1. Singular homology by cubes

2. Cubical sets: limits, colimits, tensor product and internal homs

3. Directed homotopy of cubical sets

4. Directed homology of cubical sets

5. Action of groups on cubical sets

6. Noncommutative tori, Kronecker foliations and cubical sets

7. Metric aspects by normed cubical sets

8. Similar models by equilogical and inequilogical spaces.

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1. Singular homology by cubes [Ms]

1.1. The singular cubical set of a space

1.2. The singular chain complex of a space

1.3. Homology

1.4. Elementary results

1.5. Homotopy for topological spaces

1.6. Homotopy for chain complexes of abelian groups

1.7. Homotopy Invariance of singular homology

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2. Cubical sets [G1, Section1]

2.1. Remarks on Directed Algebraic Topology

2.2. Cubical sets

2.3. Subobjects and quotients

2.4. Tensor product of cubical sets

2.5. Standard models

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3. Directed homotopy of cubical sets [G1, Section 1]

3.1. Elementary directed homotopies

3.2. Cones and suspension

3.3. Geometric realisation

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4. Directed homology of cubical sets [G1, Section 2]

4.1. Directed homology

4.2. Elementary computations

4.3. Invariance Theorem

4.4. Mayer-Vietoris and Excision

4.5. Theorem (Tensor products)

4.6. Elementary cubical tori

4.7. Theorem (Homology of suspension)

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5. Action of groups on cubical sets [G1, Section 3]

5.1. Basics

5.2. Lemma (Free actions)

5.3. Theorem (Free actions on acyclic cubical sets)

5.4. Corollary (Free actions on acyclic spaces)

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6. Noncommutative tori, Kronecker foliations and cubical sets [G1, Section 4]

6.1. Rotation algebras

6.2. Irrational rotation structures

6.3. The noncommutative two-dimensional torus

6.4. Higher foliations of codimension 1

6.5. Remarks

6.6. Lemma

6.7. Theorem (Directed homology of the irrational rotation cubical sets)

6.8. Classification Theorem (For the cubical sets of irrational rotation)

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7. Metric aspects by normed cubical sets [G2]

7.1. Normed cubical sets

7.2. Elementary models

7.3. Normed circles and irrational rotation structures

7.4. Normed abelian groups and chain complexes

7.5. Normed directed homology

7.6. Normed homology of circles and tori

7.7. Theorem (Normed homology of the normed cubical sets of irrational rotation)

7.8. Classification Theorem (For the normed cubical sets of irrational rotation)

7.9. An extension

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8. Similar models by equilogical and inequilogical spaces [G3, G4]

8.1. Equilogical spaces

8.2. Limits

8.3. Equilogical circles and spheres

8.4. Local maps

8.5. Singular homology of equilogical spaces

8.6. Actions of groups

8.7. Equilogical spaces and irrational rotations

8.8. Inequilogical spaces

8.9. Directed homology of inequilogical spaces

8.10. Classification of the inequilogical spaces of irrational rotation

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[G1] M. Grandis, Directed combinatorial homology and noncommutative tori, Math. Proc. Cambridge Philos. Soc., to appear. [Dip. Mat. Univ. Genova, Preprint 480 (2003), 29 p.]

[G2] M. Grandis, Normed combinatorial homology and noncommutative tori, Dip. Mat. Univ. Genova, Preprint 484 (2003), 14 p.

[G3] M. Grandis, Equilogical spaces, homology and noncommutative geometry, Dip. Mat. Univ. Genova, Preprint 493 (2003), 24 p.

[G4] M. Grandis, Inequilogical spaces, directed homology and noncommutative geometry, Dip. Mat. Univ. Genova, Preprint 494 (2004), 22 p.

[G5] M. Grandis, Directed combinatorial homology and noncommutative geometry (course notes).

[Ms] W. Massey, Singular homology theory, Springer, Berlin 1980.