H_{n-1}(Y))-1. \]

Using any method you like, compute the homology groups of the space \(X\) from Problem 8.

State the Excision Theorem for homology.

Let \(X\) denote the solid torus \(S^1\times D^2\). Using Mayer-Vietoris, compute the homology groups of the double of \(X\), obtained by taking two copies of \(X\) and gluing them along the boundary via the identity map. Also compute \(H_i(X,\partial X)\).

Describe an explicit cell decomposition of \(\mathbb {C}P^n\) (including the precise subsets corresponding to cells; you don’t have to check all properties of cell structures). From the cell decomposition read off all homology and cohomology groups with \(\mathbb {Z}\) coefficients.

Use the Mayer-Vietoris sequence to compute homology groups (with \(\mathbb {Z}\)-coefficients) of the space \(X_{2,3}\) from problem #17. Do the same for the space \(X_{2,2}\) defined similarly except that on \(S^1\times \{1\}\) antipodal points are identified.

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.

Define a \(\Delta \)-complex structure on the Klein bottle and compute its homology with \(\mathbb {Z}\) and \(\mathbb {Z}/2\) coefficients.

Let \(X\) be the quotient space of the 2-sphere \(S^2\) under the identifications \(x\sim -x\) for \(x\) in the equator \(S^1\). Compute the homology groups \(H_i(X)\) using any method you like.

Carefully construct a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to compute the homology groups of \(\mathbb {R}P^2\) with \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

Let \(K\) be the \(\Delta \)-complex obtained by identifying all the 2-dimensional faces of a 3-simplex. Calculate the simplicial homology of \(K\).

Construct a \(\Delta \)-complex for the Klein bottle and calculate its simplicial homology for both \(\mathbb {Z}\) and \(\mathbb {Z}_2\) coefficients.

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 \(i\ge 0\).

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

Construct carefully a \(\Delta \)-complex structure on \(\mathbb {R}P^2\) and use it to compute homology groups of \(\mathbb {R}P^2\) with \(\mathbb {Z}\) and with \(\mathbb {Z}_2\) coefficients.

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 \(t^2\subset S^5\) be a homeomorph of the two torus \(T^2\). Find the singular homology \(H_*(S^5-t^2;\mathbb {Z})\).

Find the singular homology groups \(H_*(\mathbb {R}P^5;\mathbb {Z})\).

Let \(X=\Delta -\bigcup _{j=1}^{\infty }D_j^\circ \) where \(\Delta \) is the closed unit disk and \(D_j^\circ \subset \Delta \) are open round disks with pairwise disjoint closures. Prove that the homology of \(X\) is infinitely generated.

Using homology prove Brouwer’s fixed-point theorem: If \(f:I^n\to I^n\) is a continuous map then there exists \(x\in I^n\) such that \(f(x)=x\). (Here and below \(I\) denotes the closed unit interval.)

a. Let \(S^3\) be the 3-sphere, and let \(T^2=S^1\times S^1\) be the 2-torus. Compute the singular homology of \(S^3\times T^2\) using its CW-complex structure.

b. Recompute the singular homology of \(S^3\times T^2\) using the Eilenberg-Zilber and Künneth Theorems.

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.

Find the singular homology \(H_*(S^5-(A\cup B),\mathbb {Z})\) where \(A\) is a homeomorph of \(S^2\), \(B\) a homeomorph of \(S^3\), and \(A\cap B=\{z_0\}\) is a single point.