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

**5**

votes

**1**answer

201 views

### endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra:
Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let ...

**6**

votes

**3**answers

356 views

### Simplest example of non-trivial Toda bracket in spaces

Many sources give an easy definition of a Toda bracket $\{f,g,h\}$ of appropriate maps $W \to X \to Y \to Z$ in spaces as a subset of the homotopy classes of maps $[\Sigma W, Z]$ (for example, ...

**1**

vote

**1**answer

161 views

### Universal bundles: construction of the map associated to a group homomorphism

For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG ...

**12**

votes

**0**answers

318 views

### What is the determinant of Poincare duality?

For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant
$$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$
functorial with respect to ...

**3**

votes

**1**answer

471 views

### How nilpotent is the ring of stable homotopy groups of spheres?

Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?

**4**

votes

**1**answer

254 views

### Which framed manifolds are in the image of J?

The stable homotopy groups of spheres are in natural correspondence with framed cobordism classes of framed manifolds.
Is there a good way to tell if the class of framed manifold is in the image of ...

**4**

votes

**1**answer

263 views

### Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold

Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a ...

**5**

votes

**1**answer

180 views

### Analytic maps in homotopy classes

Let $\mathcal M$ be a compact connected real-analytic manifold. It is well known that every continuous map $f\colon\mathcal M\to\mathbb S^1$ is homotopic to a smooth map. My question is the following. ...

**2**

votes

**1**answer

109 views

### Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...

**0**

votes

**0**answers

83 views

### extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

**4**

votes

**2**answers

229 views

### stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...

**0**

votes

**0**answers

53 views

### Cap product in strong homology theories

I'm looking for a (co)homology theory for compact metric pairs with cup/cap product that equals to singular (co)homology for CW complexes, the cap product equals to the "singular" cap product for CW ...

**4**

votes

**1**answer

188 views

### Question about the fundamental group of rational homology 3-spheres

By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that ...

**10**

votes

**1**answer

216 views

### Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.
Not every ghost map ...

**2**

votes

**0**answers

60 views

### Codimension $k$ homeomorphism extensions

Let $f:D \to D$ homeomorphism of $k$ codimension manifold (closed, compact, without boundary) to itself. ($D \subset \mathbb{R}^{n+k}$). For which $f$ does homeorophism $g: \mathbb{R}^{n+k} \to ...

**0**

votes

**0**answers

108 views

### Zero Dimension Intersection

Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class ...

**0**

votes

**1**answer

147 views

### Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.
For which $M$, ...

**2**

votes

**1**answer

183 views

### Lower dimensional Pin cobordisms

I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" ...

**3**

votes

**1**answer

293 views

### On the fundamental group of closed 3-manifolds

I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...

**6**

votes

**0**answers

203 views

### Topological quotient Hopf-algebras and “change-of-rings”

One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. ...

**3**

votes

**2**answers

270 views

### If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible?

Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly ...

**2**

votes

**1**answer

117 views

### Explicit reference on generator of $H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ identified with second Chern class of standard representation

It is extremely well-known that $H^*(BQ_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and ...

**6**

votes

**1**answer

163 views

### Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...

**5**

votes

**1**answer

297 views

### Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...

**10**

votes

**2**answers

1k views

### The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis?
Please see : Most interesting mathematics mistake?
Added 1: According to their method, ...

**2**

votes

**0**answers

100 views

### Which reflexive coequalizer diagrams are projectively cofibrant?

Consider the walking reflexive pair category W,
which consists of two objects 0 and 1 and three generating
morphisms f: 0→1, g: 0→1, and h: 1→0
satisfying the relation fh=gh=id₁.
Consider the ...

**7**

votes

**1**answer

266 views

### Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize.
All manifolds are closed, smooth and have dimensions $n\ge 5$.
The Atiyah-Shapiro-Bott-Orientation gives ...

**14**

votes

**5**answers

977 views

### What's special about the Simplex category?

I have been wondering lately what makes simplicial sets 'tick'.
Edited
The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of ...

**5**

votes

**1**answer

278 views

### Obstructions to the existence of stable (and unstable?) complex structures?

Let $V$ be a real vector bundle on a space $X$, perhaps the tangent bundle of a smooth compact manifold. I'm interested in understanding the obstructions to $V$ admitting a stable complex structure, ...

**6**

votes

**0**answers

122 views

### Picard-Brauer exact sequence for infinity categories

This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).
If $f:A\to B$ is a ...

**8**

votes

**1**answer

408 views

### Why does the singular simplicial space geometrically realize to the original space?

I have seen it claimed that (for compactly generated Hausdorff spaces) the geometric realization of the singular (internal) simplicial space is homotopy equivalent to the original space. I know how to ...

