# 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

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