Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas ...

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110 views

### HKR generalized character theory question regarding a certain construction

In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...

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### Maps having the right lifting property against cofibrations of compact spaces

I would like to know the properties of the maps that have the right lifting property against cofibrations of compact spaces. By definition, they are acyclic Serre fibrations, but I would hope to be ...

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119 views

### Homotopy functor calculus vs functor calculus in additive categories

Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance).
Then ...

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**2**answers

282 views

### Is there any significance to Bousfield localization in the non-derived context?

The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...

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**1**answer

206 views

### How to define 0-sphere in a category with zero object?

The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \coprod \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...

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267 views

### Maps from 2-Torus to SO(3)

Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]

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158 views

### Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....

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298 views

### Do finite simplicial sets jointly detect isomorphisms in the homotopy category? [duplicate]

Let $\mathcal{H}$ denote the homotopy category associated with the Kan-Quillen model structure on $\mathbf{sSet}$. Suppose we have a map $f\colon X \to Y$ between Kan complexes, such that for every ...

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399 views

### Sphere spectrum, Character dual and Anderson dual

The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...

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73 views

### Axiomatization of the shuffle decomposition

I am trying to figure out an axiomatization for Reedy categories such that the product of representables admits a shuffle decomposition. Cisinski suggested imposing a condition, which we will state ...

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**1**answer

234 views

### Commutativity up to homotopy implies strict commutativity, for lifting problems

Suppose we have a commutative diagram
$\require{AMScd}$
\begin{CD}
A @>>> X \\
@VVV & @VVV \\
W @>>> Y\\
\end{CD}
where the map $A\rightarrow W$ is a cofibration and the ...

**16**

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**1**answer

275 views

### Milnor Conjecture on Lie groups for Morava K-theory

A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...

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329 views

### Questions about obstruction theory (Hatcher's book)

I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...

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**1**answer

190 views

### A question on eversion of (odd) spheres

At the right column of the page 654 of the paper,
R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of ...

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89 views

### Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory:
never perform quotients, add structure instead;
never require subobjects, take fibres instead.
...

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**2**answers

231 views

### Critical points and high homotopy groups

Is there any known or interesting relation between critical points (possibly degenerate, or maybe only nondegenerate) of a function on a manifold and generators/relations of high homotopy groups? I ...

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39 views

### sub relative cell complex

This is a question concerning the proposition 11.4.8 of P.Hirschhorn's book Model categories and their localizations.
The proposition 11.4.8 is an analogous, in my opinion, to the well known ...

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**1**answer

92 views

### Naturality of minimal model of a fibre bundle

$\require{AMScd}$
For rational fibrations $F \rightarrow E \rightarrow B$, we have a nice description in terms of sullivan minimal models, namely a commutative diagram of cdga's
$$
\begin{CD}
...

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205 views

### Dualizable objects in homotopy category of chain complexes

The proposition 1.9 from "Duality, Trace and Transfer" by Dold and Puppe states that:
Given a commutative ring $R$, a chain complex of $R$-modules is strongly dualizable in $Ho(Ch(R))$, the homotopy ...

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204 views

### Is there a citeable source for generators and relations of simplicial sets?

Simplicial sets can be specified using generators and relations,
in complete analogy with groups, rings, etc.
More precisely, a system of generators of relations for a simplicial set
consists of a ...

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232 views

### “Complementarity” between homotopy and cohomology [duplicate]

I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to have ...

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**0**answers

103 views

### Known obstruction for efficient computation of Stable homotopy groups?

Computation of stable homotopy groups (for example of sphere) is hard, but still, not as hard as unstable ones.
For unstable homotopy groups there are some results showing that there cannot be ...

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**1**answer

175 views

### Conceptual and practical reasons and consequences of inverting weak equivalences

Although dealing with this in one or other form for many years, to my shame this question only struck me now.
One of the most radical differences between categories of "algebraic" and "topological" ...

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198 views

### Did the Goerss-Hopkins manuscript “Multiplicative stable homotopy theory” ever appear?

A citation to "M. J. Hopkins and P. Goerss, Multiplicative stable homotopy theory, unpublished manuscript, 1996" appears in the Hill, Hopkins, Ravenel Annals paper on the Kervaire invariant. It was ...

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**1**answer

391 views

### Is there an explicit Dold-Thom theorem?

The Dold-Thom theorem tells us that we can recover the reduced homology of a pointed space $(X,x)$ via taking homotopy groups of the symmetric product:
$$\pi_i(\mathrm{Sym}^{\infty}(X,x)) \cong H_i(X,...

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190 views

### What is the group completion of finite sets with respect to cartesian product?

Let $\Sigma_+$ be the groupoid of finite pointed sets. The Barratt-Priddy-Quillen theorem tells us that the group completion of $\Sigma_+$ with respect to the symmetric monoidal structure given by ...

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146 views

### About a zig-zag of Quillen adjunctions

I have the following situation:
Three combinatorial model categories $\mathcal{K}_1, \mathcal{K}_2, \mathcal{K}_3$ such that all objects are fibrant (so no need to bother with the fibrant ...

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votes

**0**answers

102 views

### Explicit data for $E_n$-monoidal model and simplicial categories

The definition of a monoidal model category requires that the tensor product is biclosed (such is needed to ensure that the tensor product is derivable in both variables). Obviously, in the situation ...

