Tagged Questions

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

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6
votes
2answers
142 views

Topological relationships between family of transversal intersections of manifolds

Let $M$ and $N$ be submanifolds of $\mathbb{R}^n$ and let $a(t)$ be a smooth path in $\mathbb{R}^n$ such that $M+a(t)$ intersects $N$ transversally for all $t \in [0,1]$. Is there a nice relationship ...
4
votes
2answers
256 views

A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres." If I ...
5
votes
2answers
220 views

What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...
2
votes
1answer
184 views

Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace

The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^* : x \in \mathbb{C}^3, x^*x=1 \}$. As such, ...
1
vote
0answers
87 views

generalisation of the universal coefficient spectral sequence

Suppose I have a bounded chain complex $C_{\ast}$ over the group ring $\mathbb{Z}G$ for a finite group $G$. In topology we are usually in a situation when $C_{\ast}$ is a complex of projective ...
6
votes
1answer
301 views

Do homotopy limits compute limits in the associated quasicategory in the non-combinatorial model category case?

Suppose that $\mathcal{M}$ is a model category which is not combinatorial, does a homotopy limit in $\mathcal{M}$ correspond to a limit in the associated $\left(\infty,1\right)$-category? How about ...
5
votes
1answer
450 views

A question on fixed point theory

I asked this question in MSE, but I did not received any answer, so I repeat it here: http://math.stackexchange.com/questions/858238/a-question-on-fixed-point-property Assume that $0<k<n-1$, ...
15
votes
4answers
536 views

Multiplicative Structures on Moore Spectra

The motivation for this question is that I want "toy examples" of how to prove/disprove the existence of multiplicative structures on examples of spectra. The class of examples I am thinking of is the ...
3
votes
3answers
834 views

Why are we interested in the Fundamental Groupoid of a Space?

The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$. ...
6
votes
0answers
112 views

A Property of Generalized Equivariant Cohomology

Let $G_i$ be a compact Lie group, $i=1,2$, and let $E_{G_i}^*$ be a $\mathbb{Z}$-graded complex-oriented $G_i$-equivariant generalized cohomology theory with commutative products. Let $X_i$ be a ...
5
votes
2answers
220 views

Morphisms every pushout of which is a weak equivalence

Let $M$ be a category equipped with a class of weak equivalences $W$. Is there a name for a morphism $f$ such that every pushout of $f$ (including, of course, $f$ itself) is a weak equivalence? For ...
6
votes
2answers
194 views

Covering Spaces and Vector Bundles

Suppose $f: Y \rightarrow X$ is a covering map between compact Hausdorff spaces $X$ and $Y$. Then $f$ induces a algebra homomorphism $f^*:C(X) \rightarrow C(Y)$ and gives $C(Y)$ the structure of a ...
18
votes
0answers
477 views

From the perspective of bordism categories, where does the ring structure on Thom spectra come from?

To fix ideas, let's consider the Thom spectrum of framed bordism $M$, the spectrum whose homotopy groups are the framed bordism groups. $M$ has a ring spectrum structure inducing the product of ...
20
votes
9answers
2k views

Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
8
votes
0answers
126 views

Does simplicial localization with a 3-arrow calculus commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...
2
votes
1answer
479 views

1D TQFT in Freed-Hopkins-Lurie-Teleman

In the first section of Freed-Hopkins-Lurie-Teleman they construct a one-dimensional Topological Quantum Field Theory. $F(\circ_+)$ is a vector space and $F(\circ_-)$ is the dual. $F(\circ-\circ)$ ...
4
votes
2answers
409 views

What does Freed-Hopkins-Teleman say about finite groups?

The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available here, ...
1
vote
1answer
108 views

LS-cat of Hspaces and that of co-Hspaces

I'm searching to learn more about any results or/and proofs (new or recent) about LS-category of H-spaces and that of co-Hspaces. Any references are welcomed Thanks
4
votes
0answers
193 views

Topological fundamental group of a variety

I have an explicit question. I have a complex projective variety defined by 2X2 minors of a matrix. The entries are polynomials from a weighted projective space. In fact its a singular 3-fold, with ...
3
votes
0answers
172 views

matrix ring spectra

I am trying to understand matrix ring spectra. Apparently, I have two different definitions of those and I did not manage to show that they are equivalent - maybe they even are not in the general ...
7
votes
1answer
167 views

When does simplicial localization commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...
4
votes
0answers
141 views

Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here. Is there ...
1
vote
0answers
132 views

Reference request: modern proof of the recognition principle [closed]

I'm looking for a modern version of P. May's proof of the recognition principle (in "geometry of iterated loop spaces"). Can someone help me?
5
votes
1answer
228 views

endomorphisms of modules over symmetric ring spectra

I have a probably very basic question about modules over symmetric ring spectra: Let $R$ be a commutative symmetric ring spectrum and let $M$ and $N$ be module spectra over $R$. Moreover, let ...
6
votes
3answers
421 views

Simplest example of non-trivial Toda bracket in spaces

Many sources give an easy definition of a Toda bracket $\{f,g,h\}$ of appropriate maps $W \to X \to Y \to Z$ in spaces as a subset of the homotopy classes of maps $[\Sigma W, Z]$ (for example, ...
1
vote
1answer
173 views

Universal bundles: construction of the map associated to a group homomorphism

For a Lie group $G$ let $EG \to BG$ denote the universal bundle. A Lie group homomorphism $\rho: G \to H$ determines a map $B \rho: BG \to BH$ as the classifying map for the principal $H$-bundle $EG ...
12
votes
0answers
341 views

What is the determinant of Poincare duality?

