New answers tagged at.algebraic-topology
4
votes
Detailed exposition of construction of Steenrod squares from Haynes Miller's book
Hatcher's book also proceeds by first showing that there is the power operation $P(x)$ as you say, and he proves all the properties.
Hatcher makes quite a bit of use of the fact that cohomology is ...
- 10.9k
3
votes
Object of proven finiteness, yet with no algorithm discovered?
Here is an example where finiteness is known, an algorithm exists, but such an algorithm is ineffective, in the sense that current computers (to my knowledge) won't terminate in any reasonable amount ...
10
votes
Object of proven finiteness, yet with no algorithm discovered?
There exists a positive integer $N \le 246$ such that there are infinitely many primes $p$ for which $p + N$ is also prime. However, there is no single specific value of $N$ for which this is proven.
- 1,643
10
votes
Object of proven finiteness, yet with no algorithm discovered?
The graph minor theorem implies that any family of graphs that is closed under minors has a finite forbidden minor characterization, but the proof does not yield an algorithm for finding the minors. ...
- 80.3k
9
votes
Object of proven finiteness, yet with no algorithm discovered?
We know that the chromatic number of the plane is at most 7 but do not know its exact value. To remove any dependency on AC, we could ask for the maximum possible chromatic number of (the unit-...
- 2,848
2
votes
When can a generalized connected sum be aspherical
The fiber sum you describe is $T^4$. For you are removing $D^2 \times T^2$ from each of $M$ and $N$. But $M - D^2 \times T^2 = D^2 \times T^2$ so you're just removing $D^2 \times T^2$ from $N$ and ...
- 19.3k
11
votes
Object of proven finiteness, yet with no algorithm discovered?
Another example is Euler's idoneal number problem, one of the oldest problems in number theory. It asserts that $1848$ is the largest number $n$ such that the class group of the quadratic order $\...
- 25.9k
17
votes
Object of proven finiteness, yet with no algorithm discovered?
Heath-Brown proved that Artin's conjecture on primitive roots has at most two counterexamples. So for example, we know that the conjecture is true for at least one element of the set $\{2,3,5\}$, but ...
- 80.3k
6
votes
Object of proven finiteness, yet with no algorithm discovered?
$\DeclareMathOperator\BB{BB}$My answer is a slight cheat, because I'm going to give two examples where we have not only not discovered any algorithms but where we know there is no such algorithm. Both ...
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27
votes
Accepted
Object of proven finiteness, yet with no algorithm discovered?
Most of Diophantine approximation falls into the category. For example, Roth's theorem says that for any non-rational algebraic number $\alpha\in\overline{\mathbb Q}\smallsetminus\mathbb Q$ and any $\...
- 46.7k
4
votes
Accepted
Intersection pairing on non-compact surface
Recall the non-compact Poincar'e duality:
$$H^*(M)\cong \bar H_{m-*}(M), H^*_c(M)\cong H_{m-*}(M).$$
Here $M$ is a $\mathbb{Z}$-orientable topological manifold without boundary, $\bar H$ is the Borel-...
- 23.4k
1
vote
Nerve theorem for simplicial sets
To elaborate on my comment: Define a bisimplicial set $Y$ by $Y_m = \amalg_{i_1< \cdots < i_m} A_{i_1} \cap \cdots \cap A_{i_m}$. Define another by $Z_m=\pi_0(Y_m)$. There is a map $Y \to Z$ ...
- 62.6k
4
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Nerve theorem for simplicial sets
One aim of my recent article on the Nerve Theorem was to give a simple proof of the result with minimal hypotheses. It applies, for instance, to covers of CW complexes by subcomplexes, and in ...
- 8,568
12
votes
Accepted
Product structure in Milnor exact sequence
Let $P$ be the wedge of all the $X_i$s. Up to homotopy equivalence, $X$ is the homotopy coequalizer of the identity and the shift maps from $P$ to itself. The Milnor exact sequence arises by analyzing ...
- 10.9k
2
votes
Are monomorphisms in an $\infty$-topos preserved by $0$-truncation?
I'm not an expert in ($\infty$-)topos theory, so someone would have to check that this really translates to a proof of what you want, but you can prove the following in homotopy type theory, which is ...
- 218
7
votes
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Fundamental group of the homeomorphism group of a compact manifold
Let $X$ be a compact topological manifold. Since $\mathcal{H}(X)$ is a separable metric space and $S^n$ is compact, the mapping space $C(S^n,\mathcal{H}(X))$ in the compact-open topology is also a ...
- 5,221
9
votes
Accepted
Nilpotency of generalized cohomology
$\newcommand\smashtilde[1]{\smash{\tilde{#1}}}$Yes, by the Atiyah–Hirzebruch spectral sequence applied to the identity map: $$E^{pq}_2=\tilde H^p(X,h^q(pt))\Rightarrow \smashtilde h^{p+q}(X).$$ Here $\...
