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The completed fourfold-graded cohomology ring is natural and satisfies the ring laws

Statement

For every space $B$ , the operations of The completed cohomology ring in degrees divisible by four make ${\hat{H}}^{4 *} (B; Q)$ a commutative unital ring. For every continuous map $f : B \to C$ , componentwise pullback is a unital ring homomorphism $f^{*} : {\hat{H}}^{4 *} (C; Q) \to {\hat{H}}^{4 *} (B; Q)$ . The assertion is valid for arbitrary, possibly infinite-dimensional spaces.

Facts & Assumptions

Given: Spaces $B$ and $C$ , a continuous map $f : B \to C$ , and elements $a = (a_{j})$ , $b = (b_{j})$ , $c = (c_{j})$ of ${\hat{H}}^{4 *} (C; Q)$ .

[F1]

The completed group is ${\hat{H}}^{4 *} (B; Q) = \prod_{j \geq 0} H^{4 j} (B; Q)$ with componentwise addition, product $(a \cdot b)_{n} = \sum_{i + j = n} a_{i} ⌣ b_{j}$ , and unit $(1, 0, 0, \dots)$ ; for fixed $n$ the sum has $n + 1$ terms (The completed cohomology ring in degrees divisible by four).

[F2]

The singular cohomology ring has multiplication induced by the cochain cup product, with unit the class of the constant cochain $1$ , and its multiplication is associative and distributive over addition (Singular cohomology ring).

[F3]

Cup product is natural and unital: $f^{*} (u ⌣ v) = f^{*} u ⌣ f^{*} v$ and $f^{*} 1 = 1$ (Cup product is natural, unital and associative).

[F4]

Cup product is graded commutative: $u ⌣ v = (- 1)^{p q} v ⌣ u$ for $u \in H^{p}$ , $v \in H^{q}$ (Singular cohomology is graded commutative).

Proof

technique · direct; compare finite sums in each degree

1.1givenF1F2

Addition and multiplication are well defined: both are given by finite operations in each degree $4 n$ , since $a_{i} ⌣ b_{n - i} \in H^{4 n} (B; Q)$ for $0 \leq i \leq n$ and these are the only contributing pairs, so no infinite sum occurs. Addition is associative and commutative and has the zero sequence as neutral element because these hold in each group $H^{4 n} (B; Q)$ .

1.2givenF1F2

Associativity: for every $n$ , $((a \cdot b) \cdot c)_{n} = \sum_{i + j + k = n} (a_{i} ⌣ b_{j}) ⌣ c_{k}$ and $(a \cdot (b \cdot c))_{n} = \sum_{i + j + k = n} a_{i} ⌣ (b_{j} ⌣ c_{k})$ , two finite sums over the same triples that agree termwise by associativity of the cup product.

1.3givenF1F2

Unit: $(1 \cdot a)_{n} = 1 ⌣ a_{n} = a_{n}$ and $(a \cdot 1)_{n} = a_{n} ⌣ 1 = a_{n}$ for every $n$ , since $1 \in H^{0} (B; Q)$ is the unit of the graded cohomology ring.

1.4givenF1F2

Distributivity: $(a \cdot (b + c))_{n} = \sum_{i + j = n} a_{i} ⌣ (b_{j} + c_{j}) = \sum_{i + j = n} a_{i} ⌣ b_{j} + \sum_{i + j = n} a_{i} ⌣ c_{j}$ by bilinearity of the cup product and additivity of finite sums.

2.1step 1.1F4algebra

Commutativity: $(a \cdot b)_{n} = \sum_{i + j = n} a_{i} ⌣ b_{j} = \sum_{i + j = n} (- 1)^{16 i j} b_{j} ⌣ a_{i} = (b \cdot a)_{n}$ , because $4 i \cdot 4 j = 16 i j$ is even and so every sign is $+ 1$ .

2.2step 1.1F1F3

Naturality: for each $n$ , $f^{*} ((a \cdot b)_{n}) = \sum_{i + j = n} f^{*} (a_{i} ⌣ b_{j}) = \sum_{i + j = n} f^{*} a_{i} ⌣ f^{*} b_{j} = (f^{*} a \cdot f^{*} b)_{n}$ , and $f^{*} 1 = 1$ componentwise; hence $f^{*}$ is a unital ring homomorphism.

3.1step 1.2step 1.3step 1.4step 2.1step 2.2given∎

Steps 1.1-1.4 and 2.1 give the commutative unital ring laws, and step 2.2 gives naturality, in every degree and hence componentwise; nothing was assumed about the dimension or CW type of $B$ , and the statements include the empty space and the zero ring, where all groups vanish and the same finite computations apply with zero elements. Only the prescribed formulas are used, so no choice principle is invoked.

Depends on

Used by

Cited to discharge well-definedness by The completed cohomology ring in degrees divisible by four.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Milnor and James Stasheff, Characteristic Classes (re-typeset scan; original pagination) (standard reference, not scraped)