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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.

30 votes
2 answers
2k views

Unstable homotopy groups of spheres beyond Toda's range

In 1962 Toda published his book "Composition methods in homotopy groups of spheres", which contains computations of $\pi_{n+k}(S^n)$ for $k\le 19$ and $n\le 20$. The values of these groups are conveni …
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16 votes
1 answer
1k views

Which cohomology theories are real- and complex-orientable?

A complex-oriented cohomology theory $E^*$ is a multiplicative cohomology theory with a choice of Thom class $x\in\tilde{E}^2(\mathbb{C}P^\infty)$ for the universal complex line bundle (which can be u …
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12 votes
1 answer
445 views

Extending a weak version of Sullivan's generalized conjecture

Apologies for the title. Miller's theorem (formerly Sullivan's conjecture) gives that for a finite group $G$ and a finite dimensional connected CW complex $X$, the based mapping space $\operatorname{ …
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10 votes
0 answers
483 views

Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology. The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to …
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10 votes
2 answers
1k views

Does the Borel functor take equivariant fibrations to fibrations?

Let $p\colon X\to B$ be a fibration. Let $G$ be a topological group acting continuously on $X$ and $B$, and assume that the map $p$ is $G$-equivariant. We can apply the Borel functor $EG\times_G-$ f …
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9 votes
0 answers
182 views

Aspherical fibrations and group epimorphisms

Let $\mathsf{Top}$ denote the category of pointed spaces having the pointed homotopy type of a pointed CW-complex. Let $\mathsf{Grp}$ denote the category of groups. It is well documented that for grou …
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8 votes
4 answers
1k views

What is the homotopy fiber of a fold map?

If $X$ and $Y$ are based spaces, let $p_X: X\vee Y\to X$ be the fold map, or projection, onto $X$. What is the homotopy fiber $F$ of $p_X$? I think I have an argument that $F$ is the half-smash …
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8 votes
1 answer
270 views

When is the diagonal inclusion a $\Sigma_2$-cofibration?

Recall that a space $X$ is called locally equiconnected or LEC if the diagonal map $d:X\hookrightarrow X\times X$ is a cofibration. For example, CW-complexes are LEC. There is some discussion of this …
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7 votes
3 answers
376 views

Classification of sections of free loop fibration over the two-sphere

For any space $X$ there is a fibration $$ \Omega X\to LX\stackrel{ev}{\to} X $$ where $LX=Map(S^1,X)$ is the free loop space, $\Omega X = Map_*(S^1,X)$ is the based loop space, and $ev:LX\to X$ is the …
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6 votes
1 answer
420 views

Homotopy dimension of a mapping

The homotopy dimension $h \dim X$ of a space $X$ is the minimal dimension of a CW-complex $Z$ homotopy equivalent to $X$. I am interested in the generalisation to maps $f\colon X\to Y$. Here's what I …
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6 votes
0 answers
197 views

Actions of cofibrations and induced maps of cofibres

Working in some nice category of based topological spaces (compactly generated with CW homotopy type, say) suppose we have a homotopy commutative diagram $$ \begin{array}{ccccc} & & j & & \newline & …
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6 votes
0 answers
465 views

When is the cohomology cross product square nonzero?

Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ …
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5 votes
1 answer
445 views

Pontrjagin ring structure on homology of Eilenberg-Mac Lane spaces

Is there any good reference for the Pontrjagin ring structure on $$ H_\ast(K(\mathbb{Z}/2,k);\mathbb{Z}/2)\cong H_\ast(\Omega K(\mathbb{Z}/2,k+1);\mathbb{Z}/2)? $$ I am familiar with Serre's theorem …
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4 votes
1 answer
371 views

fibre-preserving homotopy equivalence

Let $p:E\to B$ and $p':E'\to B$ be fibrations. It is well known that if $f:E\to E'$ a fibrewise map that is also a homotopy equivalence, then it is a fibrewise homotopy equivalence. What about the mor …
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3 votes
1 answer
159 views

Geometric vs cohomological dimension with families - on a proof of Lueck and Meintrup

Let $G$ be a discrete group, and let $\mathcal{F}$ be a family of subgroups of $G$ (closed under conjugation and taking subgroups). Then we may define the geometric and cohomological dimensions of $G$ …
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