EXPOSITIO

A home for expository scholarship

Expositio is a repository for expository papers in a wide range of academic disciplines, supporting preprints, revisions, optional peer review, saved papers, and overlay journals.

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What is Expositio?

A public repository for expository academic papers, with an integrated scholarly record.

Expositio is a place to read and cite expository work. Papers are published as stable, citable records, while remaining connected to later versions, editorial decisions, and related context.

Some papers are early drafts; others have been peer reviewed or formally published. Each paper remains readable on its own, while preserving the history behind it.

The result is both a reading room and a record: a place to browse by field, read papers in stable form, and follow their development over time.

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Algebraic topology on manifolds and differential topology

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Mathematics Manifolds and cell complexes Differential topology Algebraic topology on manifolds and differential topology
Domain: Mathematics × Branch: Manifolds and cell complexes × Branch: Differential topology × Branch: Algebraic topology on manifolds and differential topology ×

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Expositio paper

Poincaré-Hopf Theorem and Its Applications

Lucy Zhang & Ruoli Bai

Preprint
v1

This paper builds up and proves the Poincaré-Hopf Theorem for compact smooth manifolds. Beginning with the introduction of key concepts in differential topology, including smooth manifolds, coordinate charts, tangent spaces, and mappings, we prove foundational theorems like the …

Mathematics / Manifolds and cell complexes
Mathematics > Manifolds and cell complexes > Algebraic topology on manifolds and differential topology Primary