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The smash product is itself a pointed space, with basepoint being the equivalence class of (x 0, y 0). The smash product is usually denoted X ∧ Y or X ⨳ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous). One can think of X and Y as sitting inside X × Y as the subspaces X × {y 0} and {x 0} × Y.
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.
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 ...
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.
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/2Z. This was the first ...
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.
For example, if E is the trivial bundle , then is and, writing + for B with a disjoint basepoint, () is the smash product of + and ; that is, the n-th reduced suspension of +. Alternatively, [ citation needed ] since B is paracompact, E can be given a Euclidean metric and then T ( E ) {\displaystyle T(E)} can be defined as the quotient of ...