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Results tagged with homotopy-theory
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user 23571
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.
9
votes
Non-zero homotopy/homology in diffeomorphism groups
If ${\rm Diff}(M)$ is contractible then the question of course has a negative answer. Examples where this happens are known in dimension three but not in higher dimensions. For $M$ a closed hyperboli …
- 20k
13
votes
Accepted
How can I endow a "locally product" CW structure on a vector bundle over a CW complex?
The authors of this book are attempting to use CW structures to justify certain cohomology isomorphisms, but this seems to be the wrong approach since some of their claims about CW structures are just …
- 20k
3
votes
Mapping Class Group action on triangulated $S^2\times S^1$?
(This is a long comment rather than a complete answer.)
As Igor Rivin points out, the mapping class group is not ${\mathbb Z}_2$. There is another ${\mathbb Z}_2$ direct summand coming from a homeomor …
- 20k
16
votes
Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$
The mapping class groups of all compact orientable 3-manifolds are essentially known. A fairly detailed summary of the results, focusing on the nonprime case and with references to proofs in the lite …
- 20k
5
votes
Accepted
Homotopy classes of homeomorphisms of a multiple pointed space
The answer is No, a homotopy relative to $P$ cannot in general be improved to an isotopy. To see this, consider the fibration
$$
{\rm HomEq}^+(M\ {\rm rel} \ P)\to {\rm HomEq}^+(M)\to {\rm Map}(P,M)
…
- 20k
79
votes
Accepted
Maps which induce the same homomorphism on homotopy and homology groups are homotopic
Take the composition of a degree one map $f:T^3\to S^3$ with the Hopf map $g:S^3\to S^2$, where $T^3$ is the 3-torus. This composition is trivial on homotopy groups since $T^3$ is aspherical and $\pi_ …
- 20k
11
votes
Accepted
Homotopy versus path-homotopy on punctured surface
The special feature of $X$, a sphere with three or more punctures, that is being used here is that the space $E(X)$ of all homotopy equivalences $X\to X$ has $\pi_1 E(X)=0$. (Here we take the identity …
- 20k
22
votes
Accepted
Detecting homotopy nontriviality of an element in a torsion homotopy group
How about thinking about framed cobordism, which in this case gives an isomorphism between $\pi_4(S^3)$ and the group of cobordism classes of normally framed 1-manifolds in $S^4$. Since your map is c …
- 20k
28
votes
Accepted
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one nonco …
- 20k
43
votes
Accepted
nontrivial $\pi_2(\textrm{Diff}(M))$
$\newcommand{\Diff}{\mathrm{Diff}}$Probably the simplest such manifold is $S^1 \times S^2$, whose diffeomorphism group has the homotopy type of $O(2) \times O(3) \times \Omega SO(3)$. This has $\pi_2$ …
- 20k
20
votes
Accepted
The Wedge Sum of path connected topological spaces
A counterexample is shown on the cover of the paperback edition of the classic textbook Homology Theory by Hilton and Wylie. This can be viewed on the amazon webpage for the book. The example consis …
- 20k
3
votes
Good reference for homology of $K(\mathbb{Z}, 2n)$?
Nice question. Unfortunately the answer I posted an hour ago is wrong because I switched the structures of homology and cohomology for the James reduced product. It is the cohomology that is a divide …
- 20k
15
votes
Accepted
Is geometric realization of the total singular complex of a space homotopy equivalent to the...
The map from the (realization of the) singular complex of a space $X$ to $X$ is a homotopy equivalence if and only if $X$ is homotopy equivalent to a CW complex, so to get examples where the map is no …
- 20k
32
votes
Accepted
Survey articles on homotopy groups of spheres
While my Algebraic Topology book and my unfinished book on spectral sequences (referred to in other answers to this question) contain some information about homotopy groups of spheres, they don't real …
- 20k
16
votes
Killing the torsion in homotopy
I don't have an answer to this question, but for the analogous question for homology it looks like it can't be done. By the universal coefficient theorem, a construction like this for homology would g …
- 20k