Intereting Posts

Cartan and Eilenberg Homological Algebra
Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$
Equivalence of Archimedian Fields Properties
Winding number question.
Evaluating an Integral by Residue Theorem
Incorrect proof of the infinities between 0 and 1 and 0 and 2
The complement of every countable set in the plane is path connected
What is a real world application of polynomial factoring?
An incorrect answer for an integral
An Ordinary Differential Equation with time varying coefficients
Show that the area under the curve of a measurabe function is Lebesgue measurable
Integration with exponential constant
How to solve permutation group equations?(discrete mathematics,group theory)
Nonlinear simultaneous equations
Inverting the Cantor pairing function

To what degree can we dualize theorems regarding homotopy into theorems about cohomotopy (or is there a good source that tries to do this)?

For instance, is there some kind of Hurewicz theorem relating cohomotopy and ordinary cohomology? Is there a “cohomotopy extension property” (something that applies when relative cohomotopy groups are trivial)? If two spaces are cohomologically equivalent and have some property in cohomotopy analogous to simply-connected, are they cohomotopy equivalent?

Thanks, this is primarily a reference request, however there is the possibility that all this is impossible so no such reference exists, which would also be an acceptable answer.

- Group structure on Eilenberg-MacLane spaces
- invariance of integrals for homotopy equivalent spaces
- Construction of Hodge decomposition
- The empty set in homotopy theoretic terms (as a simplicial set/top. space)
- Proof of the Borsuk-Ulam Theorem
- Question on Good Pairs

- The empty set in homotopy theoretic terms (as a simplicial set/top. space)
- Higher homology group of Eilenberg-Maclane space is trivial
- Applications of algebraic topology
- Orientation reversing diffeomorphism
- Fundamental group of mapping torus?
- Visualizing a homotopy pull back
- What is the space obtained by identifying boundary $\mathbb T^2$ of a solid torus
- Brouwer's fixed-point theorem and the intermediate value theorem?
- invariance of integrals for homotopy equivalent spaces
- Topology knowledge for CW complexes $\oplus$ Reference request

The homotopy groups can be written as covariant homotopy invariant functors $\pi_n:\mathrm{Top}_\ast\to\mathrm{Set}$. If we were to consider contravariant homotopy invariant functors $\pi^n:\mathrm{Top}_\ast^{op}\to\mathrm{Set}$, we would obtain the cohomotopy sets. How dual is it? Well, $\pi^n(S^m)=\pi_m(S^n)$. If $X$ is a CW-complex of dimension (at most) $n$, then $\pi^p(X)\to H^p(X)$ is a bijection. See the nlab. As for whether or not a “cohomotopy extension property” exists, I don’t know; it seems like an interesting thing!

- Fundamental group of multiplicative group in Zariski topology
- Defining the Complex numbers
- Reduction formulae
- Socle of submodule relative to the module
- If $\mathbb E=0$ for all $G\in \mathcal G$, does $X=0$?
- Why isn't the volume formula for a cone $\pi r^2h$?
- Preimage of Intersection of Two Sets = Intersection of Preimage of Each Set : $f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)$
- The product of all the elements of a finite abelian group
- An exercise in infinite combinatorics from Burris and Sankappanavar
- a.s. Convergence and Convergence in Probability
- Evaluate $ \sum\limits_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}$
- Easy to read books on Graph Theory
- Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$?
- $f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$
- Counterexamples in Double Integral