Smash product ( $X \land Y$ )

For based spaces, the smash product is

X \land Y = (X \times Y) / (X \lor Y),

where the wedge

X \lor Y

consists of pairs having at least one basepoint coordinate.

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- Smash product of CW pairs Smash product

Smash product of CW pairs ( $(X, A) \land (Y, B) = (X \times Y, (A \times Y) \cup (X \times B))$ )

The quotient of the displayed CW pair is

(X / A) \land (Y / B)

. In a multiplicative generalized cohomology theory, external products of relative classes of degrees

k, m

take values in degree

k + m

of this pair. Pullback along a diagonal produces the relative cup product.

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In mathematics, particularly in category theory and algebraic topology, the smash product is a specific operation that combines two pointed spaces or pointed sets. The smash product is denoted by \( X \wedge Y \), where \( X \) and \( Y \) are pointed spaces, meaning that each has a distinguished 'base point.

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