Cochain level May-Steenrod operations. (English) Zbl 1487.55026
In this paper, the authors provide effective constructions for Steenrod operations at the cochain level, based on May’s approach [J. P. May, Lect. Notes Math. 168, 153–231 (1970; Zbl 0242.55023)] via May-Steenrod structures (in the terminology of this paper).
The authors fix a commutative ring \(R\) (usually \(\mathbb{Z}\) or \(\mathbb{F}_p\), for \(p\) a prime) and work in chain complexes over \(R\). They choose an \(E_\infty\)-operad \(\mathcal{R}\) (in particular, \(\mathcal{R} (0) = R\) and, for \(t \geq 0\), \(\mathcal{R}(t)\) is a free \(R[S_t]\)-resolution of the trivial \(S_t\)-module \(R\), where \(S_t\) denotes the symmetric group); \(\mathcal{W}(t)\) denotes the usual ‘minimal’ free \(C_t\)-resolution of the trivial \(C_t\)-module \(R\), where \(C_t \subset S_t\) is the cyclic group of order \(t\).
As the authors recall, the key step in constructing Steenrod operations via \(\mathcal{R}\) is to give a May-Steenrod structure. This boils down to specifying, for each \(t \geq 0\), a quasi-isomorphism \[ \mathcal{W} (t) \stackrel{\simeq}{\rightarrow} \mathcal{R} (t) \] of complexes over \(R[C_t]\), where the codomain is given the restricted structure via \(C_t \subset S_t\).
The main contribution of the paper is to exhibit such structures on some standard combinatorial \(E_\infty\)-operads, including the Barratt-Eccles operad and the surjection operad.
The results are applied to construct a natural May-Steenrod structure on the normalized cochains of any simplicial (respectively cubical) set. This can be implemented to give the effective computation of Steenrod operations.
The authors fix a commutative ring \(R\) (usually \(\mathbb{Z}\) or \(\mathbb{F}_p\), for \(p\) a prime) and work in chain complexes over \(R\). They choose an \(E_\infty\)-operad \(\mathcal{R}\) (in particular, \(\mathcal{R} (0) = R\) and, for \(t \geq 0\), \(\mathcal{R}(t)\) is a free \(R[S_t]\)-resolution of the trivial \(S_t\)-module \(R\), where \(S_t\) denotes the symmetric group); \(\mathcal{W}(t)\) denotes the usual ‘minimal’ free \(C_t\)-resolution of the trivial \(C_t\)-module \(R\), where \(C_t \subset S_t\) is the cyclic group of order \(t\).
As the authors recall, the key step in constructing Steenrod operations via \(\mathcal{R}\) is to give a May-Steenrod structure. This boils down to specifying, for each \(t \geq 0\), a quasi-isomorphism \[ \mathcal{W} (t) \stackrel{\simeq}{\rightarrow} \mathcal{R} (t) \] of complexes over \(R[C_t]\), where the codomain is given the restricted structure via \(C_t \subset S_t\).
The main contribution of the paper is to exhibit such structures on some standard combinatorial \(E_\infty\)-operads, including the Barratt-Eccles operad and the surjection operad.
The results are applied to construct a natural May-Steenrod structure on the normalized cochains of any simplicial (respectively cubical) set. This can be implemented to give the effective computation of Steenrod operations.
Reviewer: Geoffrey Powell (Angers)
MSC:
| 55S05 | Primary cohomology operations in algebraic topology |
| 55U05 | Abstract complexes in algebraic topology |
| 55S10 | Steenrod algebra |
| 55S12 | Dyer-Lashof operations |
| 55-04 | Software, source code, etc. for problems pertaining to algebraic topology |
| 55U15 | Chain complexes in algebraic topology |
| 55S15 | Symmetric products and cyclic products in algebraic topology |
Keywords:
Steenrod operations; operads; \(E_\infty\) operad; May-Steenrod structure; simplicial cochains; cubical cochainsCitations:
Zbl 0242.55023Software:
ComCHReferences:
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