# Tagged Questions

Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

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17 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

**0**answers

99 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

**4**answers

200 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

**1**answer

187 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

**1**answer

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

**2**answers

282 views

### Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [on hold]

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

**2**answers

517 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

**0**answers

75 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

**1**answer

230 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**

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**0**answers

196 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 $B_0,\dots,B_k$, ...

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**0**answers

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

**1**answer

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 ...

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**2**answers

352 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

**1**answer

213 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

**1**answer

555 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 ...

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votes

**0**answers

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 ...

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**0**answers

162 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

**2**answers

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 ...

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vote

**0**answers

70 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**

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**0**answers

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 ...

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196 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

**2**answers

248 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 ...

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votes

**2**answers

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 ...

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304 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 ...

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vote

**2**answers

157 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 ...

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vote

**0**answers

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 ...

**4**

votes

**1**answer

194 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
$$K(X)\otimes K(S^2) \cong K(X\times S^2)$$

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votes

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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 ...

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votes

**1**answer

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.
...

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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

**1**answer

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 $L$ ...

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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

**1**answer

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**

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**6**answers

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 ...

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votes

**1**answer

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), ...

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**1**answer

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 ...

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votes

**3**answers

366 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 ...

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votes

**4**answers

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 ...

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votes

**2**answers

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 ...

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votes

**2**answers

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 ...

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votes

**0**answers

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 ...

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votes

**0**answers

203 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

**1**answer

99 views

### Practical application of lattice knots

I am looking for examples of practical applications of lattice knots. Any help?

**1**

vote

**1**answer

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

**1**answer

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

**1**answer

106 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

**2**answers

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 ...

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votes

**0**answers

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

**1**answer

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

**1**answer

188 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 ...