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# Overview article for algebraic topology

# 1 An Overview of Algebraic Topology topics

# 1.1 Introduction

*Algebraic topology* (AT) utilizes algebraic approaches to solve topological problems,
such as the classification of surfaces, proving duality theorems for manifolds and
approximation theorems for topological spaces. A central problem in algebraic topology
is to find algebraic invariants of topological spaces, which is usually carried out by means
of homotopy, homology and cohomology groups. There are close connections between algebraic topology,
Algebraic Geometry (AG), and Non-commutative Geometry/NAAT. On the other hand, there are also close ties between algebraic geometry and number theory.

# 1.2 Outline

1. Homotopy theory and fundamental groups

2. 3. Homology and cohomology theories

4. Duality

5. Category theory applications in algebraic topology

6. 7. 8. 9. 10. 11. Quantum algebraic topology (QAT)

12. 13. Non-Abelian algebraic topology (NAAT)

# 1.3 Homotopy theory and fundamental groups

1. Homotopy

2. Fundamental group of a space

3. Fundamental theorems

4. 5. 6. Postnikov towers

# 1.4 Topology and Groupoids

1. 2. 3. 4. 5. van Kampen theorem for groupoids

6. Groupoid pushout theorem

7. Double groupoids and crossed modules

8. new4

# 1.5 Homology theory

# 1.6 Cohomology theory

1. Cohomology group

2. Cohomology sequence

3. DeRham cohomology

4. new4

# 1.7 Non-Abelian Algebraic Topology

1. Crossed Complexes

2. Modules

3. Cross-modules

4. Omega-Groupoids

5. Double Groupoids: Homotopy Double Groupoid of a Hausdorff Space

6. 7. 8. Algebroids

9. Higher Homotopy van Kampen Theorem

*…more to come*

## Mathematics Subject Classification

57R19*no label found*57N65

*no label found*11F23

*no label found*11E72

*no label found*18-00

*no label found*55N30

*no label found*55N15

*no label found*55N99

*no label found*55N40

*no label found*55N20

*no label found*55-01

*no label found*

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