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 ${\mathbf{R}}^3$. 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?

• $C_1 (X) = \left\{{0}\right\}$.

• $C_1 (X) \neq \left\{{0}\right\}$ 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 finite-dimensional so that their Euler characteristics are well defined.

Spring 2021 #6

Spring 2021 #7

Spring 2021 #9