For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant $$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$ functorial with respect to ...
5
votes
2answers
618 views

How nilpotent is the ring of stable homotopy groups of spheres?

Are there any known or conjectured bounds on the exponent $d(r)$ such that $x^{d(r)} = 0$ for all $x \in \pi_r^S(S^0)$?
4
votes
1answer
265 views

Which framed manifolds are in the image of J?

The stable homotopy groups of spheres are in natural correspondence with framed cobordism classes of framed manifolds. Is there a good way to tell if the class of framed manifold is in the image of ...
4
votes
1answer
281 views

Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold

Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a ...
5
votes
1answer
187 views

Analytic maps in homotopy classes

Let $\mathcal M$ be a compact connected real-analytic manifold. It is well known that every continuous map $f\colon\mathcal M\to\mathbb S^1$ is homotopic to a smooth map. My question is the following. ...
2
votes
1answer
112 views

Equivalence of the total spaces of two Serre fibrations with equivalent fibers

Let $B$ be a connected pointed CW complex, let $E$ and $E'$ be two CW complexes and let $f\colon E\to B$ and $f'\colon E'\to B$ be two Serre fibrations. Let $g\colon E\to E'$ be a continuous map such ...
1
vote
0answers
109 views

extension problem for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
4
votes
2answers
256 views

stability results for the Atiyah-Hirzebruch spectral sequence

For a generalized homology theory $h$ and a Serre fibration $F\rightarrow E\rightarrow B$, we can define an Atiyah-Hirzebruch spectral sequence\begin{equation}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow ...
0
votes
0answers
63 views

Cap product in strong homology theories

I'm looking for a (co)homology theory for compact metric pairs with cup/cap product that equals to singular (co)homology for CW complexes, the cap product equals to the "singular" cap product for CW ...
4
votes
1answer
207 views

Question about the fundamental group of rational homology 3-spheres

By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that ...
10
votes
1answer
234 views

Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map. Not every ghost map ...
2
votes
0answers
61 views

Codimension $k$ homeomorphism extensions

Let $f:D \to D$ homeomorphism of $k$ codimension manifold (closed, compact, without boundary) to itself. ($D \subset \mathbb{R}^{n+k}$). For which $f$ does homeorophism $g: \mathbb{R}^{n+k} \to ...
0
votes
0answers
109 views

Zero Dimension Intersection

Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class ...
0
votes
1answer
162 views

Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd. For which $M$, ...
3
votes
1answer
194 views

Lower dimensional Pin cobordisms

I'm studying Pin cobordism groups of a point for some low dimensions. I found a general result by Anderson, Brown, Peterson in Theorem 5.1 of their paper "Pin cobordism and related topics" ...
4
votes
1answer
323 views

On the fundamental group of closed 3-manifolds

I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...
6
votes
0answers
211 views

Topological quotient Hopf-algebras and “change-of-rings”

One way of constructing coalgebra objects in the homotopy theorist's category of spectra is to take the suspension spectrum of a space, with the diagonal providing a cocommutative comultiplication. ...
3
votes
2answers
283 views

If $E$ maps onto a contractible space with contractible fibers, must $E$ be contractible?

Let $p\colon E\to C$ be a continuous, surjective map between topological spaces with $C$ contractible. Suppose that $p^{-1}(c)$ is contractible for each $c\in C$. Is it true that $E$ is weakly ...
2
votes
1answer
125 views

Explicit reference on generator of $H^4(BQ_8;\mathbb{Z})\cong \mathbb{Z}_8$ identified with second Chern class of standard representation

It is extremely well-known that $H^*(BQ_8;\mathbb{Z})=\mathbb{Z}[\alpha,\beta,\gamma]$ with relations $2\alpha=2\beta=8\gamma=\alpha^2=\beta^2=\alpha\beta-4\gamma=0$, $|\alpha|=|\beta|=2$ and ...
6
votes
1answer
177 views

Rational homology and finite group actions

I'm looking for examples of the following phenomena. Let $X$ be a reasonable space (say, a CW complex) and $G$ be a finite group acting on $X$. For all $k \geq 1$, the projection map $X \rightarrow ...
5
votes
1answer
330 views

Dennis trace map K----> THH

I have some questions about Dennis trace map in algebraic K-Theory. I was wondering if there is some conceptual way to look at this map $K(-)\rightarrow THH(-)$ (natural transformation from K-Theory ...
11
votes
2answers
1k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see : Most interesting mathematics mistake? Added : According to their method, ...
2
votes
0answers
107 views

Which reflexive coequalizer diagrams are projectively cofibrant?

Consider the walking reflexive pair category W, which consists of two objects 0 and 1 and three generating morphisms f: 0→1, g: 0→1, and h: 1→0 satisfying the relation fh=gh=id₁. Consider the ...
7
votes
1answer
279 views

Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize. All manifolds are closed, smooth and have dimensions $n\ge 5$. The Atiyah-Shapiro-Bott-Orientation gives ...