Algebraic topology Homology and cohomology Concept
Singular homology
Summary
Singular homology assigns to every topological space a sequence of abelian groups , built from all continuous maps of simplices into . It is defined for every space without any choice, it is functorial, and it is invariant under homotopy equivalence. It satisfies the Eilenberg–Steenrod axioms, and on CW complexes and simplicial complexes it agrees with cellular and simplicial homology, which makes it computable.
Where this leads
42 pages build on this one.
- Eilenberg–Steenrod axioms
- Virtual fundamental class
- Singular cohomology
- Künneth theorem
- Cap product
- Poincaré duality
- Long exact sequence of a pair
- Borel–Moore homology
- Relative homology
- Degree of a continuous map
- Cohomology ring
- Cellular homology
- Lefschetz fixed-point theorem
- Universal coefficient theorem
- Fundamental class
- Alexander duality
- Morse inequalities
- Orientation of a manifold
- Bockstein spectral sequence
- Brouwer fixed-point theorem
- Intersection form
- Lefschetz duality
- Vanishing cycle
- Localization of a space
- Linking number
- Reduced homology
- Excision theorem
- Homology sphere
- Euler characteristic
- Equivariant index theorem
- Orientability
- Homotopy invariance of homology
- Eilenberg–Zilber theorem
- Betti number
- Rational homotopy theory
- G-signature theorem
- Wang sequence
- Mayer–Vietoris sequence
- Thurston norm
- Homology with coefficients
- Hurewicz theorem
- Barratt–Priddy–Quillen theorem
Further reading
About this page
| Tag | M0Q5CFJ7ZH |
|---|---|
| URI | m2://M0Q5CFJ7ZH |
| Kind | concept |
| Writing | W0 stub |
| MSC | 55N10 |
| Same as |
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What links here
Used by
- Eilenberg–Steenrod axioms
- Virtual fundamental class
- Singular cohomology
- Künneth theorem
- Cap product
- Poincaré duality
- Long exact sequence of a pair
- Borel–Moore homology
- Relative homology
- Degree of a continuous map
- Cohomology ring
- Cellular homology
- Lefschetz fixed-point theorem
- Universal coefficient theorem
- Fundamental class
- Morse inequalities
- Bockstein spectral sequence
- Vanishing cycle
- Localization of a space
- Linking number
- Reduced homology
- Excision theorem
- Homology sphere
- Euler characteristic
- Orientability
- Homotopy invariance of homology
- Eilenberg–Zilber theorem
- Betti number
- Rational homotopy theory
- Wang sequence
- Mayer–Vietoris sequence
- Thurston norm
- Homology with coefficients
- Hurewicz theorem
- Barratt–Priddy–Quillen theorem
Results about this
- Künneth theorem
- Poincaré duality
- Universal coefficient theorem
- Homotopy invariance of homology
- Eilenberg–Zilber theorem
- Wang sequence
- Mayer–Vietoris sequence
- Hurewicz theorem