# THE GEOMETRIC REALIZATION OF A SEMI-SIMPLICIAL COMPLEX

@article{Milnor1957THEGR, title={THE GEOMETRIC REALIZATION OF A SEMI-SIMPLICIAL COMPLEX}, author={J. Milnor}, journal={Annals of Mathematics}, year={1957}, volume={65}, pages={357} }

homology and homotopy groups. The terminology for semi-simplicial complexes will follow John Moore [7]. In particular the face and degeneracy maps of K will be denoted by d: Kn Kn-1 and si:K. -> Kn+1 respectively. 1. The definition

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#### References

SHOWING 1-10 OF 13 REFERENCES

ON THE EQUIVALENCE OF TWO SINGULAR HOMOLOGY THEORIES

- Mathematics
- 1950

Eilenberg in 1944 ([E])2 removed some of the difficulties associated with the classical concept of "singular cell" and obtained a new singular homology theory. In 1948, Hurewicz, Dugundji, and Dowker… Expand

Construction of Universal Bundles, II

- Mathematics
- 1956

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and… Expand

Construction of universal bundles I , Ann. of hlath

- Construction of universal bundles I , Ann. of hlath
- 1956

From Lemmas 5 and G it can be proved, using a relative Hurewica theorem, that the honlomorphisms are isomorphisms for all n . (The proof of the relative EIuren-icz theorem

- Ann. of hlath,
- 1950

ZILBER, Semi-simplicial complexes and singular homology, Ann

- Ann. of hlath.,
- 1950

Math. Soc

- Math. Soc
- 1949