The Lumenary · First-year graduate

Algebraic topology

Name: ______________________

Date: ______________________

You will learn to

Define singular chains, homology and chain homotopies, and prove that $\partial \circ \partial = 0$ and that chain-homotopic maps induce the same map on homology
Prove that homology is a homotopy-invariant functor and deduce that $\R^m$ and $\R^n$ are not homeomorphic for $m \neq n$ , and the Brouwer fixed-point theorem
Compute the homology of spheres, projective spaces and surfaces with Mayer–Vietoris and cellular homology, and relate it to the Euler characteristic
Compute cohomology rings of simple spaces and use cup products to distinguish $\C P^2$ from $S^2 \vee S^4$ , which have isomorphic homology
State and apply the Lefschetz fixed-point theorem

1.
What is the rank of $H_1(M_3)$ , where $M_3$ is the closed orientable surface of genus $3$ ?
2.
Which list gives $H_0, H_1, H_2, H_3$ of $\R P^3$ (integer coefficients)?
- A. $\Z,\ \Z/2,\ 0,\ \Z$
- B. $\Z,\ \Z/2,\ \Z/2,\ \Z/2$
- C. $\Z,\ \Z/2,\ 0,\ 0$
- D. $\Z,\ 0,\ 0,\ \Z$
3.
We try to compute $H_\bullet(S^n)$ with Mayer–Vietoris using the two closed hemispheres $U$ and $V$ . Which hypothesis of Theorem 5, as stated in the lesson, fails?
- A.The interiors of $U$ and $V$ do not cover $S^n$ .
- B. $U \cap V$ is not path connected.
- C. $U$ and $V$ are not contractible.
- D.Singular homology is only defined for open subsets.
4.
In the proof that $\R^m \not\cong \R^n$ for $m \neq n$ , why do we remove a point before taking homology?
- A.Because $\R^m$ and $\R^n$ are contractible, so their homology is the same and cannot tell them apart; removing a point leaves spaces homotopy equivalent to $S^{m-1}$ and $S^{n-1}$ , whose homology differs.
- B.Because homology is only defined for compact spaces.
- C.Because a homeomorphism must send $0$ to $0$ .
- D.To make the spaces simply connected.
5.
For which pairs do the homology groups alone prove that the two spaces are not homotopy equivalent? (Select all that apply.)
- A.Torus $T^2$ and Klein bottle $K$
- B. $\C P^2$ and $S^2 \vee S^4$
- C. $T^2$ and $S^1 \vee S^1 \vee S^2$
- D. $\R P^2$ and a point
- E. $S^3$ and Poincaré's homology sphere
6.
With H : ChainHomotopy φ ψ, which Lean statement correctly formalises chain-homotopic maps induce the same map on homology at the level of elements?
- A.∀ n z, IsCycle K n z → IsBoundary L n (φ.f n z - ψ.f n z)
- B.∀ n z, φ.f n z = ψ.f n z
- C.∀ n z, IsBoundary L n (φ.f n z - ψ.f n z)
- D.∀ n z, IsCycle K n z → IsCycle L n (φ.f n z - ψ.f n z)
7.
A map $f : \C P^2 \to \C P^2$ satisfies $f^*(x) = 3x$ on $H^2(\C P^2; \Z) = \Z x$ . What is its Lefschetz number $L(f)$ ?
8.
Write euler_characteristic(triangles) that takes a list of triangles (3-tuples of vertex labels) forming a $2$ -dimensional simplicial complex and returns $V - E + F$ , where $V$ counts distinct vertices, $E$ distinct edges (unordered pairs of vertices of a triangle) and $F$ triangles.
Computational — Practice

Answer key — Algebraic topology

1.
In the cellular chain complex $\Z \to \Z^{6} \to \Z$ both differentials vanish, so $H_1(M_3) \cong \Z^6$ and the rank is $6$ . Check: $\chi = 1 - 6 + 1 = -4 = 2 - 2g$ .
2.
With $d_k = 1 + (-1)^k$ : $H_0 = \Z$ , $H_1 = \Z / \operatorname{im}(d_2) = \Z/2$ , $H_2 = \ker(d_2) / \operatorname{im}(d_3) = 0$ , $H_3 = \ker d_3 = \Z$ .
3.
Theorem 5 requires $\operatorname{int} U \cup \operatorname{int} V = X$ . The interiors of the closed hemispheres are the open hemispheres, whose union misses the equator. The fix is to use slightly larger open sets, as in the proof of Theorem 6.
4.
$\tilde H_\bullet(\R^n) = 0$ for every $n$ , so homology alone cannot distinguish them. A homeomorphism $h$ restricts to $\R^m \setminus \{0\} \cong \R^n \setminus \{h(0)\}$ , and these are homotopy equivalent to $S^{m-1}$ and $S^{n-1}$ , whose reduced homology is concentrated in degrees $m - 1$ and $n - 1$ .
5.
Homology distinguishes $T^2$ from $K$ (torsion) and $\R P^2$ from a point. The other three pairs have isomorphic homology: $\C P^2$ and $S^2 \vee S^4$ , and $T^2$ and $S^1 \vee S^1 \vee S^2$ , are separated by cup products; $S^3$ and the homology sphere by $\pi_1$ .
6.
The correct statement is (a), which is Lumenary.AlgebraicTopology.homotopic_maps_agree_on_homology. For a cycle $z$ , $f(z) - g(z) = \partial h(z) + h(\partial z) = \partial h(z)$ is a boundary.
7.
$f^* = 1$ on $H^0$ , $3$ on $H^2$ , and $f^*(x^2) = (3x)^2 = 9x^2$ on $H^4$ . So $L(f) = 1 + 3 + 9 = 13 \neq 0$ , and $f$ has a fixed point. In general $L = 1 + m + m^2$ , which is never zero.
8.
```
def euler_characteristic(triangles):
    vertices = {v for t in triangles for v in t}
    edges = set()
    for a, b, c in triangles:
        edges |= {frozenset((a, b)), frozenset((b, c)), frozenset((a, c))}
    return len(vertices) - len(edges) + len(triangles)
```
The tetrahedron gives $4 - 6 + 4 = 2$ , the six-vertex $\R P^2$ gives $6 - 15 + 10 = 1$ , the torus $9 - 27 + 18 = 0$ . By Theorem 10 these numbers are $b_0 - b_1 + b_2$ , independent of the triangulation.