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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 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.
Tables of homotopy groups of spheres are most conveniently organized by showing π n+k (S n). The following table shows many of the groups π n+k (S n). The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following ...
In algebraic topology, Hilton's theorem, proved by Peter Hilton , states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres. John Milnor ( 1972 ) showed more generally that the loop space of the suspension of a wedge of spaces can be written as an infinite product of loop spaces of ...
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 .
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.
The map from the sum to the homology group of the product is called the cross product. More precisely, there is a cross product operation by which an i -cycle on X and a j -cycle on Y can be combined to create an ( i + j ) {\displaystyle (i+j)} -cycle on X × Y {\displaystyle X\times Y} ; so that there is an explicit linear mapping defined from ...