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Results tagged with homotopy-theory
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user 4991
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.
13
votes
What would be the ramifications of homotopy theory being as easy as homology theory?
Those who reply that being able to compute homotopy groups is akin to being able to compute values of L-functions are perhaps unwittingly referring to a great parallel: stable homotopy groups of spher …
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15
votes
How can I visualize the nontrivial element of $\pi_4(S^3)$ and $\pi_5(S^3)$ ?
Through the Pontrjagin-Thom construction, a framed $n-k$ manifold in $S^n$ determines a map from $S^n$ to $S^{n-k}$. $\eta$ is represented by $S^1$ in $S^3$ with framing which "twists around once". …
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9
votes
1
answer
511
views
Models for P map in EHP sequence
The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its suspens …
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5
votes
homotopy invariant and coinvariant
These chain complexes are modeling a standard construction in equivariant homotopy theory. Let $S^\infty$ have the standard $S^1$ action (this is a model for what is called $ES^1$) and add a disjoint …
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16
votes
How to get product on cohomology using the K(G, n)?
Yes, this iterated bar construction model should be better known, and can be expressed even more geometrically than Ravenel and Wilson do.
If $A$ is an abelian group then $K(A,n)$ is modeled by a spa …
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14
votes
2
answers
1k
views
Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop space...
The homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more acc …
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11
votes
1
answer
564
views
The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex.
Let $X$ be a finite (CW or simplicial - doesn't matter) complex and consider the space of all compact subspaces of $R^\infty$ which are homotopy equivalent to $X$, topologized say as a subspace of the …
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40
votes
Why does one think to Steenrod squares and powers?
Here's how I explain Steenrod squares to geometers. First, if $X$ is a manifold of dimension $d$ then one can produce classes in $H^n(X)$ by proper maps $f: V \to X$ where $V$ is a manifold of dimens …
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6
votes
Accepted
Whitehead products and Framed Manifolds
By Pontryagin's Theorem, you are asking for the preimages of any chosen points
in each wedge factor of $S^p \vee S^q \vee S^r$ for the iterated Whitehead product map $p : S^{p+q+r-2} \to S^p \vee S^q …
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19
votes
Modern survey of unstable homotopy groups?
Behrens's monograph "The Goodwillie tower and the EHP sequence" reproduces some of the Toda calculations (out to the k~20 range as you cite) using a modern toolset, as named in the title. Depending o …
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15
votes
What is the intuition for higher homotopy groups not vanishing?
Homotopy groups of spheres correspond to framed submanifolds of Euclidean space through the Pontrjagin-Thom construction. For example, the Hopf map corresponds to a circle in $\mathbb{R}^3$ framed “w …
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22
votes
Accepted
Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?
I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question:
CW complexes conne …
16
votes
References for homotopy colimit
Dan Dugger wrote the following intended for grad students (just a draft - not on the arxiv yet): http://www.uoregon.edu/~ddugger/hocolim.pdf
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