querent

Spam yourself with math PhD qualifying-exam questions.

Browse Practice

Stanford Spring 2026, 3 Topology, fixed point theorems, Lefschetz fixed point theorem, real projective space

Let $𝑓 : ℝ ℙ^{𝑛} \to ℝ ℙ^{𝑛}$ be a smooth map.

(a) When $𝑛$ is even show that $𝑓$ has a fixed point.

(b) When $𝑛$ is odd give an example when $𝑓$ does not have any fixed points.

Stanford Spring 2026, 3 Topology, manifolds with boundary, retractions, homology

Let $𝑀$ be a compact manifold with boundary $𝜕 𝑀$ . Prove that there is no retraction $𝑟 : 𝑀 \to 𝜕 𝑀$ .

Stanford Spring 2026, 4 Topology, Euler characteristic, manifolds with boundary, Poincaré duality

Let $𝑀$ be a compact topological manifold of dimension $2 𝑛 + 1$ . Prove that

(a) if $𝜕 𝑀 = \emptyset$ then $𝜒 (𝑀) = 0$ ;

(b) if $𝜕 𝑀 \neq \emptyset$ then $𝜒 (𝜕 𝑀)$ is even.

Stanford Spring 2026, 4 Topology, Lie derivative, volume forms, Stokes' theorem

Let $𝑀$ be a closed orientable smooth manifold, and let $𝜔$ be a volume form on $𝑀$ . Prove that there is no vector field $𝑋$ on $𝑀$ such that ${ℒ︀}_{𝑋} 𝜔 = 𝜔$ .

Stanford Spring 2026, 5 Topology, embeddings, Stiefel-Whitney classes, real projective space

Prove that there is no smooth embedding of $ℝ ℙ^{3}$ into $ℝ^{4}$ .

Stanford Spring 2026, 5 Topology, Poincaré-Alexander duality, submanifolds, separation

Let $𝑀$ be an $𝑛$ -dimensional smooth connected manifold with $𝐻_{𝑛 - 1} (𝑀; ℤ) = 0$ . Let $\emptyset \neq 𝑋 \subset 𝑀$ be a smooth connected orientable closed codimension 1 submanifold.

(a) Show that $𝑋$ separates $𝑀$ into exactly 2 connected components.

(b) Show that $𝑋$ is the boundary of an open connected subset of $𝑀$ with compact closure.

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