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

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5
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1answer
324 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 ...
11
votes
2answers
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 : According to their method, ...
2
votes
0answers
106 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
1answer
273 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
5answers
1k 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
1answer
289 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
0answers
132 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 ...
9
votes
1answer
425 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 ...
15
votes
1answer
275 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
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0answers
78 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
0answers
208 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
2answers
376 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
1answer
222 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
4answers
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
1answer
136 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
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0answers
69 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
1answer
171 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
1answer
227 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) ...
10
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0answers
210 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 ...
7
votes
0answers
95 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 ...
10
votes
1answer
383 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 ...
2
votes
1answer
172 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 ...
4
votes
0answers
173 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
3answers
265 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 ...
4
votes
2answers
183 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, ...
4
votes
0answers
112 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
1answer
115 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 ...
0
votes
1answer
138 views

Gauss Sums over “semisimple spherical tensor category”?

I read on the arXiv the following: Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let $\mathbf{\Gamma}$ be the set of isomorphism classes of simple ...
8
votes
0answers
216 views

Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...
9
votes
0answers
102 views

Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?

It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the ...
2
votes
0answers
154 views

Generator of $\pi_3(SU(4))$ in Mimura-Toda

In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as ...
26
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0answers
702 views

Finite complexes whose homotopy groups are not “finitely generated”

I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$. It seems likely that ...
0
votes
0answers
84 views

Dimension of category of sheaves [duplicate]

Let $k$ be a field. Consider the category $Shv(\mathbb{R}^n)$ of sheaves of $k$-vector spaces on $\mathbb{R}^n$. What is the cohomological dimension $d$ of $Shv(\mathbb{R}^n)$? I know that $d \in ...
2
votes
1answer
213 views

How to calculate the first and second homotopy groups of the following space constructed from $U(4)$

In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how ...
4
votes
0answers
226 views

The general Smith homomorphism in bordism

The Smith homomorphisms are a family of homomorphisms between equivariant bordism groups in different dimensions. One example that is known to be an isomorphism is the map $$ \tilde\Omega_{d+1}^{\rm ...
5
votes
0answers
82 views

A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...
8
votes
1answer
223 views

Shriek push-forward for parameterized spectra

In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow ...
3
votes
3answers
267 views

Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
3
votes
1answer
126 views

Does every canonical decomposition of the intersection form come from a canonical homology basis?

Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...
12
votes
3answers
551 views

Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?

Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact ...
4
votes
1answer
200 views

Generators of cohomology groups of higher push-forward sheaves

Let $\phi:S\rightarrow \mathbb{P}^1$ be an elliptic fibration of a compact complex surface. Assume that there is a multiple section $s$ of $\phi$. Is it true that $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...
2
votes
1answer
158 views

Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...
5
votes
1answer
194 views

Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...
16
votes
2answers
557 views

What's the relationship between these two isomorphisms involving G and T?

Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms. Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ...
5
votes
1answer
215 views

How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class

As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class. I know that $H_2(\mathbb{C}P^2 \# \mathbb{C}P^2; ...
13
votes
1answer
354 views

Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...
7
votes
0answers
126 views

$v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...
11
votes
3answers
746 views

Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...
3
votes
0answers
163 views

Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds

Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...
1
vote
2answers
145 views

Pontryagin square of Stiefel-Whitney classes and Pontryagin classes

On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class: $\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; ...