querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Practice, 3 Topology, isotopy, degree theory, spheres

Let $𝑆^{𝑛} \subset ℝ^{𝑛 + 1}$ be the unit sphere. Determine the values of $𝑛 \geq 0$ for which the antipodal map $𝑆^{𝑛} \to 𝑆^{𝑛}$ , $𝑥 \mapsto - 𝑥$ is isotopic to the identity.

Stanford Practice, 3 Topology, immersions and embeddings, tori, fundamental group

Let $𝑀$ be a compact, simply connected smooth manifold of dimension $𝑛$ . Prove that there is no smooth immersion $𝑓 : 𝑀 \to 𝑇^{𝑛}$ , where $𝑇^{𝑛} = 𝑆^{1} \times \dots \times 𝑆^{1}$ is the $𝑛$ -torus.

Stanford Practice, 4 Topology, Lie derivative, Cartan's formula, volume forms

(a) State Cartan's formula for the Lie derivative of a differential form with respect to a vector field.

(b) Use this formula to show that the flow generated (locally) by a vector field $𝑋$ on $ℝ^{3}$ preserves volume if and only if the divergence of $𝑋$ is everywhere zero. (Here, divergence is the classical operator that takes a vector field with components $𝑓, 𝑔, ℎ$ to the function $𝜕_{𝑥} 𝑓 + 𝜕_{𝑦} 𝑔 + 𝜕_{𝑧} ℎ$ , where $𝑥, 𝑦, 𝑧$ are the usual coordinates on $ℝ^{3}$ .)

Stanford Practice, 4 Topology, differential forms, de Rham cohomology, Stokes' theorem

Consider the form

𝜔 = (𝑥^{2} - 𝑥 + 𝑦 𝑧) d 𝑦 \land d 𝑧

on $ℝ^{3}$ . Let $𝑆^{2} = {𝑥^{2} + 𝑦^{2} + 𝑧^{2} = 1} \subset ℝ^{3}$ be the unit sphere, and $𝑖 : 𝑆^{2} \to ℝ^{3}$ the inclusion.

(a) Calculate $\int_{𝑆^{2}} 𝜔$ .

(b) Construct a closed form $𝛼$ on $ℝ^{3}$ such that $𝑖^{*} 𝛼 = 𝑖^{*} 𝜔$ , or show that such a form $𝛼$ does not exist.

Stanford Practice, 5 Topology, covering spaces, de Rham cohomology, pullback

Let $𝑓 : 𝑋 \to 𝑌$ be a smooth, finite covering map between smooth $𝑛$ -dimensional manifolds. Show that the induced map on de Rham cohomology

𝑓^{*} : 𝐻_{dR}^{𝑘} (𝑌; ℝ) \to 𝐻_{dR}^{𝑘} (𝑋; ℝ)

is injective, for all $𝑘 \geq 0$ .

Stanford Practice, 5 Topology, Morse theory, spheres

Let $𝑀$ be a closed, smooth manifold and $𝑓 : 𝑀 \to ℝ$ a Morse function with only two critical points. Show that $𝑀$ is homeomorphic to a sphere.

Stanford Spring 2024, 1 Topology, differential forms, exterior derivative, volume forms

Let $𝑀$ be a closed $𝑛$ dimensional smooth manifold. Show that for any $𝑛 - 1$ form $𝛼$ on $𝑀$ there exists a point $𝑝 \in 𝑀$ such that ${(d 𝛼)}_{𝑝} = 0$ .

Stanford Spring 2024, 1 Topology, mapping degree, fundamental group, covering spaces

If $𝑓 : 𝑀 \to 𝑁$ is a degree one map between compact connected oriented manifolds without boundary, show that the induced map on $𝜋_{1}$ is surjective.

Stanford Spring 2024, 2 Topology, fixed point theorems, Lefschetz fixed point theorem, Euler characteristic

Let $𝑓 : 𝑋 \to 𝑋$ be a continuous function such that $𝑓 \circ 𝑓 \circ 𝑓 = {id}_{𝑋}$ but $𝑓 (𝑥) \neq 𝑥$ for all $𝑥 \in 𝑋$ .

(a) give an example of such a function $𝑓$ when $𝑋$ is the 2 dimensional torus $𝑇^{2}$ .

(b) show that such a function $𝑓$ does not exist when $𝑋$ is the 2 dimensional sphere $𝑆^{2}$ .

Stanford Spring 2024, 2 Topology, Poincaré duality, de Rham cohomology, tori

Let $𝑀 = ℝ^{2} / ℤ^{2}$ be the 2 dimensional torus, $𝐿$ the line $2 𝑥 = 5 𝑦$ in $ℝ^{2}$ , and let $𝑆$ be the image of $𝐿$ under the projection map $𝜋 : ℝ^{2} \to ℝ^{2} / ℤ^{2} = 𝑀$ . Find a differential form on $𝑀$ that represents the Poincaré dual of $𝑆$ , and show it does so.

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