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cup product

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Idea

Recall from the discussion at cohomology that every notion of cohomology (e.g. group cohomology, abelian sheaf cohomology, etc) is given by Hom-spaces in an (∞,1)-topos H\mathbf{H}. Cohomology on an object XHX \in \mathbf{H} with coefficients in an object AHA \in \mathbf{H} is

H(X,A):=π 0H(X,A). H(X,A) := \pi_0 \mathbf{H}(X,A) \,.

The cup product is an operation on cocycles with coefficients A 1A_1 and A 2A_2 that is induced from a pairing morphism

A 1×A 2A 3 A_1 \times A_2 \to A_3

in H\mathbf{H}. In applications this is often a pairing operation with A 1=A 2A_1 = A_2, i.e. A×AAA \times A \to A', and typically it is the product morphism A×AAA \times A \to A for a ring object structure on the coefficients AA.

If g 1:XA 1g_1 : X \to A_1 and g 2:XA 2g_2 : X \to A_2 are two cocycles in H(X,A 1)\mathbf{H}(X,A_1) and H(X,A 2)\mathbf{H}(X,A_2), respectively, then their cup product with respect to this pairing is the cocycle

g 1g 2:XId×IdX×Xg 1×g 2A 1×A 2A 3 g_1 \cdot g_2 : X \stackrel{Id \times Id}{\to} X \times X \stackrel{g_1 \times g_2}{\to} A_1 \times A_2 \to A_3

in H(X,A 3)\mathbf{H}(X,A_3).

Examples

Via the Dold-Kan correspondence

When the coefficient object AA \in ∞Grpd is “sufficiently abelian” in that under the Dold-Kan correspondence it is represented by a chain complex then using the lax monoidalness of the Dold-Kan correspondence (see at monoidal Dold-Kan correspondence) one obtains a chain complex model for the cup product which makes the origin of the typical grading shift manifest.

Write

Now:

Using all this, then for

f:V W Z f \;\colon\; V_\bullet \otimes W_\bullet \longrightarrow Z_\bullet

a given chain map, then this induces a map of the corresponding Kan complexes

DK:DK(V )×DK(W )DK(Z ) \cup_{DK} \;\colon\; DK(V_\bullet) \times DK(W_\bullet) \longrightarrow DK(Z_\bullet)

as the following composite

DK:DK(V )×DK(W )=U(Γ(V ))×U(Γ(W ))U(Γ(V )×Γ(W ))U(p)U(Γ(V )Γ(W ))U(γ)U(Γ(V W ))U(Γ(f))U(Γ(Z ))=DK(Z ). \cup_{DK} \;\colon\; DK(V_\bullet)\times DK(W_\bullet) = U(\Gamma(V_\bullet)) \times U(\Gamma(W_\bullet)) \stackrel{\simeq}{\to} U(\Gamma(V_\bullet) \times \Gamma(W_\bullet)) \stackrel{U(p)}{\to} U(\Gamma(V_\bullet)\otimes \Gamma(W_\bullet)) \stackrel{U(\gamma)}{\longrightarrow} U(\Gamma(V_\bullet \otimes W_\bullet)) \stackrel{U(\Gamma(f))}{\to} U(\Gamma(Z_\bullet)) = DK(Z_\bullet) \,.

With this in hand then for XX any homotopy type, the cup product on its cohomology with coefficients in DK(V )DK(V_\bullet) and DK(W )DK(W_\bullet) is induced by just homming XX into this morphism:

H(X,DK(V ))×H(X,DK(W ))H(X,DK(V )×DK(W ))H(X, DK)H(X,DK(Z )). \mathbf{H}(X, DK(V_\bullet)) \times \mathbf{H}(X, DK(W_\bullet)) \simeq \mathbf{H}(X, DK(V_\bullet) \times DK(W_\bullet)) \stackrel{\mathbf{H}(X,\cup_{DK})}{\longrightarrow} \mathbf{H}(X, DK(Z_\bullet)) \,.

