Products of Cocycles and Extensions of Mappings

Steenrod, N. E.

doi:10.2307/1969172

Cited by 255 publications

(222 citation statements)

References 8 publications

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“…Obtaining those strange sign factors is another breakthrough that allows us to obtain topological fermionic path integral beyond (1+1) dimensions. It appears that those sign factors are related to a deep mathematical structure called Steenrod squares [72,73] Different vertex orders give rise to different branching structures. It turns out that all the valid branching structures give rise to the same condition (47).…”

Section: (3+1)d Casementioning

confidence: 99%

“…− n 3 (g 0 ,g 1 ,g 2 ,g 3 )m 2 (g 0 ,g 3 ,g 4 ) − n 3 (g 1 ,g 2 ,g 3 ,g 4 )m 2 (g 0 ,g 1 ,g 4 ) − n 3 (g 0 ,g 1 ,g 2 ,g 3 )m 2 (g 0 ,g 3 ,g 5 ) − n 3 (g 1 ,g 2 ,g 3 ,g 5 )m 2 (g 0 ,g 1 ,g 5 ) − n 3 (g 0 ,g 1 ,g 2 ,g 4 )m 2 (g 0 ,g 4 ,g 5 ) − n 3 (g 1 ,g 2 ,g 4 ,g 5 )m 2 (g 0 ,g 1 ,g [72,73]. This realization will allow us to generalize Eq.…”

Section: Appendix F: Calculate Group Supercohomology Classmentioning

confidence: 99%

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Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T 2 = 1 time-reversal Z T 2 and fermion-number-parity Z f 2 symmetries described by a full symmetry group Z T 2 × Z f 2 ). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group Z 2 × Z f 2 . Those 2D fermionic SPT phases all have central-charge c = 1 gapless edge excitations, if the symmetry is not broken.

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Section: (3+1)d Casementioning

confidence: 99%

Section: Appendix F: Calculate Group Supercohomology Classmentioning

confidence: 99%

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

2014

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“…As was suggested by one of the referees, for the case d = 2k − 1, the existence of an equivariant map is characterized by (an equivariant version of) a theorem of Steenrod [Ste47], which involves the first cohomological obstruction plus another apparently computable invariant (a Steenrod square operation in cohomology). This might also yield even a polynomial-time decision algorithm, but some computational issues still need to be checked.…”

Section: Appendix B the Deleted Product Obstructionmentioning

confidence: 99%

Hardness of embedding simplicial complexes in $\mathbb{R}^d$

Matoušek¹,

Tancer²,

Wagner³

2010

J. Eur. Math. Soc.

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Abstract. Let EMBED k→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into R d ?Known results easily imply the polynomiality of EMBED k→2 (k = 1, 2; the case k = 1, d = 2 is graph planarity) and of EMBED k→2k for all k ≥ 3.We show that the celebrated result of Novikov on the algorithmic unsolvability of recognizing the 5-sphere implies that EMBED d→d and EMBED (d−1)→d are undecidable for each d ≥ 5. Our main result is the NP-hardness of EMBED 2→4 and, more generally, of EMBED k→d for all k, d with d ≥ 4 and d ≥ k ≥ (2d − 2)/3. These dimensions fall outside the metastable range of a theorem of Haefliger and Weber, which characterizes embeddability using the deleted product obstruction. Our reductions are based on examples, due to Segal, Spież, Freedman, Krushkal, Teichner, and Skopenkov, showing that outside the metastable range the deleted product obstruction is not sufficient to characterize embeddability.

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“…In 1947, Steenrod [21] introduced the Steenrod squares Sq k in terms of cocycles in simplicial cochain complex by modifying the Alexander-Ĉech-Whitney formula for the cup product construction. Serre [16] showed that they generate all stable operations in cohomology over F 2 under composition.…”

Section: Preliminariesmentioning

confidence: 99%

A Note on the Unstability Conditions of the Steenrod Squares on the Polynomial Algebra

Janfada¹

2009

Journal of the Korean Mathematical Society

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Abstract. We extend some results involved the action of the Steenrod operations on monomials and get some corollaries on the hit problem. Then, by multiplying some special matrices, we obtain an efficient tool to compute the action of these operations. PreliminariesIn 1947, Steenrod [21] introduced the Steenrod squares Sq k in terms of cocycles in simplicial cochain complex by modifying the Alexander-Ĉech-Whitney formula for the cup product construction. Serre [16] showed that they generate all stable operations in cohomology over F 2 under composition. For an overview on algebraic topology we cite [4]. Cartan [2] discovered a formula for working out a Steenrod square on a product of cohomology classes f , g.Adem[1] and Serre [16] established a faithful representation of A by its action on the cohomology of a test space consisting of infinite real projective spaces whose cohomology is the polynomial algebra P(n) = F 2 [x 1 , x 2 , . . . , x n ] = d≥0 P d (n), viewed as a graded module over the Steenrod algebra A at prime 2. The grading is by the homogeneous polynomials P d (n) of degree d in the variables x 1 , x 2 , . . . , x n of grading 1. We cite to [5] and [11] in cohomology operations and to [11] and [22] in the Steenrod algebra.The Steenrod algebra A is defined to be the graded algebra over the field F 2 , generated by the Steenrod squares Sq k , in grading k ≥ 0, subject to the Adem relations [7,24]. From a topological point of view, the Steenrod algebra is the algebra of stable cohomology operations for ordinary cohomology H * over F 2 .For present purpose we only need to know that the Steenrod algebra acts by composition of linear operators on P(n) and the action of the Steenrod squares

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Products of Cocycles and Extensions of Mappings

Cited by 255 publications

References 8 publications

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinearσmodels and a special group supercohomology theory

Hardness of embedding simplicial complexes in $\mathbb{R}^d$

A Note on the Unstability Conditions of the Steenrod Squares on the Polynomial Algebra

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