Questions tagged [at.algebraic-topology]

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

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15
votes
2answers
254 views

Which elements of $H^2(M,\mathbb{Z}/2)$ are the $w_2(E)$ for a real bundle $E$?

Any element of $H^1(M,\mathbb{Z}/2)$ is the $w_1(E)$ of a real line bundle $E$ over $M$. I wonder how to characterize (probably using the Steenrod squares) which elements of $H^2(M,\mathbb{Z}/2)$ are ...
0
votes
0answers
78 views

Modules over quasiisomorphic DG algebras

Suppose there is a quasiisomorphism $q: A \to B$ between DG algebras. Is there some reasonable description of induced functor $q^*: B-mod \to A-mod$? Can we say something better if it was a $B$-...
12
votes
1answer
396 views

Explicit isomorphism $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_1(\mathbb{RP}^{n-1})$

From covering space theory we know that $\pi_{n+1}(\mathbb{RP}^n) \cong \pi_{n+1}(\mathbb{S}^n)$. From wikipedia I can notice that $\pi_{n+1}(\mathbb{S}^n) \cong \pi_1(\mathbb{RP}^{n-1})$.* My ...
10
votes
2answers
470 views

Does injectivity of $\pi_1(\partial U) \to \pi_1(M)$ imply injectivity of $\pi_1(U) \to \pi_1(M)$?

Let $M$ be a smooth compact manifold of dimension $n$, and let $U$ be a smooth compact manifold with boundary, of the same dimension $n$, embedded in $M$. The embedding induces maps on $\pi_1$. If $...
5
votes
0answers
133 views

Equivariant Venice Lemma

In the paper J. Simons and D. Sullivan. Structured vector bundles define differential K-theory, one of the key ideas is the so called Venice Lemma, which essentially can be stated as Theorem: For ...
3
votes
3answers
276 views

Which $\infty$-groupoids correspond to simplicial abelian groups?

Kan complexes model $\infty$-groupoids, so since every simplicial abelian group is a Kan complex, every simplicial abelian group yields an $\infty$-groupoid. What sort of $\infty$-groupoids do you ...
1
vote
0answers
78 views

Lower bounds on “size” of Whitehead covers?

Let $X$ be a nonzero finite spectrum, connective say, and consider the Whitehead tower of $n$-connected covers $\dots \to X\langle n \rangle \to X\langle n-1 \rangle \to \dots \to X\langle 0 \rangle = ...
3
votes
0answers
163 views

Composition of prolongations of $\Gamma$-spaces

Let $S,T:\Gamma^{\text{op}}\to \mathsf{Top}_*$ be two $\Gamma$-spaces ($\Gamma^{\text{op}}$ being the category of finite based sets $r_+=\{*,1,\dotsc,r\}$ with based maps as morphisms. The “op” has ...
5
votes
0answers
156 views

How much of the category of motives can be recovered from automorphisms of the Betti functor

Say we are working with schemes over a field $k\subset \mathbb{C}.$ A motive in the sense of Voevodsky is a functor $Sch\to D^bVect$ from (an appropriate category of) schemes to the DG category of ...
3
votes
0answers
109 views

Extra Algebraic $(1,1)$ cycles on a complex surface

Suppose $x,y,w,z$ are homogeneous coordinates of $\mathbb{CP}^3$, and \begin{eqnarray} X_t := \left(F_t = x f_2 +y g_2 +t F_3 = 0 \right) \end{eqnarray} be a family of degree 3 hypersurfaces in $\...
3
votes
1answer
168 views

Homotopy type of linear isometric self-isomorphisms of ${\mathbb R}^\infty$

In the paper "Orbispaces, orthogonal spaces, and the universal compact Lie group" by Stefan Schwede, he studies (spaces with an action of) the topological monoid $\mathbf{L}(\mathbb R^\infty,\mathbb R^...
5
votes
0answers
99 views

On the weak homotopy type of a differentiable (Chen) space

Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ). Assume that $M$ also has the structure of a topological space and that the two ...
4
votes
0answers
131 views

K-theory for a (geometric) stack

There is a notion of $K$-theory for a manifold $M$. Is there a notion of $K$-theory for a stack $\mathcal{D}\rightarrow \text{Man}$ that is representable by a Lie groupoid $\mathcal{G}$; that is $...
10
votes
0answers
181 views

For which $k$-types are $E_{n,m}$-algebras automatically $E_{n+1}$ algebras?

