1 (Spring ’09) #topology/qual/work
Compute the homology of the onepoint union of \(S^1 \times S^1\) and \(S^1\).
2 (Fall ’06) #topology/qual/work

State the MayerVietoris theorem.

Use it to compute the homology of the space \(X\) obtained by gluing two solid tori along their boundary as follows. Let \({\mathbb{D}}^2\) be the unit disk and let \(S^1\) be the unit circle in the complex plane \({\mathbf{C}}\). Let \(A = S^1 \times {\mathbb{D}}^2\) and \(B = {\mathbb{D}}^2 \times S^1\).
Then \(X\) is the quotient space of the disjoint union \(A {\textstyle\coprod}B\) obtained by identifying \((z, w) \in A\) with \((zw^3 , w) \in B\) for all \((z, w) \in S^1 \times S^1\).
3 (Fall ’12) #topology/qual/work
Let \(A\) and \(B\) be circles bounding disjoint disks in the plane \(z = 0\) in \({\mathbf{R}}^3\). Let \(X\) be the subset of the upper halfspace of \({\mathbf{R}}^3\) that is the union of the plane \(z = 0\) and a (topological) cylinder that intersects the plane in \(\partial C = A \cup B\).
Compute \(H_* (X)\) using the Mayer–Vietoris sequence.
4 (Fall ’14) #topology/qual/work
Compute the integral homology groups of the space \(X = Y \cup Z\) which is the union of the sphere \begin{align*} Y = \left\{{x^2 + y^2 + z^2 = 1}\right\} \end{align*} and the ellipsoid \begin{align*} Z = \left\{{x^2 + y^2 + {z^2 \over 4} = 1}\right\} .\end{align*}
5 (Spring ’08) #topology/qual/work
Let \(X\) consist of two copies of the solid torus \({\mathbb{D}}^2 \times S^1\), glued together by the identity map along the boundary torus \(S^1 \times S^1\). Compute the homology groups of \(X\).
6 (Spring ’17) #topology/qual/work
Use the circle along which the connected sum is performed and the MayerVietoris long exact sequence to compute the homology of \({\mathbf{RP}}^2 # {\mathbf{RP}}^2\).
7 (Fall ’15) #topology/qual/work
Express a Klein bottle as the union of two annuli.
Use the Mayer Vietoris sequence and this decomposition to compute its homology.
8 (Spring ’09) #topology/qual/work
Let \(X\) be the topological space obtained by identifying three distinct points on \(S^2\). Calculate \(H_* (X; Z)\).
9 (Fall ’05) #topology/qual/work
Compute \(H_0\) and \(H_1\) of the complete graph \(K_5\) formed by taking five points and joining each pair with an edge.
10 (Fall ’18) #topology/qual/work
Compute the homology of the subset \(X \subset {\mathbf{R}}^3\) formed as the union of the unit sphere, the \(z{\hbox{}}\)axis, and the \(xy{\hbox{}}\)plane.
11 (Spring ’05/Fall ’13) #topology/qual/work
Let \(X\) be the topological space formed by filling in two circles \(S^1 \times \left\{{p_1 }\right\}\) and \(S^1 \times \left\{{p_2 }\right\}\) in the torus \(S^1 \times S^1\) with disks.
Calculate the fundamental group and the homology groups of \(X\).
12 (Spring ’19) #topology/qual/work
 Consider the quotient space \begin{align*} T^2 = {\mathbf{R}}^2 / \sim {\quad \operatorname{where} \quad} (x, y) \sim (x + m, y + n) \text{ for } m, n \in {\mathbf{Z}} ,\end{align*} and let \(A\) be any \(2 \times 2\) matrix whose entries are integers such that \(\operatorname{det}A = 1\).
Prove that the action of \(A\) on \({\mathbf{R}}^2\) descends via the quotient \({\mathbf{R}}^2 \to T^2\) to induce a homeomorphism \(T^2 \to T^2\).
 Using this homeomorphism of \(T^2\), we define a new quotient space \begin{align*} T_A^3 \coloneqq{T^2\times{\mathbf{R}}\over \sim} {\quad \operatorname{where} \quad} ((x, y), t) \sim (A(x, y), t + 1) \end{align*}
Compute \(H_1 (T_A^3 )\) if \(A=\left(\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right).\)
13 (Spring ’12) #topology/qual/work
Give a selfcontained proof that the zeroth homology \(H_0 (X)\) is isomorphic to \({\mathbf{Z}}\) for every pathconnected space \(X\).
14 (Fall ’18) #topology/qual/work
It is a fact that if \(X\) is a single point then \(H_1 (X) = \left\{{0}\right\}\).
One of the following is the correct justification of this fact in terms of the singular chain complex.
Which one is correct and why is it correct?

