Contents
Context
Cohomology
cohomology
Special and general types
Special notions
Variants
Operations
Theorems
Homological algebra
Contents
Idea
A Bockstein homomorphism is a connecting homomorphism induced from a short exact sequence whose injective map is given by multiplication with an integer.
The archetypical examples are the Bockstein homomorphisms induced this way from the short exact sequence
These relate notably degree- cohomology with coefficients in (such as Stiefel-Whitney classes) to cohomology with integral coefficients in degree (such as integral Stiefel-Whitney classes).
Definition
Let be an abelian group and be an integer. Then multiplication by
induces a short exact sequence of abelian groups
where is the subgroup of -torsion elements of , and so a long fiber sequence
of ∞-groupoids, where denotes the -fold delooping (hence is the Eilenberg-MacLane object on in degree ).
This induces, in turn, for any object (for the ambient (∞,1)-topos, such as Top ∞Grpd) , a long fiber sequence
of cocycle ∞-groupoids.
Here the connecting homomorphisms are called the Bockstein homomorphisms.
Notice that often this term is used to refer only to the image of the above in cohomology, hence to the image of under 0-truncation/0th homotopy group :
Examples
Example
(mod 2 Bockstein homomorphism into integral cohomology)
The Bockstein homomorphism for the sequence
serves to define integral Stiefel-Whitney classes
in degree from -valued Stiefel-Whitney classes in degree .
Example
(first Steenrod square)
The Bockstein homomorphism for the sequence
is also called the first Steenrod square, denoted .
This is often equivalently denoted , as in example . The difference between the two is just the mod-2 reduction in their codomain:
(1)
More generally, for any prime number the multiplication by on induces the short exact sequence . The corresponding Bockstein homomorphism appears as one of the generators of the mod Steenrod algebra.
Example
(integral Steenrod squares)
For odd defines the integral Steenrod squares to be
By example and by the first Adem relation (this example) these indeed are lifts of the odd Steenrod squares:
because, by (1) we have
When , the equivalence (which holds in ambient contexts such as ETop∞Grpd or Smooth∞Grpd under geometric realization ) identifies the morphisms with the morphisms induced by the inclusion of the subgroup of -th roots of unity into . This identifies the Bockstein homomorphism with the natural homomorphism .
More in detail:
Example
(mod 2 Bockstein homomorphism and the exponential exact sequence)
Let
-
be the ordinare Bockstein homomorphism
-
the canonical inclusion;
-
the classifying map.
Then
Because
Proposition
(Deligne-Beilinson cup product on odd-degree ordinary differential cohomology)
Let
be a class in ordinary differential cohomology with underlying class in odd degree
This implies that its Beilinson-Deligne cup product with itself satisfies
hence
hence
hence that the ordinary cup product is a 2-torsion class. Let then
with from example .
Then
This is a differential cohomology-refinement of the first Adem relation on the Steenrod squares (this example) in that, by example , its image in ordinary cohomology with coefficients in is
This was first observed in (Gomi 08). Streamlined proofs are given in (Bunke 12, propblem 3.106, Grady-Sati 16, prop. 22).
References
Original references include
-
Meyer Bockstein,
Universal systems of -homology rings, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 243–245, MR0008701
A complete system of fields of coefficients for the -homological dimension , C. R. (Doklady) Acad. Sci. URSS (N.S.) (1943), 38: 187–189, MR0009115
-
Meyer Bockstein, Sur la formule des coefficients universels pour les groupes d’homologie , Comptes Rendus de l’académie des Sciences. Série I. Mathématique (1958), 247: 396–398, MR0103918
The relation to the Beilinson-Deligne cup product is discussed in
-
Kiyonori Gomi, Differential characters and the Steenrod squares, In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 297?308. Math. Soc. Japan, Tokyo, 2008
-
Ulrich Bunke, problem 3.106 in Differential cohomology (arXiv:1208.3961)
-
Daniel Grady, Hisham Sati, prop. 22 in: Primary operations in differential cohomology, Adv. Math. 335 (2018), 519-562 (arXiv:1604.05988, doi:10.1016/j.aim.2018.07.019)