For example if

V =[n 1],W =[n 2] V_\bullet = \mathbb{Z}[n_1] \,, \;\;\; W_\bullet = \mathbb{Z}[n_2]

is the chain complex concentrated in degree n 1n_1 and n 2n_2, respectively, on the group of integers, then

DK(V )B n 1K(n 1,) DK(V_\bullet) \simeq B^{n_1} \mathbb{Z} \simeq K(n_1,\mathbb{Z})

is the corresponding Eilenberg-MacLane space which classifies ordinary cohomology (singular cohomology) with integral coefficients in the given degree. By the nature of the tensor product of chain complexes one has

V W [n 1+n 2]. V_\bullet \otimes W_\bullet \simeq \mathbb{Z}[n_1 + n_2] \,.

Hence we may take Z [n 1+n 2]Z_\bullet \coloneqq \mathbb{Z}[n_1 + n_2] and f=idf = id and we get a cup product

:H n 1(X,)×H n 2(X,)H n 1+n 2(X,). \cup \;\colon\; H^{n_1}(X, \mathbb{Z}) \times H^{n_2}(X, \mathbb{Z}) \to H^{n_1 + n_2}(X, \mathbb{Z}) \,.

On Moore complexes of cosimplicial algebras

For A=(A )A = (A^\bullet) any cosimplicial algebra, its dual Moor cochain complex N (A)N^\bullet(A) naturally inherits the structure of a dg-algebra under the cup product.

The general formula is literally the same as that for the case where A A^\bullet is functions on the singular complex of a space, which is discussed below. For the moment, see below.

This cup product operation on N (A)N^\bullet(A) is not in general commutative. However, it is a standard fact that it becomes commutative after passing to cochain cohomology.

This suggests that the cup product should be, while not commutative, homotopy commutative in that it makes N (A)N^\bullet(A) a homotopy commutative monoid object.

This in turn should mean that N (A)N^\bullet(A) is an algebra over an operad for the E-∞ operad.

That this is indeed the case is the main statement in (Berger-Fresse 01)

In singular cohomology

A special case of the cup product on Moore complexes is the complex of singular cohomology, which is the Moore complex of the cosimplicial algebra of functions on the singular simplicial set of a topological space.

Often in the literature by cup product is meant specifically the realization of the cup product on singular cohomology.

For XX a topological space, let Π(X) :=X Δ Top \Pi(X)_\bullet := X^{\Delta_{Top}^\bullet} be the simplicial set of nn-simplices in XX – the fundamental ∞-groupoid of XX.

For RR some ring, let Maps(Π(X),R) Maps(\Pi(X),R)^{\bullet} be the cosimplicial ring of RR-valued functions on the spaces of nn-simplices. The corresponding Moore cochain complex C (X)C^\bullet(X) is the cochain complex whose cochain cohomology is the singular cohomology of the space XX: a homogeneous element ω pC p(X)\omega_p \in C^p(X) is a function on pp-simplices in XX.

Write, as usual, for pp \in \mathbb{N}, [p]={0<1<<p}[p] = \{0 \lt 1 \lt \cdots \lt p\} for the totally ordered set with p+1p+1 elements. For μ:[p][p+q]\mu : [p] \to [p+q] an injective order preserving map and KK some cosimplicial object, write d μ *K:K pK p+qd_\mu^* K : K^p \to K^{p+q} for the image of this map under KK.

Specifically, for p,qp,q \in \mathbb{N} let L:[p][p+q]L : [p] \to [p+q] be the map that sends i[p]i \in [p] to i[p+q]i \in [p+q] and let R:[q][p+q]R : [q] \to [p+q] be the map that sends i[q]i \in [q] to i+q[p+q]i+q \in [p+q].

Then the cup product

:C (X)C (X)C (X) \smile : C^\bullet(X) \otimes C^\bullet(X) \to C^\bullet(X)

is the cochain map that on homogeneous elements abC p(X)C q(X)C (X)C (X)a \otimes b \in C^p(X) \otimes C^q(X) \subset C^\bullet(X) \otimes C^\bullet(X) is defined by the formula

ab=(d L *a)(d R *b). a \smile b = (d_L^* a) \cdot (d_R^* b) \,.