\(C_1 (X) = \left\{{0}\right\}\).

\(C_1 (X) \neq \left\{{0}\right\}\) but \(\ker \partial_1 = 0\) with \(\partial_1 : C_1 (X) \to C_0 (X)\).
 \(\ker \partial_1 \neq 0\) but \(\ker \partial_1 = \operatorname{im}\partial_2\) with \(\partial_2 : C_2 (X) \to C_1 (X)\).
15 (Fall ’10) #topology/qual/work
Compute the homology groups of \(S^2 \times S^2\).
16 (Fall ’16) #topology/qual/work
Let \(\Sigma\) be a closed orientable surface of genus \(g\). Compute \(H_i(S^1 \times \Sigma; Z)\) for \(i = 0, 1, 2, 3\).
17 (Spring ’07) #topology/qual/work
Prove that if \(A\) is a retract of the topological space \(X\), then for all nonnegative integers \(n\) there is a group \(G_n\) such that \(H_{n} (X) \cong H_{n} (A) \oplus G_n\).
Here \(H_{n}\) denotes the \(n\)th singular homology group with integer coefficients.
18 (Spring ’13) #topology/qual/work
Does there exist a map of degree 2013 from \(S^2 \to S^2\).
19 (Fall ’18) #topology/qual/work
For each \(n \in {\mathbf{Z}}\) give an example of a map \(f_n : S^2 \to S^2\).
For which \(n\) must any such map have a fixed point?
20 (Spring ’09) #topology/qual/work
 What is the degree of the antipodal map on the \(n\)sphere?
(No justification required)
 Define a CW complex homeomorphic to the real projective \(n{\hbox{}}\)space \({\mathbf{RP}}^n\).
 Let \(\pi : {\mathbf{RP}}^n \to X\) be a covering map. Show that if \(n\) is even, \(\pi\) is a homeomorphism.
21 (Fall ’17) #topology/qual/work
Let \(A \subset X\). Prove that the relative homology group \(H_0 (X, A)\) is trivial if and only if \(A\) intersects every path component of \(X\).
22 (Fall ’18) #topology/qual/work
Let \({\mathbb{D}}\) be a closed disk embedded in the torus \(T = S^1 \times S^1\) and let \(X\) be the result of removing the interior of \({\mathbb{D}}\) from \(T\) . Let \(B\) be the boundary of \(X\), i.e. the circle boundary of the original closed disk \({\mathbb{D}}\).
Compute \(H_i (T, B)\) for all \(i\).
23 (Fall ’11) #topology/qual/work
For any \(n \geq 1\) let \(S^n = \left\{{(x_0 , \cdots , x_n )\mathrel{\Big}\sum x_i^2 =1}\right\}\) denote the \(n\) dimensional unit sphere and let \begin{align*}E = \left\{{(x_0 , . . . , x_n )\mathrel{\Big}x_n = 0}\right\}\end{align*} denote the “equator”.
Find, for all \(k\), the relative homology \(H_k (S^n , E)\).
24 (Spring ’12/Spring ’15) #topology/qual/work
Suppose that \(U\) and \(V\) are open subsets of a space \(X\), with \(X = U \cup V\). Find, with proof, a general formula relating the Euler characteristics of \(X, U, V\), and \(U \cap V\).
You may assume that the homologies of \(U, V, U \cap V, X\) are finitedimensional so that their Euler characteristics are well defined.
Spring 2021 #6
For each of the following spaces, compute the fundamental group and the homology groups.

The graph \(\Theta\) consisting of two edges and three vertices connecting them.

The 2dimensional cell complex \(\Theta_2\) consisting of a closed circle and three 2dimensional disks each having boundary running once around that circle.
Spring 2021 #7
Prove directly from the definition that the 0th singular homology of a nonempty pathconnected space is isomorphic to \({\mathbf{Z}}\).
Spring 2021 #9
Prove that for every continuous map \(f: S^{2n} \to S^{2n}\) there is a point \(x\in S^{2n}\) such that either \(f(x) = x\) or \(f(x) = x\).
You may use standard facts about degrees of maps of spheres, including that the antipodal map on \(S^{2n}\) has degree \(d=1\).