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Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2025, 3 Topology, principal bundles, classifying spaces, homotopy groups of spheres

Show that any principal $SU (2)$ -bundle over the sphere $𝑆^{2}$ is trivial.

Stanford Fall 2025, 3 Topology, distributions, Frobenius theorem, integrability

Consider the rank 2 distribution on $ℝ^{3}$ spanned by the vector fields

𝑋_{1} = \frac{𝜕}{𝜕} 𝑥 and 𝑋_{2} = \frac{𝜕}{𝜕} 𝑦 + 𝑥 \frac{𝜕}{𝜕} 𝑧 .

Determine whether or not this distribution is integrable.

Stanford Fall 2025, 4 Topology, Poincaré-Hopf theorem, Euler characteristic, vector fields

Use the Poincaré-Hopf index theorem to calculate the Euler characteristic of the $𝑛$ -dimensional sphere $𝑆^{𝑛}$ . (You must compute the indices in local coordinates. Drawings do not suffice.)

Stanford Fall 2025, 4 Topology, fiber bundles, Euler class, Gysin sequence, homology

Let $𝜋 : 𝐸 \to Σ_{𝑔}$ be an oriented $𝑆^{1}$ -bundle over a closed oriented surface of genus $𝑔$ , with Euler class $𝑒 = 𝑘 \cdot [Σ_{𝑔}] \in 𝐻^{2} (Σ_{𝑔}; ℤ) ≅ ℤ$ .

(a) Compute the homology groups $𝐻_{𝑖} (𝐸; ℤ)$ .

(b) For what values of $𝑘$ does $𝐸$ admit a section?

Stanford Fall 2025, 5 Topology, smooth manifolds, Morse theory, CW complexes

Show that every connected smooth manifold admits a contractible dense open subset.

Stanford Fall 2025, 5 Topology, fixed point theorems, complex projective space, Lefschetz fixed point theorem

For what values of $𝑛$ does there exist a continuous map $𝑓 : ℂ ℙ^{𝑛} \to ℂ ℙ^{𝑛}$ with no fixed points?

Stanford Practice, 1 Topology, suspension, homology, CW complexes

Let $𝑋$ be a topological space. Define the suspension $𝑆 (𝑋)$ to be the space obtained from $𝑋 \times [0, 1]$ by contracting $𝑋 \times {0}$ to a point, and contracting $𝑋 \times {1}$ to another point. Describe the relation between the homology groups of $𝑋$ and $𝑆 (𝑋)$ .

Stanford Practice, 1 Topology, CW complexes, cellular homology, classification of surfaces

For $𝑛 \geq 2$ , let $𝑋_{𝑛}$ be the space obtained from a regular $(2 𝑛)$ -gon by identifying the opposite sides with parallel orientations. This description produces a cell decomposition of $𝑋_{𝑛}$ .

(a) Write down the associated cellular chain complex.

(b) Show that $𝑋_{𝑛}$ is a surface, and find its genus.

Stanford Practice, 2 Topology, connected sum, covering spaces, real projective space

(a) Consider the space $𝑌$ obtained from $𝑆^{2} \times [0, 1]$ by identifying $(𝑥, 0)$ with $(- 𝑥, 0)$ and also identifying $(𝑥, 1)$ with $(- 𝑥, 1)$ , for all $𝑥 \in 𝑆^{2}$ . Show that $𝑌$ is homeomorphic to the connected sum $ℝ ℙ^{3} #︎ ℝ ℙ^{3}$ .

(b) Show that $𝑆^{2} \times 𝑆^{1}$ is a double cover of the connected sum $ℝ ℙ^{3} #︎ ℝ ℙ^{3}$ .

Stanford Practice, 2 Topology, Poincaré-Lefschetz duality, torsion, homology

Let $𝑀$ be an $𝑚$ -dimensional manifold ( $𝑚 \geq 3$ ), compact, simply connected, and with non-empty boundary. Show that the homology groups $𝐻_{𝑚 - 1} (𝑀; ℤ)$ and $𝐻_{𝑚 - 2} (𝑀; ℤ)$ are torsion free.

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