Recall that an $E_{n,m}$ algebra is an $A_m$ algebra in $E_n$ algebras. Here I index my $A_m$ algebras so that an $A_1$ algebra is pointed, an $A_2$ algebra has a unital multiplication, $A_3$ is ...
1
vote
1answer
106 views

Injectivity of homomorphism between homology groups of manifold and its boundary

I asked similar question before, after some modification, I have a new question. Suppose $M^{n\geq 4}$ is a connected compact smooth manifold with connected nonempty boundary. Suppose $i_*: H_1(\...
2
votes
0answers
98 views

Principal symbol of a non-local operator and Atiyah–Singer index formula

I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the ...
3
votes
0answers
84 views

Properties of a generalization (regularization) of the Euler characteristic?

Intro: This question is about a version of the Euler characteristic for infinite dimensional chain complexes. I have no idea if this is a pre-existing concept, that's essentially what my query is ...
8
votes
1answer
256 views

Torsion in the cohomology of Fano varieties of lines

Let $\mathrm{X}$ be a cubic $d$-fold, and $\mathrm{F}(\mathrm{X})$ its Fano variety of lines. Is the integral cohomology of $\mathrm{F}(\mathrm{X})$ torsion-free? For $d=3$ A. Collino (`The ...
7
votes
1answer
182 views

Higher-dimensional version of the “Magic Cube Lemma” for homotopy pushouts/pullbacks

The "Magic Cube Lemma" is a surprising (to me) relationship between (homotopy) pushouts and (homotopy) pullbacks of spaces: Consider a cubical diagram $I^3\to \mathcal{S}$ in the $\infty$-category of ...
8
votes
0answers
177 views

When does a locally symmetric space have no odd degree Betti numbers?

Let $G$ be a semisimple real lie group, $K$ be a maximal compact subgroup of $G$, $\Gamma$ be a torsion-free cocompact discrete subgroup. The Betti number the locally symmetric space $X_{\Gamma}:=\...
5
votes
0answers
138 views

The fundamental loopoid?

Let $X$ be a homotopy type (modeled as either a topological space or a simplicial set). We can construct a category as follows: The objects are maps $f,g : S^1 \to X$. A morphism $f \to g$ is a map $S^...
3
votes
1answer
101 views

Example: closed path not homotopic to path in subset

I am looking for an example for the following setting: Given an open subset $U$ of $M$, both path-connected, such that there is a closed path in $M$ that is not homotopic to a closed path in $U$, ...
1
vote
0answers
117 views

Homology of homotopy fiber of inclusion

We consider an inclusion $j: A \hookrightarrow X$. Let $A_j = A \times_X PX$ be the homotopy fiber (which is the fiber of the fibration associated to $j$). The space $PX$ there is the Moore path space ...
5
votes
0answers
112 views

Continuous functors, spectra and homology theories

Let $T:\mathbf{Top}_*\to \mathbf{Top}_*$ be a continuous functor and $E$ a spectrum with maps $\sigma_n:E_n\wedge S^1\to E_{n+1}$. We have a new spectrum $TE$ with structure maps $$(TE_n)\wedge S^1\...
11
votes
0answers
355 views

The $\infty$-category of $n$-manifolds and open embeddings determined homotopically from that of topological manifolds?

Let $\mathrm{Diff}_n$, $\mathrm{PL}_n$, $\mathrm{Top}_n$ denote the $\infty$-categories of $n$-manifolds which are respectively smooth/PL/topological, and open embeddings (for instance by taking the ...
10
votes
0answers
183 views

Subvarieties with isomorphic complements

Let $X$ be a smooth irreducible projective variety over $\mathbb C$, $Y_1, Y_2$ are two closed smooth subvarieties. Assume $X-Y_1 \cong X-Y_2$, what do $Y_1$ and $Y_2$ have in common (at least ...
2
votes
1answer
122 views

Pontryagin square of first Stiefel-Whitney class

Let $w_1$ be the 1st Sitefel-Whitney classes of the tangent bundle of a 4-manifold $M$ ($M$ is non orientable). My question is Is $$\exp\left(\frac{i\pi}{2}\int_{M_4} \mathcal{P}(w_1^2)\right)$$ ...
2
votes
2answers
198 views

Measuring failure of a setup to preserve some structure giving interesting notions

I am looking for some examples of failure of some structures giving interesting notions. For example, we have the following situation: Let $P(M,G)$ be a principal bundle. Let $\Gamma\subseteq TP$ be ...
12
votes
1answer
227 views

What is the smallest class of spaces closed under finite homotopy colimits, finite homotopy limits, and splitting of homotopy-coherent idempotents?

