CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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A tangent intersection whose set-theoretic intersection is not of the expected dimension

Statement refuted

False claim: even without transversality, tangent intersections still have the expected dimension.

Facts & Assumptions

Given: In $R^{2}$ , the embedded submanifolds $S = T = R \times {0}$ .

[L1]

Intersecting submanifolds need not be transverse (Intersecting submanifolds need not be transverse).

Counterexample

technique · direct

1.1

The intersection $S \cap T$ is the whole $x$ -axis, so it is $1$ -dimensional.

given

2.1

Each of $S$ and $T$ has codimension $1$ in $R^{2}$ , so a transverse intersection would have expected codimension $2$ and thus expected dimension $0$ . Step 1.1 shows the actual intersection dimension is larger. This is precisely the nontransverse situation highlighted in [L1].

L1step 1.1algebra

3.1

Therefore tangent intersections need not have the expected set-theoretic dimension.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)