Practice: Homotopy theory

Homotopy theory

Warm-upHuman

For which $k$ is $\pi_k(S^3) = 0$ ? (Select all that apply.)

Choose all that apply

Homotopy theory

CoreHuman

Which list gives $\pi_1, \pi_2, \pi_3$ of $\R P^3$ ?

Choose one

\Z/2,\ 0,\ \Z

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

\Z,\ 0,\ \Z

\Z/2,\ 0,\ 0

Homotopy theory

CoreHuman

The quaternionic Hopf fibration $S^3 \to S^7 \to S^4$ has null-homotopic fibre inclusion (since $\pi_3(S^7) = 0$ ), and this splits its long exact sequence: $\pi_n(S^4) \cong \pi_n(S^7) \oplus \pi_{n-1}(S^3)$ . Given $\pi_6(S^3) \cong \Z/12$ and $\pi_7(S^7) \cong \Z$ , what is the order of the torsion subgroup of $\pi_7(S^4)$ ?

Homotopy theory

CoreHuman

The Eckmann–Hilton argument proves that $\pi_n(X)$ is abelian for $n \ge 2$ . Why does it not apply to $\pi_1(X)$ ?

Choose one

Loops have only one coordinate, so there is only one way to concatenate them and no second operation to interchange with.The constant loop is not an identity element for concatenation.Homotopy classes of loops cannot be concatenated in a well-defined way.The interchange law fails for loops because the unit interval is not a cube.

Homotopy theory

CoreHuman

Which example shows that the CW hypothesis in Whitehead's theorem cannot be dropped?

Choose one

The map from the Warsaw circle to a pointThe map from

\R

to a pointThe degree-two map

S^1 \to S^1

The inclusion

S^1 \hookrightarrow D^2

Homotopy theory

StretchHuman

Let $\Sigma$ be Poincaré's homology sphere and $c : \Sigma \to S^3$ a degree-one map. Which statement is correct?

Choose one

c

induces isomorphisms on all homology groups, but it is not a homotopy equivalence, because

\pi_1(\Sigma) \neq 0

; so the homology Whitehead theorem really needs simple connectivity.

c

is a homotopy equivalence by the homology Whitehead theorem.

c

cannot induce an isomorphism on

H_3

, since

\Sigma \not\cong S^3

.

c

induces isomorphisms on all homotopy groups.

Homotopy theory

CoreFormal

Let S3 z w abbreviate Complex.normSq z + Complex.normSq w = 1. Which Lean statement says that the fibre of the Hopf map through a point of $S^3$ is exactly the circle through it?

Choose one

S3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → ∃ l : ℂ, Complex.normSq l = 1 ∧ z' = l * z ∧ w' = l * w∀ l : ℂ, Complex.normSq l = 1 → hopf (l * z, l * w) = hopf (z, w)S3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → ∃ t : ℝ, z' = t * z ∧ w' = t * wS3 z w → S3 z' w' → hopf (z, w) = hopf (z', w') → z = z' ∧ w = w'

Homotopy theory

CoreComputational

Write hopf(z, w) returning the triple (x, y, t) of real numbers with $x + iy = 2 z \bar w$ and $t = |z|^2 - |w|^2$ . The tests check that points of $S^3$ go to $S^2$ , that the map is constant on circles $\lambda(z, w)$ , and two special values.

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