There is some glue missing here to connect this back to the above general definition, something involving the Eilenberg-Zilber map.

This means that (ab) i 0,,i p+q=a i 0,,i pb i p,,i p+q(a \smile b)_{i_0, \cdots, i_{p+q}} = a_{i_0, \cdots, i_p} \cdot b_{i_p, \cdots, i_{p+q}}.

Proposition

This cup product enjoys the following properties:

  • it is indeed a cochain complex morphism as claimed, in that it respects the differential: for any homogeneous abC p(X)C q(X)a\otimes b \in C^p(X) \otimes C^q(X) as above we have

    d(ab)=(da)b+(1) pa(db). d(a \smile b) = (d a) \smile b + (-1)^p a \smile (d b) \,.
  • the image of the cup product on cochain cohomology

    :H (C (X))H (C (X))H (C (X)) \smile : H^\bullet(C^\bullet(X))\otimes H^\bullet(C^\bullet(X)) \to H^\bullet(C^\bullet(X))

    is associative and distributes over the addition in H (C (X))H^\bullet(C^\bullet(X)).

Accordingly, the cup product makes H (C (X))=H (X,R)H^\bullet(C^\bullet(X)) = H^\bullet(X,R) into a ring: the cohomology ring on the ordinary cohomology of XX.

See for instance section 3.2 of

  • Hatcher, Algebraic Topology (web pdf)

In abelian sheaf cohomology

Traditionally the cup product is considered for abelian cohomology, such as generalized (Eilenberg-Steenrod) cohomology and more generally abelian sheaf cohomology.

In that case all coefficient objects A iA_i are complexes (A i) (A_i)_\bullet of sheaves and the pairing that one usually considers is the tensor product of chain complexes

(A 1) ×(A 2) (A 1A 2) (A_1)_\bullet \times (A_2)_\bullet \to (A_1 \otimes A_2)_\bullet

where

(A 1A 2) n= p(A 1) p(A 2) np. (A_1 \otimes A_2)_n = \oplus_p (A_1)_p \otimes (A_2)_{n-p} \,.

with differential

d(a 1a 2)=(da 1)a 2+(1) |a 1|a 1da 2. d (a_1 \otimes a_2) = (d a_1) \otimes a_2 + (-1)^{|a_1|} a_1 \otimes d a_2 \,.

In abelian Čech cohomology

The cup product has a simple expression in abelian Čech cohomology.

For A 1A_1 and A 2A_2 two chain complexes (of sheaves of abelian groups) construct a morphism of Čech complexes

ϕ:C ({U i},A 1)C ({U i},A 2)C ({U i},A 1A 2) \phi : C^\bullet(\{U_i\}, A_1) \otimes C^\bullet(\{U_i\}, A_2) \to C^\bullet(\{U_i\}, A_1 \otimes A_2)

by sending αC p(U,A 1) \alpha \in C^p(U,A_1)_\bullet and βC q(U,A 2) \beta \in C^q(U,A_2)_\bullet to

ϕ(αβ) i 0,,i p+q:=α i 0,,i pβ i p,i p+q. \phi(\alpha \otimes \beta)_{i_0, \cdots , i_{p + q}} \;:=\; \alpha_{i_0, \cdots, i_p} \otimes \beta_{i_p, \cdots i_{p+q}} \,.

For instance (Brylinski, section (1.3)) spring

In Čech-Deligne cohomology (ordinary differential cohomology)

For the case that of Čech hypercohomology with coefficients in Deligne complexes the above yields the Beilinson-Deligne cup-product for ordinary differential cohomology.

References

The cup product in Čech cohomology is discussed for instance in section 1.3 of

Recall from the discussion at models for ∞-stack (∞,1)-toposes that all hypercomplete ∞-stack (∞,1)-toposes are modeled by the model structure on simplicial presheaves. Accordingly understanding the cup product on simplicial presheaves goes a long way towards the most general description. For a bit of discussion of this see around page 19 of

An early treatment of cup product can be found in this classic

  • Whitney, On Products in a Complex (JSTOR)

See also

Revised on May 19, 2014 02:30:27 by Urs Schreiber (109.144.208.68)