Math 2.0

Algebraic topology Definition

Cap product

Builds on Singular cohomologySingular homologyCup productCommutative ring

Definition

Let XX be a topological space and let RR be a commutative ring with 11. For a singular simplex σ ⁣:Δn→X\sigma \colon \Delta^n \to X with vertices v0,…,vnv_0, \ldots, v_n and a cochain φ∈Ck(X;R)\varphi \in C^k(X; R) with k≤nk \le n, the cap product is the chain σ⌢φ=φ(σ∣[v0,…,vk]) σ∣[vk,…,vn]∈Cn−k(X;R).\sigma \frown \varphi = \varphi\big(\sigma|_{[v_0, \ldots, v_k]}\big) \, \sigma|_{[v_k, \ldots, v_n]} \in C_{n-k}(X; R). It extends linearly to a bilinear map Cn(X;R)×Ck(X;R)→Cn−k(X;R)C_n(X; R) \times C^k(X; R) \to C_{n-k}(X; R). It satisfies ∂(σ⌢φ)=(−1)k(∂σ⌢φ−σ⌢δφ)\partial(\sigma \frown \varphi) = (-1)^k \big(\partial\sigma \frown \varphi - \sigma \frown \delta\varphi\big). Hence it induces a bilinear map ⌢ ⁣:Hn(X;R)×Hk(X;R)→Hn−k(X;R)\frown \colon H_n(X; R) \times H^k(X; R) \to H_{n-k}(X; R) between singular homology and singular cohomology, also called the cap product.

Remarks

Cap and cup product are related by ψ(σ⌢φ)=(φ⌣ψ)(σ)\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma) for φ∈Ck\varphi \in C^k and ψ∈Cn−k\psi \in C^{n-k}. On homology, (c⌢φ)⌢ψ=c⌢(φ⌣ψ)(c \frown \varphi) \frown \psi = c \frown (\varphi \smile \psi), so H∙(X;R)H_\bullet(X; R) is a module over H∙(X;R)H^\bullet(X; R). It is natural in the form f∗(c⌢f∗φ)=f∗(c)⌢φf_*(c \frown f^*\varphi) = f_*(c) \frown \varphi for a continuous map ff.

Example: for k=nk = n, σ⌢φ=φ(σ) σ(vn)\sigma \frown \varphi = \varphi(\sigma)\, \sigma(v_n) is a 00-chain. Its image under the augmentation C0(X;R)→RC_0(X; R) \to R is φ(σ)\varphi(\sigma), so the cap product in top degree followed by augmentation is the Kronecker pairing of cohomology with homology.

Non-example: for k>nk > n the formula does not apply, since an nn-simplex has no face of dimension kk; the product Hn×Hk→Hn−kH_n \times H^k \to H_{n-k} is only defined for k≤nk \le n.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Commutative ring, concept, no evidence levelCommutative ringconceptCup product, concept, no evidence levelCup productconceptSingular cohomology, concept, no evidence levelSingular cohomologyconceptSingular homology, concept, no evidence levelSingular homologyconceptCap product, definition, no evidence levelCap productdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Commutative ring concept
  2. Cup product concept
  3. Singular cohomology concept
  4. Singular homology concept
  5. Cap product definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cap_product
About this page
Tag8TPW8B9G48
URIm2://8TPW8B9G48
Kinddefinition
WritingW0 stub
MSC55N45
DefinesCap product
UsesSingular cohomology, Singular homology, Cup product, Commutative ring
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.