Tagged Questions
Homotopy, stable homotopy, homology and cohomology, homotopical algebra.
21
votes
3answers
568 views
Are the higher homotopy groups of the Hawaiian earring trivial?
The fundamental group of the Hawaiian earring is very complicated, but since it's "1-dimensional" one might guess that the higher homotopy groups vanish. Do they? Since the Hawaiian earring does not ...
0
votes
0answers
101 views
Where are there defined objects between gerbes and bundle gerbes?
Consider a special kind of/something like a gerbe where there is first given local trivialization data with equivalence over 2-fold overlaps but not isomorphism.
Does this exist in the literature?
1
vote
4answers
202 views
Homology of infinite intersection
If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...
5
votes
1answer
189 views
Can eta invariant be written in terms of topological data?
The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
4
votes
1answer
147 views
Section of the homology functor on spectra
Consider the (reduced) homology functor $H_*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ ...
2
votes
2answers
285 views
Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [closed]
There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that ...
4
votes
2answers
519 views
Why Cech cohomology does not compute sheaf cohomology on an open annulus
Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets:
$U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...
3
votes
0answers
77 views
understanding the definition of $\infty$-operad of module objects
I'm just trying to understand the following definition:
Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:
Let $O^\otimes$ be ...
5
votes
1answer
234 views
Finite group acting on sphere
Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
8
votes
0answers
198 views
Lift chain complex from $F_2$ to $Z$
We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...
-2
votes
0answers
58 views
Complexity of a function [closed]
I am looking for a natural definition of the complexity a function. If the image is discrete, i was thinking it could be: consider the preimage of an element of the image, count the number of ...
3
votes
1answer
108 views
In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?
Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...
8
votes
2answers
353 views
Eilenberg-MacLane Spaces of “large” groups
It is well-known that if $G$ is a discrete group, then $BG=K(G,1)$. I'm interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups but equipped ...
3
votes
1answer
214 views
sphere bundles over spheres
Localized at an odd prime there is a space $B_k$ which sits in a fibration $S^{2k+2p-3}\rightarrow B_k \rightarrow S^{2k-1}$ and has homology $H_{\ast}(B_k;\mathbb{Z}/p\mathbb{Z})\cong ...
27
votes
1answer
558 views
For which maps $S^1\to S^1$ is the winding number defined?
There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number:
• Continuous maps:
Using the unique path lifting property of the universal covering map $\mathbb R\to ...
0
votes
0answers
106 views
Homotopy equivalent type of a knot complement [closed]
Let $S$ denote the bounded complement of a tame knot in $S^3$,then $S$ is homotopy equivalent to a finite 2-dimensional simplicial complex $K$ [Milnor's paper "infinite cyclic covering"],I do not ...
13
votes
0answers
163 views
Unoriented bordism and homology, reference?
The following has undoubtedly been known to the experts for years, but I only noticed it the other day. Can anyone give a reference?
One can prove Thom's theorem to the effect that every mod $2$ ...
11
votes
2answers
340 views
Truncations of E_infinity algebras
In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an ...
1
vote
0answers
71 views
Homology of the fixed points of the singular complex of a G-space
I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me.
Suppose $X$ is a topological space and $G$ a ...
-1
votes
0answers
154 views
Algebraic topology vs. category theory [migrated]
I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
11
votes
0answers
199 views
p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)
I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
4
votes
2answers
249 views
Fundamental group of a manifold with an $S^1$-action
Let $M$ be a compact connected manifold with an $S^1$-action. Suppose that $S^1$ has a fixed point in $M$. Is it true that $\pi_1(M)=\pi_1(M/S^1)$?
I is there some reference or a short proof of this ...
3
votes
2answers
137 views
Nielsen-Thurston classification of homeomorphisms for open surfaces?
In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
14
votes
0answers
307 views
What is to tmf as KR is to KO?
The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
1
vote
2answers
159 views
How can I prove that Hopf fibrations are the only ones with fiber, total space and base space homeomorphic to spheres?
I know that Hopf fibrations (the four ones) are the only ones that have the form
$S^k \to S^m \to S^n$, but I never seen a proof. Could anyone link me a paper or text where this is proved, or prove it ...
1
vote
0answers
154 views
Units of a ring spectrum
Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$?
A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...
5
votes
1answer
196 views
Equivalent fomulations of Bott periodicity
Is there an easy way to see the equivalence of the two statements of Bott periodicity.
$$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and
$$K(X)\otimes K(S^2) \cong K(X\times S^2)$$
4
votes
0answers
125 views
Third cohomology of mapping class group
I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...
