We will present proofs for two conjectures stated in Rot (Homotopy classes of proper maps out of vector bundles, 2020. arXiv:1808.08073 ). The first one is that for an arbitrary manifold W , the homotopy classes of proper maps $$W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}$$ W × R n → R k + n stabilise as $$n\rightarrow \infty $$ n → ∞ , and the second one is that in a stable range there is a Pontryagin–Thom type bijection for proper maps $$W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}$$ W × R n → R k + n . The second one actually implies the first one and we shall prove the second one by giving an explicit construction.

中文翻译：

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用于正确映射的稳定 Pontryagin-Thom 构造

我们将证明 Rot 中陈述的两个猜想（向量丛中的真映射的同伦类，2020. arXiv:1808.08073）。第一个是对于任意流形 W ，真映射的同伦类 $$W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}$$ W × R n → R k + n 稳定为 $$n\rightarrow \infty $$ n → ∞ ，第二个是在一个稳定的范围内，对于真映射 $$W\times \mathbb {R}^ 存在 Pontryagin–Thom 类型的双射n\rightarrow \mathbb {R}^{k+n}$$ W × R n → R k + n 。第二个实际上暗示了第一个，我们将通过给出一个明确的构造来证明第二个。