**3**

votes

**1**answer

213 views

### Motivation for classifying vector bundles

The statement I am familiar with regarding classification of vector bundles is :
Given a paracompact space $X$. The set of isomorphism classes of rank $n$ vector bundles over $X$ is in bijective ...

**4**

votes

**1**answer

192 views

### Simplicial model categories and simplicial equivalence

Let $M$ and $N$ two very nice simplicial model categories and let $F:N\rightarrow M$ be a (nice) simplicial functor which induces an equivalence of homotopy categories, i.e. $Ho(F): Ho(N)\rightarrow ...

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**1**answer

151 views

### Homotopy of paths at the boundary

Denote by $\Gamma$ a hypersurface in $\mathbb{C}^2$, i.e. the zero locus of a polynomial of two complex variables. Denote by $X$ the complement of $\Gamma$ in $\mathbb{C}^2$. I am trying to define a ...

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**2**answers

271 views

### Connectivity of suspension-loop adjunction

Let $X$ be a $k$-connected spectrum for $k \in \Bbb{Z}$.
I want to deduce how connected the counit of $(\Sigma^\infty, \Omega^\infty)$- adjunction is, that is, how connected is the map
$$
\Sigma^\...

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165 views

### Quotient space, a fundamental group, and higher homotopy groups 2

Previously, I ask for comments/suggestions on setting up the calculation in Quotient space, homogeneous space, and higher homotopy groups. There, however, I was looking for whatever methods and tools ...

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votes

**1**answer

356 views

### detecting weak equivalences in a simplicial model category II

The question is related to the question: detecting weak equivalences in a simplicial model category
Suppose that we have a simplicial model category $M$ and denote by $M^{f}$ the full simplicial ...

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votes

**0**answers

234 views

### Non-Abelian fundamental group? — a bizarre example

For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(G_1) \to \pi_n(G_0) \to \...

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votes

**0**answers

65 views

### Bar construction and space of homotopy invariant functionals

I am introducing myself to the topic of iterated integrals using these notes
http://www.ihes.fr/%7Ebrown/ColombiaNotes7.pdf
On pag 22 is defined the following operator
$$ D : (X^1)^n \to (\mathcal{A}...

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**1**answer

169 views

### A question about Wall's construction for CW-complexes

For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...

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votes

**0**answers

197 views

### Quotient space, homogeneous space, and higher homotopy groups

Preparation and my input:
For the quotient space $G/H$, knowing the homotopy
groups of $G$ and $H$ one can determine homotopy groups from the long
exact sequence
$$
...
\to \pi_n(H) \to \pi_n(G) ...

**0**

votes

**1**answer

118 views

### detecting weak equivalences in a simplicial model category

Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. ...

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votes

**1**answer

165 views

### Localization of a model category

Let $M$ be a very nice model category (cofibrantly generated, combinatorial or cellular and left proper simplicial model category). Let $f: X\rightarrow Y$ and $g: X\rightarrow Z $ be two morphisms ...

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votes

**1**answer

90 views

### Are the fibrations between $\mathbb{A}^{1}$-local objects $\mathbb{A}^{1}$-fibration?

Let $S$ be Noetherian scheme and $(Sm/S)_{Nis}$ is the Nisnevich site of smooth schemes over $S$. The category of simplicial sheaves on $(Sm/S)_{Nis}$ is denoted
$Spc(S)$ and this category has two ...

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votes

**0**answers

70 views

### Reference Request: Equivariant Symplectic bordism

Non-equivariantly, symplectic bordism has been developed extensively by Ray, Gorbunov, and specially S. Kochman in this memoir: http://dx.doi.org/10.1090/memo/0496 Yet the coefficients ...

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**2**answers

623 views

### $(\infty,1)$ 2d TFTs

2d topological field theories $Z : \mathrm{Cob}(2) \to \mathrm{Vect}$ are classified by commutative Frobenius algebras.
What can be said about $(\infty,1)$ 2d TFTs $Z: \mathrm{Cob}(2) \to \mathcal{S}$...

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**2**answers

109 views

### Excellent monoidal model categories admit enriched fibrant replacement functors?

Let $\mathbf{S}$ be an excellent model category in which all objects are cofibrant, viewed as an $\mathbf{S}$-enriched category by its canonical self-enrichment. Then we know that there is an obvious ...

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**0**answers

108 views

### Enriched homotopy colimit and space of paths

I work in the category of $\Delta$-generated spaces. Let $G$ be the group of strictly increasing continuous bijections from $[0,1]$ to itself (they necessarily take $0$ to $0$ and $1$ to $1$). I call ...

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votes

**1**answer

212 views

### Proposition in HTT on cofibrations of categories

Proposition A.3.3.9. in Higher Topos Theory is as follows:
Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...

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votes

**1**answer

169 views

### Coefficient (or target) category for factorization homology

In the article "Factorization homology of topological manifolds" by Ayala and Francis, a symmetric monoidal $\infty$-category $\mathcal{V}$ is fixed as the target or coefficient category. This ...

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votes

**1**answer

99 views

### Two model structures of some category

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called ...

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vote

**0**answers

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### Deformation sublevel sets of functions which preserve boundary

I'm interested in proving the following fact, which seems to naturally arise from gradient flow deformations, but appears to be a bit tricky.
Consider a smooth family
$$f_s : M \to \mathbb{R}, \quad ...

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**2**answers

332 views

### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...