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The smash product of any pointed space X with a 0-sphere (a discrete space with two points) is homeomorphic to X. The smash product of two circles is a quotient of the torus homeomorphic to the 2-sphere. More generally, the smash product of two spheres S m and S n is homeomorphic to the sphere S m+n.
A further construction in A 1-homotopy theory is the category SH(S), which is obtained from the above unstable category by forcing the smash product with G m to become invertible. This process can be carried out either using model-categorical constructions using so-called G m -spectra or alternatively using infinity-categories.
The unit circle is a (,).; The infinite-dimensional complex projective space is a model of (,).; The infinite-dimensional real projective space is a (/,).; The wedge sum of k unit circles = is a (,), where is the free group on k generators.
The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail.
MO is a rather weak cobordism theory, as the spectrum MO is isomorphic to H(π * (MO)) ("homology with coefficients in π * (MO)") – MO is a product of Eilenberg–MacLane spectra. In other words, the corresponding homology and cohomology theories are no more powerful than homology and cohomology with coefficients in Z /2 Z .
The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy.
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One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S 1. Σ X ≅ S 1 ∧ X {\displaystyle \Sigma X\cong S^{1}\wedge X} For well-behaved spaces, such as CW complexes , the reduced suspension of X is homotopy equivalent to the unbased suspension.