2
votes
1answer
341 views
Two questions about the grassmannian
There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference.
...
-1
votes
0answers
36 views
connected components of a real algebraic variety and its hyperplane section
Let $X$ be a smooth projective variety of dimension at least $2$ over the real numbers $\mathbb{R}$ and $H \subset X$ a smooth hyperplane section. Assume that the set of real points is non-empty for ...
2
votes
1answer
337 views
Topological degree and polynomial degree
Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
0
votes
0answers
136 views
Symplectic submanifolds in $\mathbb{R}^4$
Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
3
votes
1answer
211 views
Compactly supported cohomology of homotopy equivalent manifolds
Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?
21
votes
6answers
1k views
Down-to-earth expositions of Hodge theory
What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory?
Of course, since I haven't found a (for me) readable introduction, I don't know what I ...
-2
votes
1answer
218 views
Can you overcome the 6th degree obstruction?
I read and am still thinking about a 3-year old paper from the Danish-Norwegian "Niels Abel Journal". Two authors, named Somethingson (not Jacobson) and another Somethingelseson (still not Jacobson), ...
0
votes
1answer
81 views
Extending binary operation used by homotopy classes
There is this operation you learn in algebraic topology when working with homotopy groups and loops i.e. paths on a topological space $X$, $p:[0,1]\rightarrow X$ with $p(0) = p(1)$. Basically it is ...
11
votes
3answers
367 views
rationalization of classifying spaces
This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:
Let $G$ be a simply-connected topological group. In particular, it is an ...
17
votes
4answers
1k views
Quillen's motivation of higher algebraic K-theory
Almost the same question was already asked on MO Motivation for algebraic K-theory?
However, to my taste, the answers there consider the subject from a more modern point of view.
When I open a book ...
8
votes
2answers
386 views
The cohomology plus what characterizes the rational homotopy type?
For simplicity let me work only with connected and simply connected spaces. "Space" will mean a space of this type.
A space is rational if its homotopy groups are rational vector spaces ...
2
votes
2answers
111 views
Which graphs generate a matroidal independence complex?
The independence complex $I(G)$ of a graph $G=(V,E)$ has as point set the vertex set $V$ and as simplices the independent sets of $G$.
Now, if $G$ is a well-covered graph (where all maximal ...
3
votes
0answers
86 views
Special representations for morphisms of spectra from a smash product
I follow the definitions of spectra, function, morphism, found on Switzer, chapter 8.
After definition 8.15 where he defines homotopies of spectra, he says:
In terms of cofinal subspectra we can ...
5
votes
0answers
204 views
Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?
Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex:
We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by ...
2
votes
1answer
99 views
Practical application of lattice knots
I am looking for examples of practical applications of lattice knots. Any help?
1
vote
1answer
196 views
Possible homotopy-theoretical approach to Gauss-Bonnet
Consider a vector bundle $V\to E\to X$ with fiber $V$, with structure group $G$, and $X$ path-connected. Consider a connection $\nabla$ on $E$. Then for any loop $L$ in $X$, based at $p$, we have a ...
14
votes
1answer
1k views
What have simplicial complexes ever done for graph theory?
(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...)
Are there examples of results in "classical" [*] graph theory that have
been achieved by using simplicial ...
3
votes
1answer
107 views
Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures
Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...
2
votes
2answers
189 views
Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra
Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum ...
5
votes
0answers
122 views
Is Euler-characteristic of a simplicial complex on $n$ vertices and $f$ facets at most $n^{O(\log f)}$?
(Definition: Facet = Maximal Face)
This question is a continuation of the previous one that I had asked a couple of years ago: Is Euler characteristic of a simplicial complex upper bounded by a ...
1
vote
1answer
96 views
Is the equivariant Gysin map an $H_G^*(\text{pt})$-module morphism?
Let $G$ be a complex reductive group, $X$ a smooth projective variety on which $G$ acts algebraically, and $Y \subseteq X$ a $G$-invariant smooth closed subvariety such that $X\setminus Y$ is also ...
6
votes
1answer
189 views
Are there models for homotopy colimits and limits of simplicial sets that generalize Kan's suspension and loop functors?
Consider the category C of pointed simplicial sets.
The pair of functors X∈C↦X∧S¹∈C and Y∈C↦Map(S¹,Y)∈C
models the suspension and loop functors on the underlying ∞-category of C.
There is another ...
