querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2024, 3 Topology, transversality, cohomology, intersection theory

Let $𝑀$ be a compact smooth manifold with boundary, and let $𝑁_{\pm} \subset 𝑀$ be a pair of smooth closed submanifolds, which intersect transversely at one point. Show that $𝐻^{𝑘} (𝑀; ℤ / 2)$ must be non-trivial in degrees $𝑘 \in {dim (𝑁_{-}), dim (𝑁_{+})}$ . Show by example that this may fail if the submanifolds are instead assumed to have boundary.

Stanford Fall 2024, 3 Topology, CW complexes, universal coefficient theorem, torsion

Let $𝑋$ be a finite CW complex.

(a) Show that the dimension of $𝐻^{*} (𝑋; 𝔽_{𝑝})$ agrees with the dimension of $𝐻^{*} (𝑋; ℚ)$ for all sufficiently large primes $𝑝$ .

(b) Give an example where the dimensions of $𝐻^{*} (𝑋; 𝔽_{𝑝})$ and $𝐻^{*} (𝑋; ℚ)$ are different for some prime $𝑝 \neq 2$ .

Stanford Fall 2024, 4 Topology, differential forms, de Rham cohomology, fundamental group

Let $𝜔$ be a 1-form on a connected smooth manifold $𝑀$ . Show that $𝜔$ is exact if and only if for all piecewise smooth closed curves $𝑐 : 𝑆^{1} \to 𝑀$ we have $\int_{𝑐} 𝜔 = 0$ .

Stanford Fall 2024, 4 Topology, vector bundles, line bundles

Let $𝑀$ be a smooth manifold. Show that every real line bundle is trivial in the complement of a smooth submanifold of strictly positive codimension.

Stanford Fall 2024, 5 Topology, fixed point theorems, involutions, Lefschetz fixed point theorem

Let $𝑋$ be a compact manifold with boundary $𝑆^{𝑛}$ for $𝑛 \geq 0$ . An involution on $𝑋$ is a continuous map $𝑓 : 𝑋 \to 𝑋$ with $𝑓 \circ 𝑓 = {id}_{𝑋}$ . Show that there is no fixed-point free involution on $𝑋$ that restricts to the antipodal map on $𝜕 𝑋 = 𝑆^{𝑛}$ .

Stanford Fall 2024, 5 Topology, CW complexes, cellular homology, homology

Let $𝐴$ be a CW complex, and $𝑛$ a positive integer. Assuming that every homology class in degree $𝑑$ greater than or equal to $𝑛$ is in the image of $𝐻_{𝑑} (𝑆^{𝑑}; ℤ)$ for some map $𝑆^{𝑑} \to 𝐴$ , show that there is a CW complex $𝑋$ , and a cellular map $𝐴 \to 𝑋$ so that the inclusion map $𝐻_{𝑖} (𝐴; ℤ) \to 𝐻_{𝑖} (𝑋; ℤ)$ is an isomorphism whenever $𝑖 < 𝑛$ , and vanishes otherwise.

Stanford Fall 2025, 1 Topology, de Rham cohomology, closed and exact forms, fundamental group

Consider the 1-form $𝜔 = \frac{- 𝑦}{𝑥^{2} + 𝑦^{2}} d 𝑥 + \frac{𝑥}{𝑥^{2} + 𝑦^{2}} d 𝑦$ on the manifold $𝑀 = ℝ^{2} - {(0, 0)}$ .

(a) Show that $𝜔$ is a closed form.

(b) By integrating over a suitable path, show that $𝜔$ is not exact on $𝑀$ .

Stanford Fall 2025, 1 Topology, immersions and embeddings, submersions, smooth manifolds

Let $Γ \subset ℝ^{2}$ be the graph of the function $𝑦 = | 𝑥 |$ .

(a) Construct a smooth function $𝑓 : ℝ \to ℝ^{2}$ whose image is $Γ$ . Can $𝑓$ be an immersion?

(b) Construct a smooth function $𝑓 : ℝ^{2} \to ℝ$ such that $𝑓^{- 1} (0) = Γ$ . Can $𝑓$ be a submersion?

Stanford Fall 2025, 2 Topology, cohomology ring, connected sum, complex projective space

Let $𝑀 = ℂ ℙ^{2} #︎ ℂ ℙ^{2}$ be the connected sum of two copies of the complex projective plane. Provide a presentation of the integer cohomology ring $𝐻^{*} (𝑀; ℤ)$ .

Stanford Fall 2025, 2 Topology, covering spaces, fundamental group, deck transformations, free groups

Let $𝑋 = 𝑆^{1} \lor 𝑆^{1}$ be the wedge of two circles, with basepoint $𝑥_{0}$ at the wedge point. Let $𝑝 : \tilde{𝑋} \to 𝑋$ be the connected covering space corresponding to the commutator subgroup $[𝜋_{1} (𝑋, 𝑥_{0}), 𝜋_{1} (𝑋, 𝑥_{0})]$ .

(a) Describe the group of deck transformations of this cover.

(b) Draw a picture of the covering space $\tilde{𝑋}$ .

(c) Compute the fundamental group $𝜋_{1} (\tilde{𝑋})$ and the homology groups $𝐻_{𝑘} (\tilde{𝑋}; ℤ)$ for all $𝑘 \geq 0$ .

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