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

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Contractible chain complex from non-contractible space

Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= ...
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0answers
25 views

Map which is null-homotopic on compacts

This is the missing ingredient towards answering my previous question. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). ...
6
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2answers
340 views

Is limit of null-homotopic maps null-homotopic?

The question is motivated by my failed comment to this one. Let $M$ and $N$ be path connected locally compact, locally contractible metric spaces (you may assume that they are manifolds). Let $\...
4
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81 views

Fibrant objects in $\mathbb{S}$-local model structure on $Top_*$

Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$. Equipping $Top_*$ with the Quillen model structure (...
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107 views

Conceptual definition of derived functors allowing for quick proof of comparison theorems for sheaf cohomology

There are several approaches of increasing sophistication and simplicity to defining derived functors. I know of universal $\delta$-functors and Kan extensions along localizations. More definitions ...
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1answer
61 views

Relation between compact vertical cohomology and local cohomology groups

I'm reading the books by Bott & Tu and Milnor & Stasheef simultaneously. The following is my doubt: The Thom isomorphism in Bott & Tu is obtained as $H_{cv}^{*+n}(E)\rightarrow H^*(M)$, ...
4
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225 views

Geometric interpretation of nonconnective, non-coconnective chain complexes / spectra?

Let's stipulate that Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about ...
3
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70 views

Building strong topologies on the space of continuous functions

Let $\{K_k\}_{k=1}^{\infty}$ be a compact exhaustion on $\mathbb{R}^n$ (ie $\bigcup_{k=1}^{\infty} K_k = \mathbb{R}^n$). Equipe each $X_k:=\{f \in C(\mathbb{R}^n):\, \operatorname{supp}(f)\subseteq ...
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1answer
131 views

Explicit description of exponentials of etale spaces

It is well known that the category $\mathit{Sh}(X)$ of sheaves of sets on a topological space $ X $ is a topos. On the other hand, there exists a natural equivalence of categories between $\mathit{Sh}(...
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0answers
116 views

Algebras of the cone monad on Top?

Let us work in Top, the category of topological spaces - although the reader is welcome to replace this by their favorite convenient category of topological spaces. If $X,Y$ are spaces, let $X\ast Y$ ...
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181 views

Applications of the Atiyah-Patodi-Singer eta-function $\eta(s)$

The eta function of a differential operator was used by Atiyah, Patodi and Singer to derive their famous index theorem, and is given by $$ \eta(s)=\sum_{\lambda\neq 0}(\mathrm{sign}\lambda)|\lambda|^...
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124 views

Where can I read about non-principal obstruction theory?

Most treatments of obstruction theory assume a principal Postnikov tower. I have the vague sense that if one uses cohomology with local coefficients, one does not need to make any assumptions on one's ...
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93 views

When the local system of coefficients are simple in the Leray-Serre spectral sequence

Let $F\to E\to B$ a fibration and $\{E_{r}^{\ast,\ast},d_{r}\}$ the Leray-Serre Spectral sequence converging to $H^{\ast}(E;R),$ such that $$E_{2}^{p,q}=H^{p}(B;\mathcal{H}^{q}(F;R))$$ is the ...
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1answer
97 views

Conditions for certain inclusion functor to preserve internal homs

Suppose that $\mathcal{C}$ is a locally cartesian closed right proper Quillen model category for which all objects are fibrant. Let $x$ be an object of $\mathcal{C}$. Let $\mathbb{F}$ denote the class ...
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2answers
119 views

About submersion and sections

Let $\pi:X \rightarrow Y$ be a surmersion (surjective submersion) between closed manifolds. 1) Is there any obstruction to the existence of a "multi-valued" section $s$ of $\pi$ such that $\pi \circ ...
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98 views

Describing the THH of function spectra?

Are there any results describing the $THH$ of spectra of the form $F(X, E)$ where $X$ is a space (say, finite CW) and $E$ is a (nice enough) ring spectrum? I'm happy to put various (further, or ...
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104 views

Foliated circle bundles whose Euler class is torsion

Let $X$ be a closed manifold. By a foliated circle bundle $E \rightarrow X$ we mean a circle bundle over $X$ with total space $E$ and structure group $Diff^+(S^1)$, and a codimension one foliation of $...
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1answer
133 views

Showing that a certain simplicial set has levelwise small cardinality

My question concerns Section 2 of the article "The Simplicial Model of Univalent Foundations (after Voevodsky)" (https://arxiv.org/pdf/1211.2851.pdf). Let $\alpha$ be a strongly inaccessible cardinal....
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192 views

Rational homotopy theory [closed]

