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Total Steenrod square

Definition

Write ordinary total mod-two cohomology as the graded direct sum

$H^{*} (X; F_{2}) = ⨁_{n \geq 0} H^{n} (X; F_{2}) .$

For a homogeneous class $x \in H^{n} (X; F_{2})$ , its total Steenrod square is

$S q (x) : = \sum_{i = 0}^{n} S q^{i} (x) .$

For an arbitrary element $x = \sum_{n} x_{n}$ of the graded direct sum, define $S q (x) = \sum_{n} S q (x_{n})$ . Both sums are finite: the first by instability and the second by the definition of direct sum. Thus this definition takes values in the same ordinary direct sum; it does not use a completed product. It is generally not degree-preserving.

Facts & Assumptions

Given: A space $X$ and a finite-support element of $H^{*} (X; F_{2})$ .

[F1]

Every square is additive and natural (Steenrod squares are well-defined and natural).

[F2]

Squares vanish above the degree of their input and $S q^{0}$ is the identity (Steenrod normalization, instability, suspension, and top square).

[F3]

Squares satisfy the internal Cartan formula, with only finitely many nonzero terms (Cartan formula for Steenrod squares).

Verification

technique · sum the finite Cartan identities

1.1givenF1F2

The definition is well-defined, additive, and natural. For each homogeneous component, [F2] leaves only indices $0 \leq i \leq n$ . An element of the direct sum has only finitely many homogeneous components, so its total image again has finite degree support. Termwise additivity and naturality follow from [F1]. No rearrangement of an infinite family is involved.

2.1F3step 1.1

The total square is multiplicative. For homogeneous $x, y$ , all sums below are finite, and [F3] gives

$S q (x ⌣ y) = \sum_{k} S q^{k} (x ⌣ y) = \sum_{i, j} S q^{i} (x) ⌣ S q^{j} (y) = S q (x) ⌣ S q (y) .$

Distributivity and the finite homogeneous support in step 1.1 extend this to arbitrary total classes.

2.2F2step 1.1

It preserves the unit. The unit $1 \in H^{0} (X; F_{2})$ satisfies $S q^{0} (1) = 1$ , and every $S q^{i} (1)$ with $i > 0$ vanishes by instability. Hence $S q (1) = 1$ .

3.1F1F2F3step 1.1step 2.1step 2.2∎

The total square sends zero to zero and is unique on the empty-space cohomology group. For the empty space the total group is zero; for the zero class the defining sum is zero. On a point only degree zero survives, and step 2.2 makes the operation the identity, including on the elements zero and one. The endpoints $i = 0$ and $i = n$ are included, while every $i > n$ is zero before summing. Degenerate singular simplices are inherited unchanged from the already well-defined component operations. The construction makes only finite sums and uses no choices, so it assumes no AC. It asserts neither a degreewise endomorphism nor a biconditional.

Depends on

Used by

Steenrod squares on complex projective space mod two Example
Steenrod squares on real projective space Example
Thom identity for Stiefel–Whitney classes Theorem

Dependency tree · two levels

13 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

Hatcher, Algebraic Topology (standard reference, not scraped)
Mosher and Tangora, Cohomology Operations and Applications in Homotopy Theory (standard reference, not scraped)