**14**

votes

**1**answer

266 views

### Cohomology class of the group extension from a principal bundle

Let $M$ be a closed connected manifold and fix a basepoint $q \in M$ and a Riemannian metric on $M$. Let $F(M)$ denote the orthonormal frame bundle of $M$. This is a principal $O(n)$-bundle over $M$ ...

**1**

vote

**0**answers

76 views

### Complex non stably trivial complex vector bundle with vanishing Chern classes

There have been some discussions of a similar question, but I'd like an example of a non-trivial class in complex K-theory of an orientable manifold, for which all Chern classes vanish. This does not ...

**0**

votes

**0**answers

206 views

### Divisibility in homology/homotopy

I have a simply-connected CW-complex $F$ of finite-type, and I know that the imprimitivity of its particular integral homology is divisible by an odd prime $p$; that is,
$$ \forall n,\exists \delta, ...

**9**

votes

**2**answers

357 views

### Is the geometric realization of a level-wise weak equivalence a weak equivalence?

For the purposes of this question a topological space will mean a compactly generated weak Hausdorff space, though I am actually somewhat flexible on what category of topological spaces we use. I ...

**3**

votes

**1**answer

212 views

### Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric
equations in $t$: $x(t)$ and $y(t)$.
Are there conditions on these two functions that guarantee
that $C$ does not self-intersect?
For example,
the ...

**23**

votes

**4**answers

1k views

### What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...

**4**

votes

**1**answer

134 views

### Homotopy groups of linearization of a space

If $X$ is a pointed space and $A$ is an abelian group, then we can form the space $A[X]$ whose points are finite formal sums $\sum a_i x_i$ with $a_i \in A, x_i \in X$ subject to some natural ...

**1**

vote

**0**answers

64 views

### restriction to the boundary in Morse theory

Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...

**3**

votes

**1**answer

160 views

### Configuration spaces of trees are Eilenberg-MacLane spaces

I'm reading Ghrist's paper "Configuration spaces and braid groups on graphs in robotics".
In this discussion, counterexamples are shown for both Theorem 2.3 and the implication "Theorem 2.3 ...

**6**

votes

**1**answer

213 views

### Group actions in a homotopy category

Let $M$ be a model category and $G$ a finite group, and equip the category $M^G$ of $G$-objects in $M$ with, say, a projective model structure. Then there is a canonical functor
$$\mathrm{Ho}(M^G) ...

**9**

votes

**0**answers

208 views

### Goodwillie calculus and morphisms of functors

Let $F,G: \mathcal{T}\to \mathcal{S}$ be two functors from topological spaces to spectra (or topological spaces) and let $s: F\to G$ be a morphism between them.
Suppose $F$ and $G$ are analytic and ...

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votes

**0**answers

92 views

### When is the projection of an induced fibration trivial on cohomology?

Let $p: E\to B$ be a fibration, and let $f: A\to B$ be a continuous map. In my applications, $E$ and $B$ are finite complexes, but $A$ need not be. Form the pullback
$$
\begin{array}{ccc} W & \to ...

**9**

votes

**1**answer

369 views

### The homotopy of universal Thom spectrum

Let $S^0_p$ be the $p$-adic sphere spectrum. Let $GL_1(S^0_p)$ be the set of unit componen of $\Omega^{\infty}S^0_p$. For any map $ X \to BGL_1(S_p^0)$ we get a Thom spectrum call it $Mf$. Now ...

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votes

**1**answer

164 views

### spectral sequence with non-trivial action on coefficients

Set-up:
Consider the trivial extension, where $p$ is the projection onto the $\mathbb{Z}_2$ component,$$1\rightarrow N\rightarrow N\times\mathbb{Z}_2\xrightarrow{p}\mathbb{Z}_2\rightarrow 1$$
Define ...

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votes

**0**answers

165 views

### Which ring spectra have some kind of exponential map turning addition into multiplication?

This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time.
What will follow is sort of vernacular but whether it can be ...

**5**

votes

**3**answers

257 views

### coend formulation of homotopy colimit

I have recently been trying to learn about homotopy colimits. In so doing, I have gone through way too many papers, and now I can't find the one which introduced homotopy colimits via a coend ...

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votes

**2**answers

178 views

### Question about lower homology class of cobordism

Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...

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votes

**0**answers

106 views

### spectral sequence differential for cobordism

From page 6 of these solutions:
the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the ...

**6**

votes

**1**answer

96 views

### Hochschild-Serre spectral sequence and non-trivial action on coefficients

Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action ...