Practice: Algebraic topology

Algebraic topology

Warm-upHuman

What is the rank of $H_1(M_3)$ , where $M_3$ is the closed orientable surface of genus $3$ ?

Algebraic topology

CoreHuman

Which list gives $H_0, H_1, H_2, H_3$ of $\R P^3$ (integer coefficients)?

Choose one

\Z,\ \Z/2,\ 0,\ \Z

\Z,\ \Z/2,\ \Z/2,\ \Z/2

\Z,\ \Z/2,\ 0,\ 0

\Z,\ 0,\ 0,\ \Z

Algebraic topology

CoreHuman

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?

Choose one

The interiors of

U

and

V

do not cover

S^n

.

U \cap V

is not path connected.

U

and

V

are not contractible.Singular homology is only defined for open subsets.

Algebraic topology

CoreHuman

In the proof that $\R^m \not\cong \R^n$ for $m \neq n$ , why do we remove a point before taking homology?

Choose one

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.Because homology is only defined for compact spaces.Because a homeomorphism must send

0

to

0

.To make the spaces simply connected.

Algebraic topology

StretchHuman

For which pairs do the homology groups alone prove that the two spaces are not homotopy equivalent? (Select all that apply.)

Choose all that apply

Torus

T^2

and Klein bottle

K

\C P^2

and

S^2 \vee S^4

T^2

and

S^1 \vee S^1 \vee S^2

\R P^2

and a point

S^3

and Poincaré's homology sphere

Algebraic topology

CoreFormal

With H : ChainHomotopy φ ψ, which Lean statement correctly formalises chain-homotopic maps induce the same map on homology at the level of elements?

Choose one

∀ n z, IsCycle K n z → IsBoundary L n (φ.f n z - ψ.f n z)∀ n z, φ.f n z = ψ.f n z∀ n z, IsBoundary L n (φ.f n z - ψ.f n z)∀ n z, IsCycle K n z → IsCycle L n (φ.f n z - ψ.f n z)

Algebraic topology

StretchHuman

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 $L(f)$ ?

Algebraic topology

CoreComputational

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.

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