1 (Spring ’09) #topology/qual/work
Compute the homology of the one-point union of S^1 \times S^1 and S^1.
2 (Fall ’06) #topology/qual/work
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State the Mayer-Vietoris theorem.
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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 half-space 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 Mayer-Vietoris 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 self-contained proof that the zeroth homology H_0 (X) is isomorphic to {\mathbf{Z}} for every path-connected 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?
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C_1 (X) = \left\{{0}\right\}.
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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 nth 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 finite-dimensional 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.
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The graph \Theta consisting of two edges and three vertices connecting them.
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The 2-dimensional cell complex \Theta_2 consisting of a closed circle and three 2-dimensional 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 path-connected 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.