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Results tagged with homotopy-theory
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user 4625
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
10
votes
How are these algebraic and geometric notions of homotopy of maps between manifolds related?
There is a simple way to understand the implication "geometric implies algebraic homotopy" if you remember that $\Omega^*(M \times I)$ is the (projective) tensor product of $\Omega^*(M)$ and $\Omega^* …
- 1,696
13
votes
Accepted
Natural examples of finite dimensional spaces with interesting 2-type
The 2-type of a 4-manifold is an extremely interesting invariant. In fact, work of Hambleton and Kreck shows that in many cases it determines the homotopy type (if one adds the intersection form as an …
- 1,696
21
votes
Accepted
Complex orientations on homotopy
The natural starting point of this story are E-orientations on, say closed, manifolds M. That's just a fundamental class $[M^n] \in E_n(M)$ such that cap product induces a (Poincare duality) isomorphi …
- 1,696
21
votes
Failure of smoothing theory for topological 4-manifolds
John, if you look at chapter 8 of Freedman-Quinn's book on topological 4-manifolds, you'll find the following computation of the homotopy groups of Top(4)/O(4):
$\pi_3 = Z/2$ and $\pi_i = 0$ for $i=0 …
- 1,696