December 1958
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14 Reads
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12 Citations
Commentarii Mathematici Helvetici
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Publications (6)
January 1956
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23 Reads
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24 Citations
Commentarii Mathematici Helvetici
April 1953
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7 Reads
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50 Citations
Proceedings of the National Academy of Sciences
April 1953
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14 Reads
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44 Citations
Proceedings of the National Academy of Sciences
January 1952
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78 Reads
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1,050 Citations
The Mathematical Gazette
The need for an axiomatic treatment of homology and cohomology theory has long been felt by topologists. Professors Eilenberg and Steenrod present here for the first time an axiomatization of the complete transition from topology to algebra.
February 1951
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24 Reads
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52 Citations
Proceedings of the National Academy of Sciences
This chapter discusses vector fields on the n-sphere. Any set of 2k continuous vector fields tangent to Sn are somewhere dependent. The case k = 0 is the classical result that a vector field on a sphere of even dimension has at least one zero. The case k = 2 was based on the erroneous assertion of π5 (S3) =0. If n and k are with r > 0, then the fiber bundle h: Vn+1, 2h+1 → Sn does not have a cross-section. If n is not of the form 2k — 1, then the fiber bundle Rn+1 → Sn does not have a cross-section.
Citations (6)
... By the Arens-Royden theorem (see, e.g., [14,Theorem,p.295]), the group A´1{e A is isomorphic to the firstČech cohomology group H 1 pMpAq, Zq of MpAq with integer coefficients. For background onČech cohomology, see, e.g., [6]. For a contractible space, all cohomology groups are trivial (see, e.g., [6, IX, Theorem 3.4]). ...
- Citing Article
January 1952
The Mathematical Gazette
... Using results of Dold (5.5 of [5]), Steenrod (5.1 of [14]) and Steenrod and Thomas ([ 15]), it follows (c.f. 96 of [lo]) that any many variable cohomology operation is generated, using composition and product, by the above elementary operations. ...
- Citing Article
December 1958
Commentarii Mathematici Helvetici
... 1.1. 2 Steenrod realized early on that a more general theory of cohomology operations could be formulated using the language of homology of groups, [25]. For any group G and a contractible free G-space EG, the homology of BG is given by the homology of the coinvariant quotient complex N * (EG) G = N * (G\EG) = N * (BG), defined by setting x ≡ g x, all g ∈ G, x ∈ N * (EG). ...
- Citing Article
January 1956
Commentarii Mathematici Helvetici
... Steenrod also introduced operations on the mod p cohomology of spaces when p is an odd prime [33,34]. To define these effectively, generalization of the cup-i coproducts were introduced in [13]. ...
- Citing Article
April 1953
Proceedings of the National Academy of Sciences
... Homotopy orbit spaces with respect to the symmetric group action on iterated products, known as extended powers, play many key roles in algebraic topology. Steenrod introduced them as a central character in the of study cohomology operations [Ste53]. Extended powers of spectra play an essential role in Nishida's proof of his Nilpotence Theorem [Nis75]. ...
- Citing Article
April 1953
Proceedings of the National Academy of Sciences
... So, we take a point from C k that is the center of an n-sphere. An n-sphere (or n-hypersphere) is a topological space that is homeomorphic to a standard n-sphere [70]. It is a set of points in (n + 1)-dimensional Euclidean space situated at a constant distance r from a fixed point. ...
- Citing Article
February 1951
Proceedings of the National Academy of Sciences