# Questions tagged [at.algebraic-topology]

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

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### Higher K-theory groups and Fredholm operators

As is well known, whenever $X$ is a compacta space, the Atiyah-Janich theorem says that there is an identification $$[X,\mbox{Fred}(H) ]\cong K^0(X) $$ between the set of homotopy classes of maps from ...

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### A question on relative equivariant cohomology

Suppose that we have defined an extraordinary $G$-equivariant cohomology theory $H$ (say $G$ is a compact group). If $X$ is a $G$-space and $A\subset X$ is a closed $G$-equivarant contractible ...

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### Minimal cell structures

Is there some reasonable obstruction theory which detects whether a finite type complex can be provided with a cell structure such that number of $i$-cells is equal to the maximal rank of $i$th ...

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### Equivalent definitions of Thom spectra

Background and notations:
Recall the classical contruction and definition of Thom spectra. To a spherical fibration $S^{n-1} \to \xi \to B$, we can associate the data of a Thom space $T_n(\xi)$, given ...

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### Question about fundamental group of complements of divisors

Let $X$ be a quasi-projective smooth algebraic variety (over $\mathbb{C}$) and $D$ an irreducible divisor, such that $(X,D)$ is log smooth.
Is it true that there is an exact sequence $\mathbb{Z}\...

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### Reference request: mod 2 cohomology of periodic KO theory

The mod 2 cohomology of the connective ko spectrum is known to be the module $\mathcal{A}\otimes_{\mathcal{A}_2} \mathbb{F}_{2}$, where $\mathcal{A}$ denotes the Steenrod algebra, and $...

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### Simplicial sets of categories as models for $(\infty,1)$-categories [duplicate]

Disclaimer: I am not knowledgeable at all about the subject of this question, so my apologies if I say something incorrect or too imprecise.
In my understanding, there are several models for $(\infty,...

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### Strøm model structure on nonnegatively graded chain complexes

Let $\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$ be the category of $\mathbb{N}$-graded chain complexes over some ring $R$, and $\Ch(R)$ the category of $\mathbb{Z}$-graded chain complexes.
The ...

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### Oriented vector bundle with odd-dimensional fibers

Is it true that for every oriented vector bundle with odd-dimensional fibers, there is always a global section that vanishes nowhere?

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### Homotopy equivalence of $K$-theory and $G$-theory

Let $X$ be regular variety then it is known that $Q(Vect(X))\cong Q(\mathcal{M}_X)$. Where $Q$ is the Quillen's q-construction and $\mathcal{M}_X$ is the category of coherent sheaves on $X$. You can ...

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### What is the intuition for higher homotopy groups not vanishing?

The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This ...

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### Replacing the Fibre of a Fibration

This was a question I first asked on stack exchange, here. In my head it seems like a fairly reasonable thing to ask for, but I'm not aware of any construction in the literature.
Let $p:E\rightarrow ...

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### Petries exotic circle action

In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set ...

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### Proj construction in derived algebraic geometry

The question
My question is easy to state:
Is there a Proj construction in derived geometry, that produces a derived stack from a “graded derived algebra”?
Given the vagueness of the question, you’...

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### Generalize $H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R)$ to fundamental Groupoid

Let $X$ be a path-connected smooth manifold, it is known that: $$H^1(X):=H^1_{dR}(X)=\mathrm{Hom} (\pi_1(X),\mathbb R).$$ Explicitly, a closed one-form $\alpha$ gives a function on $\pi_1(X)$ by $[\...