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

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3
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0answers
83 views

Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...
10
votes
2answers
429 views

Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows. Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...
2
votes
0answers
76 views

Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [on hold]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
-1
votes
0answers
52 views

Spherical Element (Simplicial Set) [on hold]

Definition: An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$. $X$ is a pointed fibrant simplicial set. I am puzzling over this definition. For instance, if ...
3
votes
2answers
303 views

Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...
1
vote
0answers
128 views

Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?
4
votes
1answer
207 views

Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question. In the knot theory, the Reidemeister moves play fundamental roles. ...
4
votes
0answers
118 views

Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
2
votes
0answers
100 views

Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$ If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
5
votes
0answers
662 views

Ordinary cohomology groups of $(\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ that I need for my string theory research

Let $X=(B^6\times T^2)/\mathbb{Z}_k\subset (\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ where $k=2,3,4,6$, where the generator of $\mathbb{Z}_k$ acts on $\mathbb{C}^3$ by the multiplication by a primitive ...
0
votes
0answers
115 views

Is there a t-structure on the homotopy category of spectra that has the sphere spectrum in its heart?

Maybe the heart of such a t-structure should be the category of abelian groups, and the t-homology functor should be given by taking usual homology groups of the spectrum. Is it impossible?
0
votes
0answers
63 views

Algebraic Topology problem on continuous functions over disk on sphere [closed]

I have come across the next problem, and I would like a little hint. Everything I'm thinking is not working or is a dead end. Let $ f,g:D^2\rightarrow S^2$ be continous maps such that $(x,y) \in S^1$ ...
10
votes
2answers
457 views

A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms ...
4
votes
0answers
55 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
1
vote
1answer
147 views

group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram If $\xi$ is a trivial bundle, ...
0
votes
0answers
127 views

Equivariant Cohomology questions [closed]

Hello I am a beginner to Cohomology theory. I have the following question and I don't know if it will fit into Mathoverflow. Let $X$ be a (smooth) projective variety on which a finite group $G$ acts ...
7
votes
1answer
139 views

(Non)-equivariant equivalence in $G$-spectra

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in ...
8
votes
1answer
257 views

Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
0
votes
1answer
57 views

Conjugation Cells [equivariant cohomology]

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5) ...
3
votes
0answers
149 views

some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus ...
9
votes
1answer
316 views

Wild half-line in a Euclidean space

Is there an $m$-dimensional simplicial complex $S$ with the following properties: The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...
30
votes
1answer
646 views

A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ...
15
votes
2answers
948 views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
6
votes
0answers
91 views

Detection tools for (reduced) suspension

I'm learning about loop spaces and the work of Stasheff on $A_{\infty}$-spaces. The broad idea that I'm getting is the following. Given a space $Y$, we want to know under which conditions there exists ...
6
votes
3answers
753 views

Does anyone know the classification of fourth order surfaces?

Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$ where $f$ is a polynomial of degree ...
4
votes
0answers
99 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
4
votes
1answer
130 views

The principal bundle of embeddings

In a paper of P. Michor, it was shown that Emb(M,N) is a smooth principal diff(M)-bundle, M and N are smooth locally compact manifolds provided dim M < dim N. My question is why there is a ...
3
votes
0answers
97 views

Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...
7
votes
1answer
342 views

higher algebraic homotopy groups for schemes?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where ...
7
votes
0answers
194 views

Two different Thom diagonals in recent literature?

Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...
1
vote
0answers
207 views

Can one prove the poincare duality for projective scheme by proving it for projective space?

It's well known the relationship between Poincare duality and Thom isomorphism(I mean cohomology purity $R^q i^! F=0$ if $q\neq c $ ) $\quad $ $Rf_!Ri_!=R(f|_Z)_!$ where f is $P_k^n\rightarrow k$ ...
12
votes
0answers
173 views

Uniqueness of connected cover of Morava K-theory

Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$ where $A$ is the Steenrod algebra and $Q_n$ is ...
10
votes
1answer
284 views

Dimension in CW-approximation

The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question. Let $X$ be a topological space, and let $\tilde{X}\to X$ be a ...
17
votes
5answers
900 views

Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas. The audience is familiar with ...
8
votes
1answer
432 views

What is the Status of Borel conjecture today?

Let me recall the conjecture: $M$ and $N$ two aspherical closed $n$-manifolds with isomorphic fundamental groups, then $M$ and $N$ are homeomorphic.
4
votes
0answers
127 views

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...
6
votes
1answer
109 views

Rational cohomology of the Rosenfeld projective planes

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes ...
6
votes
2answers
349 views

Genuine equivariant ambidexterity

A particular case of Lurie and Hopkins' ambidexterity theory is that if $G$ is a finite group acting on a $K(n)$-local spectrum $X$ then the norm map $$ X_{hG} \to X^{hG} $$ is a $K(n)$-local ...
3
votes
1answer
122 views

cohomology module of unit tangent vector bundles over spheres

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration $$ S^{m-1}\longrightarrow ...
9
votes
0answers
261 views

What does convergence of a Bousfield-Kan spectral sequence say about the homotopy type of the totalization?

Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and ...
25
votes
5answers
2k views

What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
2
votes
1answer
160 views

$n$-th cohomology of locally compact subsets in R^n

Where can I find a reference that for any locally compact (or just open) subset $U$ of $\mathbb{R}^n$, $H^n(U;\mathbb{Z})$ (the n-th Cech integral cohomology) is trivial?
6
votes
0answers
154 views

What is the paper “Structure and homology of configuration spaces” by F. Cohen and L. Taylor?

Cohen and Taylor, in their paper Computations of Gelʹfand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces (1978), cite this numerous times: [9] F. R. ...
6
votes
2answers
440 views

quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ ...
1
vote
1answer
297 views

Reference request for Godement's “Topologie algébrique et théorie des faisceaux”

Does anybody know if an english translation of this paper exists please?
3
votes
0answers
172 views

Are all 4-manifolds $Pin^{\tilde{c}}$?

It's known that all oriented 4-manifolds admit a $Spin^c$ structure, ie. a spin structure on $TX\oplus\mathcal{L}$ for some complex line bundle $\mathcal{L}$. A usual generalization of this ...
25
votes
1answer
632 views

What is, really, the stable homotopy category?

When you try to understand the fuss behind the new good categories of spectra that arose on the 90's, you read things such as the following paragraph written by Peter May (from "The Hare and the ...
3
votes
0answers
56 views

Is the functor of PA forms known to be equivalent to the functor of PL forms for noncompact spaces?

In the following paper: Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić, Real homotopy theory of semi-algebraic sets, Algebr. Geom. Topol. 11 (2011), no. 5, 2477–2545. the authors ...
16
votes
1answer
532 views

(really) basic intuition for $\mathbb A^1$-homotopy theory

Apologies in advance if this question is inappropriate for MO. I'm trying to read here and there about $\mathbb A^1$-homotopy theory in algebraic geometry. I understand some abstract machinery is ...
28
votes
2answers
391 views

If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$?

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a ...