Homology and Degree Theory

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

  • State the Mayer-Vietoris 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 \({\mathbb{R}}^3\). Let \(X\) be the subset of the upper half-space of \({\mathbb{R}}^3\) 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 \({\mathbb{RP}}^2 # {\mathbb{RP}}^2\).

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 \(X \subset {\mathbb{R}}^3\) 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 = {\mathbb{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 \({\mathbb{R}}^2\) descends via the quotient \({\mathbb{R}}^2 \to T^2\) 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{\mathbb{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?

  • \(C_1 (X) = \left\{{0}\right\}\).

  • \(C_1 (X) \neq \left\{{0}\right\}\) 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 \({\mathbb{RP}}^n\).
  • Let \(\pi : {\mathbb{RP}}^n \to X\) 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.

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

  • 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 \(d=-1\).

#topology/qual/work #6 #7 #9