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Homotopy theory, homological algebra, algebraic treatments of manifolds.

11 votes
2 answers
725 views

Do there exist "topologically significant" (and not "algebraic") triangulated categories kil...

I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably …
Mikhail Bondarko's user avatar
5 votes

Computing fundamental groups and singular cohomology of projective varieties

First we assume that your equations have rational coefficients; if this is not so then you can probably 'approximate' your variety by a variety defined over rationals without changing its topology (th …
Mikhail Bondarko's user avatar
9 votes

Chain homotopy: Why du+ud and not du+vd?

Amusingly, I have considered the category of complexes where maps of the form du+vd are killed in: Weight structures vs. $t$-structures; weight filtrations, spectral sequences, and complexes (for mo …
Mikhail Bondarko's user avatar
3 votes
1 answer
195 views

Can one define relative Hurewicz maps using the Dold-Thom theorem

Let $A\to X$ be a (Hurewicz) cofibration of path-connected topological spaces. Then we have a long homotopy sequence $$ \dots\to \pi_i(A)\to \pi_i(X)\to \pi_i(X,A)\to \dots; $$ here we fix a base poin …
Mikhail Bondarko's user avatar
6 votes
1 answer
535 views

Exceptional collections of objects in topological triangulated categories?

People often consider exceptional sets of objects (i.e. collections of objects satisfying certain strong orthogonality conditions: $Ext^{l}(P_i,P_j)$ should be zero for $l\neq 0$ + something else) in …
Mikhail Bondarko's user avatar
4 votes
1 answer
498 views

Does the (singular)cohomology of any acyclic spectrum vanish?

I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or $H\mathbb …
Mikhail Bondarko's user avatar
12 votes
1 answer
403 views

Which statements and arguments of Hovey's "Model categories" fail without functorial factori...

I would like to study the homotopy theory of the category of pro-objects over a proper model category $M$. $Pro-M$ is endowed with the strict model structure; it seems that functorial functorizations …
Mikhail Bondarko's user avatar
3 votes
0 answers
173 views

"Extending scalars" for (motivic) ring spectra and for modules over them: are the correspond...

Let $S$ be a (motivic symmetric) ring spectrum (more generally, one can possibly consider a commutative ring object in any symmetric stable model category); let $R$ be an associative commutative unit …
Mikhail Bondarko's user avatar
4 votes

Why torsion is important in (co)homology ?

There are some very important 'torsion motivic' statement: the calculation of Suslin's homology, Milnor and Bloch-Kato conjecture (proved by Voevodsky). Also, the proof of the latter statements uses a …
2 votes
0 answers
75 views

Does there exist a "Margolis-type" definition of equivariant cellular towers?

I am interested in cellular towers in the equivariant stable homotopy category $SH_G$ corresponding to a compact Lie group along with a complete universe for it. Note here that a cellular tower for a …
Mikhail Bondarko's user avatar
1 vote
0 answers
71 views

Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the "topological" stable homotopy category $SH$; a filtration of a spectrum $E$ is a sequence of compatible maps $E_{\le i}\to E$) whose …
Mikhail Bondarko's user avatar
1 vote
0 answers
86 views

Terminology: are there any names for "quotients" of cellular towers in stable categories?

A cellular tower in SH or in a "more general stable homotopy category" is a chain of morphisms $\dots X^{(n)}\stackrel{g^n}{\to} X^{(n+1)}\to \dots$ along with some more data and conditions; one of th …
Mikhail Bondarko's user avatar
0 votes
1 answer
519 views

Lefschetz fixed point formula: an 'easy' proof; cohomology with compact support

I have two questions concerning the LFPF (for etale cohomology). Is there an easy 'explanation' of this statements (that could be understood by students)? In particular, I would like to get away wit …
Mikhail Bondarko's user avatar
3 votes
1 answer
149 views

Interesting examples of a 4-torsion X in a triangulated category such that $2 End(X/2X)\neq ...

It is well-known that for the sphere spectrum $S$ in the ('topological') stable homotopy category the object $S/2S$ i.e. the cone of $S\stackrel{\times 2}{\to}S$, is not $2$-torsion. So I wonder whe …
Mikhail Bondarko's user avatar
4 votes
1 answer
431 views

If a t-truncation of the unit object in a stable homotopy category is a ring object up to ho...

Let $S$ be the unit object in a monoidal stable homotopy category $SH$ (we demand that the multiplication $S\times S\to S$ is commutative and associative on the level of spectra, and not just up to we …
Mikhail Bondarko's user avatar

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