I am trying to read the paper "Rational homotopy theory " by Quillen and am stuck with the notion of complete augmented algebra. He had defined the complete augmented algebra and I don't understand ...
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137 views

Simplicial simple homotopy vs. cellular simple homotopy

I recently started reading up on simple homotopy theory. Here is a question I stumbled upon. In his 1938 Paper Simplicial Spaces, Nuclei and m-Groups Whitehead introduced the notion of elementary ...
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1answer
335 views

About the cohomology of $BG^\delta$. Making a Lie group discrete

Let $G$ be a connected Lie group. Recall that the topological group $G^\delta$ is $G$ endowed with the discrete topology. The inclusion $G^\delta \to G$ induces a map between the classifying spaces $\...
10
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139 views

Embedding 2-complexes null homotopically into 2-complexes

Whitehead's conjecture states that if $L$ is an aspherical 2-complex and $K$ is a subcomplex of $L$, then $K$ is also aspherical. It is known by work of Howie and Luft that if the Whitehead ...
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79 views

Any cobordism invariant made of “characteristic classes”, on unorientable manifolds, must be a mod 2 class?

For any cobordism invariant (or simply bordism invariant) quantity $\omega$ that satisfy the conditions: $\omega$ can be fully decomposed from the cup product of characteristic classes (such as ...
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99 views

Clarify formula for Steifel-Whitney (Poincaré dual) homology classes in a barycentric subdivision?

Let $X$ be a triangulated manifold of dimension $n$. Let $[W_{n-p}] \in H_{n-p}(X,\mathbb{Z}_2)$, be the homology class that's Poincaré dual to the $p$-th Stiefel-Whitney class $[w_p] \in H^p(X,\...
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1answer
175 views

Riemann-Hilbert correspondence versus Simpson correspondence

I couple of days ago, I asked extensively the same question on Stack-exchange (see https://math.stackexchange.com/questions/3592151/riemann-hilbert-correspondence-versus-simpson-correspondence)and go ...
8
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1answer
288 views

Cobordism invariants: topological v.s. geometric

Some cobordism invariants are not cohomology classes. Such as the $\mathbb{Z}_{16}$-valued eta invariant $\eta$ of $\Omega_4^{Pin^+}$, the $\mathbb{Z}_8$-valued Arf-Brown-Kervaire invariant ABK of $\...
9
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1answer
135 views

Is a normal covering of the total space of a principal bundle also a principal bundle?

Let $G\to E \to B$ be a principal $G$-bundle over $B$. Take a normal covering $\bar{E}$ of $E$. Does $\bar{E}$ admit a principal bundle structure? Namely, $\bar{G}\to \bar{E}\to \bar{B}$, such that $\...
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99 views

Integration on an non-orientable manifold [closed]

Suppose $M_n$ is a $n$ dimensional non-orientable manifold. I am interesting in knowing whether the following statements are true: A characteristic class $w_{n}^{(p)} \in H^{n}(M_n, \mathbb{Z}_p)$...
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0answers
135 views

The Seiberg-Witten equations for forms

I define equations like the Seiberg-Witten equations for forms of a riemannian four-manifold $(M,g)$. $\alpha \in \Lambda_+(TM)$ and $\theta \in \Lambda^1(TM)$. $$ d\alpha+\theta \wedge \alpha=0 $$ $$ ...
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2answers
283 views

How does the Steenrod algebra act on $\mathrm{H}^\bullet(p^{1+2}_+, \mathbb{F}_p)$?

Let $p$ be an odd prime. The $\mathbb F_p$ cohomology of the cyclic group of order $p$ is well-known: $\mathrm{H}^\bullet(C_p, \mathbb F_p) = \mathbb F_p[\xi,x]$ where $\xi$ has degree 1, $x$ has ...
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0answers
88 views

Defining the cospecialization in topology

Below is an excerpt from part V of Deligne's Étale cohomology - starting points. Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
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165 views

What is a morphism of ∞-sites?