- 23.4k
2
votes
Equivariant cohomology of fixed points using the localisation theorem
Like Andy's answer, this is about alternate proofs of the inequality in Theorem 1, which is, as was mentioned, perhaps due to Ed Floyd in a 1952 paper.
This Floyd theorem obviously implies the 1941 ...
- 10.9k
2
votes
Accepted
Finding $\mathbb{C}(u,v)$ such that $\mathbb{C}(u,v,x^p+y^p)=\mathbb{C}(x,y)$, for every prime number $p$
Not much can be said about $F$. The second conditions should hold for $\mathbb C(u,v)$ for $u$ and $v$ two "generic" algebraically independent polynomials.
For an explicit construction, take ...
- 141k
1
vote
Equivariant cohomology of fixed points using the localisation theorem
Since you also asked about alternate proofs of Theorem 1, there is a proof that does not use the localization theorem of a stronger version of the inequality (due to Floyd) in my notes Smith theory ...
- 44.2k
2
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Equivariant cohomology of fixed points using the localisation theorem
I am just posting my comments as one answer.
Let $\mathbb{S}^1$ denote the one-dimensional circle Lie group. The classifying space $B\mathbb{S}^1$ is simply connected with integral cohomology ring ...
3
votes
LS category of 4-manifolds with free fundamental group
Since we have settled on an argument in the comments, let me post it as an answer.
We have to show that a closed $4$-manifold with nontrivial free $\pi_1(M)$ does not have $\mathrm{cat}(M)=1$. Indeed, ...
- 9,507
13
votes
Accepted
Low dimensional homotopy groups of $\operatorname{Top}(4)$
It is $\mathbb{Z}/2 \oplus \mathbb{Z}/2$, see this note.
- 8,098
3
votes
Accepted
Unimodular intersection form of a smooth compact oriented 4-manifold with boundary
Yes, $H_1(\partial X)$ can be torsion, in which case it has to be metabolic (with respect to the linking form).
To see this, consider any rational homology 3-sphere $Y$ that bounds a rational homology ...
- 10.6k
3
votes
Topological Properties of Subsets of $R^{m}$ induced by Smooth Manifolds in Matrix Spaces
Let me write $\mathbb M\cdot p$ for the set $\{Mp,M\in\mathbb M\}\subset\mathbb R^m$. Clearly it is a linear image of a manifold, which already gives you some amount of information. However, it may ...
- 3,174
17
votes
Is automorphism on a compact group necessarily homeomorphism? How about N-dimensional torus？
Suppose that $G$ is a compact Hausdorff group. Let's call $G$ semisimple if it is connected and perfect (i.e. $G=[G,G]$). Then, according to Theorem B in
Braun, O.; Hofmann, Karl H.; Kramer, L., ...
- 10.4k
3
votes
Accepted
Why is the Vietoris–Rips complex $\operatorname{VR}(S, \epsilon)$ a subset of the Čech complex $\operatorname{Čech}(S, \epsilon\sqrt{2})$?
$\DeclareMathOperator\Cech{Čech}\DeclareMathOperator\VR{VR}$
I'm the OP. Much thanks to @alesia for pointing me to Jung's Theorem; I have a valid proof now.
The inclusion $\VR(S, \epsilon) \subseteq \...
3
votes
Accepted
Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$
This is written in Némethi's book Normal surface singularities, Example 5.1.17. (He actually talks about the more general case of complete intersections of Brieskorn-type.)
He refers to Jankins and ...
- 10.6k
2
votes
How to determine the LS category of branched covers?
There's no relation, because every PL manifold is a branched covering of a sphere; see J. W. Alexander, Note on Riemann spaces, 1920, Bull. Amer. Math. Soc. 26.
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3
votes
Accepted
Cohomology of the complement of a subvariety
No. Take $X = \mathbb C^d$, $Y$ a point $P$. Consider a line $L$ through $P$ and let $\mathcal F$ be the pushforward of $\mathbb Q$ from $L \setminus P $ to $U = \mathbb C^d \setminus P$, so that $j_* ...
- 141k
13
votes
Accepted
Detecting a PL sphere and decompositions
To answer question 1:
There are fast (and easy to implement) algorithms to recognize the zero-, one-, and two-dimensional spheres.
Recognising the three-sphere is fast in practice. An algorithm to do ...
- 27k
6
votes
Accepted
A topological space has the homotopy-type of a CW-complex, then is it locally contractible?
The best you can do is that if $X$ is homotopy equivalent to a locally contractible space, then $X$ is semilocally contractible: every $x\in X$ has a neighborhood $U$ whose inclusion $U\subseteq X$ is ...
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4
votes
Accepted
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
Sometimes quotients behave poorly with respect to homotopy. Thatfor it is better to take the homotopy quotient, e.g. the mapping cone of the inclusion. Its homotopy type only depends on the homotopy ...
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