Finite CW complexes fail spectacularly to be closed under finite homotopy limits (e.g. $\Omega S^1 = \mathbb Z$). More subtly, they fail to be closed under homotopy retracts (by the Wall finiteness ...
11
votes
1answer
505 views

Manifolds with nonwhere vanishing closed one forms

I am trying to find examples of closed manifolds $M$ admitting a nowhere vanishing closed one form. I am wondering if there are any examples beyond $N\times S^1$.
2
votes
0answers
95 views

Killing cohomology classes from $\operatorname{BSO}(2n)$

We know that the mod-2 cohomology of $\operatorname{BSO}(2n)$ is the polynomial algebra of Stiefel-Whitney classes $w_i$ with $\mathbb Z/2$-coefficients, $2\le i\le 2n$. Is it possible to kill off ...
6
votes
1answer
493 views

How to construct the Moore spectrum?

I am trying to understand how the Moore spectrum is constructed. And in reading Foundations of Stable Homotopy Theory by David Barnes and Constanze Roitzheim, I see that in example 8.4.7 (pg 340) they ...
2
votes
0answers
89 views

Pushforward of covering maps

Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid. Given a covering map of $A$ and a ...
9
votes
2answers
457 views

How are characteristic classes morphisms of infinite loop spaces? (if they are)

The direct sum of real vector bundles endows $BO=\mathrm{colim} BO(n)$ with a natural structure of abelian group up to homotopy. The same applies to the classifying spaces of all groups in the ...
5
votes
0answers
130 views

Topology of different special fibers of a smooth projective variety over $\mathbb C((t))$

Motivation: Let $Y_i$ be two smooth projective varieties over $\mathbb R$ which are isomorphic over $\mathbb C$, although $Y_i(\mathbb R)$ (under analytic topology) can be different, they do have some ...
6
votes
1answer
137 views

Morphisms from $bstring$ to $X\otimes \mathbb{Q}$ and sequences $s_n\in\pi_n(X)\otimes \mathbb{Q}$

I'm currently studying Ando-Hopkins-Rezk's work Multiplicative orientations of KO-theory and of the spectrum of TMFs. At a point a presumably obvious isomorphism is mentioned, which I'm however not ...
12
votes
2answers
632 views

Image of a map on cohomology rings

The following seems like an extremely basic algebraic topology question, but it's not something I ever learned, nor does it look familiar to the algebraic topologists I've asked. Let $f:X\to Y$ be ...
7
votes
1answer
178 views

Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
4
votes
0answers
86 views

When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
6
votes
0answers
128 views

Is there an $\infty$-topos of monochromatic spaces?

Fix (a prime $p$ and) a chromatic height $h$. Recall that the Bousfield-Kuhn functor $\Phi_h: \mathcal M_h^f \to Sp_{T(h)}$ is monadic, where $\mathcal M_h^f \subseteq Top_\ast$ is a certain ...
29
votes
1answer
661 views

Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?

One of the old classic MO questions is a big-list of proofs of the fundamental theorem of algebra. Here is a second big-list question about this big list: Which of the FTA proofs can, even in ...
1
vote
0answers
82 views

Based loops objects in model categories

If I have a pointed model category then for I can define based loops objects as homotopy pullbacks: $\require{AMScd}$ \begin{CD} \Omega X @>>> *\\ @V V V @VV V\\ * @>>>...
2
votes
0answers
163 views

Simply put Floer homology

I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would ...
5
votes
1answer
182 views

Kähler manifold with even-only singular cohomology

Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all ...
7
votes
1answer
311 views

Descent properties of topological Hochschild homology

Question: What is the finest topology on $\mathrm{CAlg}$ (commutative ring spectra) for which THH (Topological Hochschild Homology) satisfies descent? Adaptations of the arguments appearing in ...
4
votes
1answer
148 views

Homotopy fibre sequence and left Bousfield localization

Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...
3
votes
0answers
129 views

Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?

There's a canonical functor $i: \Delta^{op} \to Fin_\ast$. For example, one uses the pullback $i^\ast$ to turn a $\Gamma$-space into a simplicial space, and then takes a geometric realization to ...
1
vote
0answers
46 views

manifold bounded by compact manifold with $b_1=0$

Let $X$ be a non-compact manifold without boundary. Suppose that $b_1(X)=0$. Suppose $Y$ is a codimension zero compact submanifold with corner. Q Can we find a compact submanifold $Z$ with smooth ...
3
votes
1answer
227 views

Lectures on triangulations of manifolds by Robion Kirby

I was looking for the book mentioned in the title. Seemingly it was not published, but copies are available in several mathematical libraries. Google books does not provide preview. I am wondering if ...
12
votes
2answers
319 views

The $K$-theory homology of the Eilenberg-MacLane spectrum

Let $KU$ be the complex $K$-theory spectrum and $H\mathbb{Z}$ be the Eilenberg-MacLane spectrum. For $n\in \mathbb{Z}$, it is known what the homology groups $KU_{n}(H\mathbb{Z})$ are?