Recall that a morphism of sites is a covering-flat functor that preserves covering families. Morphisms of sites can be identified with those geometric morphisms of induced toposes for which the ...
2
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1answer
71 views

Persistent homology: maximum/upper bound of number of points in point cloud

I am interested in the topic of persistent homology (topological data analysis). According to what I read, there is some roadblock in the analysis of "big data" using persistent homology as it is ...
11
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1answer
309 views

Complex cobordism and Chern numbers

Let $L$ be the Lazard's universal ring, and $R=\mathbb{Z}[b_1,b_2,\cdots,b_n,\cdots]$, regarded as a graded ring with the degree of $b_i$ equal to $2i$. Let $\theta: L\rightarrow R$ be the ...
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0answers
87 views

Explicit action of the Dehn twist in the homology of punctured sphere with local coefficients

Let $X=\mathbb{P}^1\setminus S$, where $S=\{a_1,\dots, a_k\}$ is a finite subset of $\mathbb{P}^1$ and we may assume that $|S|\geq 4$. Let $\mathbb{L}$ be a local system on $X$ given by a monodromy ...
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98 views

Tangent bundle of symmetric product of surface

Hello everybody please help me with this doubt: Let $f:\mathbb{P}^{1} \rightarrow \mathrm{Sym}^{d}(X)$, where $X$ is a smooth projective surface, $\mathbb{P}^{1}$ is a projective line and $\mathrm{...
4
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1answer
306 views

Every _______ $d$-manifold has an $S$-structure

I am looking for some analogous nontrivial but known statements and references about statements of the form: Every _______ $d$-manifold has an $S$-structure. Here _______ is a placeholder for ...
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0answers
120 views

A confusion about geometric fixed points via spectral Mackey functors and smashing localisations

Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\...
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4answers
1k views

The homology of the universal covering space, why so difficult to compute

Let suppose that we are given a connected CW-complex $X$, such that we know All its homology groups. All its homotopy groups, in particular we know $\pi_{1}(X)$. As far as I know there is no ...
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0answers
86 views

Does the self-homeomorphism group of a finite CW complex have CW homotopy type?

Let $X$ be a finite CW complex and form the group $\mathcal{H}(X)$ of self-homeomorphisms $X\xrightarrow{\cong}X$, furnishing it with the compact-open topology. Under the assumptions on our space $\...
3
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1answer
137 views

Semi-cocartesian operads

Context: In this interesting blog post, Mike Shulman indicates an approach for defining generalized types of operads. If I interpret the details correctly, (edit: which I apparently did not,) the idea ...
2
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1answer
161 views

Branched coverings of surfaces over an infinite branching locus

So let me start by reviewing my understanding of branched coverings of surfaces in a simple case. If I start with a disk, and two marked points with integer values, say $n$ and $m$ assigned to them, ...
11
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1answer
403 views

Realizing inner automorphisms on Eilenberg-MacLane spaces

Let $G$ be a discrete group and let $(X,x_0)$ be a based Eilenberg-MacLane space for $G$, so there is a fixed isomorphism $\pi_1(X,x_0) = G$ and the universal cover $\widetilde{X}$ is contractible. ...
4
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2answers
449 views

How to compute fundamental groups of closed surfaces without using Van-Kampen theorem?

Denote $X=mP^2$ the sphere glued with $m$ Mobius bands. It has a polygon representation $a_1a_1...a_ma_m$, i.e. it's a quotient by a $2m$-sides polygon $P$. Let $o$ be the center point of $P$, $x_0$ a ...
7
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1answer
207 views

How to identify cup product with intersection

What's the standard generalization and reference for the following statement: If two oriented submanifolds $L$, $L'$ of an oriented compact manifold $M$ intersect transversally, then the Poincare ...
4
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1answer
237 views

Understanding the proof of Proposition 5.1 of Segal's paper: Classifying spaces and spectral sequences

Let A be a semi-simplicial space and $k^*$ be a generalised cohomology theory as in This paper proposition 5.1. Using the natural filtration of the realisation of $A$ and then using the staircase ...
3
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1answer
175 views

$G$ uncountable implies $K(G,1)$ is not a finite CW complex

I have read that $H^i(K(\mathbb{R},1)$) has rank $2^\omega$ for any $i\in \mathbb{N}$ (see Thurston's comment here Nontrivial finite group with trivial group homologies?) therefore $K(\mathbb{R},1)$ ...
4
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0answers
96 views

When is a manifold boundary a deformation retract of its open neighborhood?

For this question a manifold is a locally upper-Euclidean Hausdorff space; paracompactness or second-countability is not assumed, and boundary may be present. Let $M$ be a manifold. What are ...
8
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1answer
563 views

For which G is BLG weak homotopy equivalent to LBG?

Let $G$ be a (Edit: path-)connected topological group. Under what additional hypotheses on $G$ is it true that $LBG$ is a classifying space for $LG$? (or, I guess equivalently, when is $LBG \sim BLG$?)...
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0answers
194 views

Homological and homotopical equivalence of complex analytic varieties

Consider a map between two complex analytic varieties of finite type $f:X\to Y$. Suppose that $f$ induces isomorphisms on cohomology with (constant) integral coefficients. Under what reasonable ...

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