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

learn more… | top users | synonyms (1)

3
votes
2answers
236 views

Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [on hold]

There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that ...
4
votes
1answer
140 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
4
votes
1answer
127 views

Section of the homology functor on spectra

Consider the (reduced) homology functor $H_*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ ...
4
votes
2answers
495 views

Why Cech cohomology does not compute sheaf cohomology on an open annulus

Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets: $U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...
8
votes
0answers
182 views

Lift chain complex from $F_2$ to $Z$

We start with a finite dimensional chain complex over $F_2$, equipped with a basis. That is, we have finitely many finite dimensional $F_2$-vector spaces $C_0,\dots,C_k$ with bases $B_0,\dots,B_k$, ...
3
votes
0answers
71 views

understanding the definition of $\infty$-operad of module objects

I'm just trying to understand the following definition: Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following: Let $O^\otimes$ be ...
5
votes
1answer
213 views

Finite group acting on sphere

Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
3
votes
1answer
205 views

sphere bundles over spheres

Localized at an odd prime there is a space $B_k$ which sits in a fibration $S^{2k+2p-3}\rightarrow B_k \rightarrow S^{2k-1}$ and has homology $H_{\ast}(B_k;\mathbb{Z}/p\mathbb{Z})\cong ...
1
vote
1answer
471 views

euler class of the normal bundle and self intersection number [duplicate]

Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via ...
14
votes
1answer
1k views

What have simplicial complexes ever done for graph theory?

(I am asking in a somewhat tongue-in-cheek fashion, of course, but nevertheless...) Are there examples of results in "classical" [*] graph theory that have been achieved by using simplicial ...
-2
votes
0answers
55 views

Complexity of a function [on hold]

I am looking for a natural definition of the complexity a function. If the image is discrete, i was thinking it could be: consider the preimage of an element of the image, count the number of ...
3
votes
1answer
102 views

In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In Meyer, Werner Die Signatur von ...
8
votes
2answers
345 views

Eilenberg-MacLane Spaces of “large” groups

It is well-known that if $G$ is a discrete group, then $BG=K(G,1)$. I'm interested in comparing classifying spaces of topological groups with the classifying spaces of the same groups but equipped ...
27
votes
1answer
532 views

For which maps $S^1\to S^1$ is the winding number defined?

There are two classes of maps $S^1\to S^1$ for which I know how to define the winding number: • Continuous maps: Using the unique path lifting property of the universal covering map $\mathbb R\to ...
0
votes
0answers
106 views

Homotopy equivalent type of a knot complement [on hold]

Let $S$ denote the bounded complement of a tame knot in $S^3$,then $S$ is homotopy equivalent to a finite 2-dimensional simplicial complex $K$ [Milnor's paper "infinite cyclic covering"],I do not ...
17
votes
4answers
1k views

Quillen's motivation of higher algebraic K-theory

Almost the same question was already asked on MO Motivation for algebraic K-theory? However, to my taste, the answers there consider the subject from a more modern point of view. When I open a book ...
13
votes
0answers
154 views

Unoriented bordism and homology, reference?

The following has undoubtedly been known to the experts for years, but I only noticed it the other day. Can anyone give a reference? One can prove Thom's theorem to the effect that every mod $2$ ...
1
vote
2answers
189 views

Freely Periodic map of $(S^{3} , K) $ and a fixed loop in the induced isomorphism of $\pi_{1} ( S^{3} \backslash K )$

Let $K$ be a link in $S^{3}$ and $f: S^{3} \rightarrow S^{3} $ a freely periodic map of order $n$ with $f(K) = K$. Let $\psi_{f} : \pi_{1} ( S^{3} \backslash K ) \rightarrow \pi_{1} ( S^{3} ...
11
votes
2answers
337 views

Truncations of E_infinity algebras

In section 4.1 of Lurie's DAG VIII, he implies the existence of an $E_\infty$-ring spectrum $A$ such that the coconnective truncation $\tau_{\leq 0} (A)$ does not admit the structure of an ...
18
votes
2answers
1k views

Open Problems in Algebraic Topology and Homotopy Theory

Some time ago (I see it was initially written before 1999?) Mark Hovey assembled a list of open problems in algebraic topology. The list can be found here. Some of the problems I know about have been ...
1
vote
0answers
68 views

Homology of the fixed points of the singular complex of a G-space

I posted the following to stackexchange a while ago [1], without any answers. Maybe the question is too unmotivated, but it seems very natural to me. Suppose $X$ is a topological space and $G$ a ...
-1
votes
0answers
154 views

Algebraic topology vs. category theory [migrated]

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
14
votes
5answers
2k views

What is the equivariant cohomology of a group acting on itself by conjugation?

This question makes sense for any topological group $G$, but I'd particularly like to know the answer for $G$ a compact, connected Lie group. $G$ acts on itself by conjugation. One has the ...
4
votes
2answers
508 views

Is a Lie group equivariantly formal under conjugation by a maximal torus?

Given an action of a group $G$ on a topological space $X$, the associated homotopy quotient is $$X_G := (EG \times X)/G,$$ where $EG$ is the total space of a universal principal $G$-bundle $EG \to BG$ ...
2
votes
1answer
291 views

Endomorphisms of degree d on a sphere with infinite fibers on a dense subset

Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$ disjoint ...
6
votes
1answer
240 views

Iterated Milnor fibrations and Thom's a_f condition

Ok so there's a lot of litterature about nearby cycles functor since it was introduced by Grothendieck and Deligne but I couldn't find any clear answer to the following natural question: Problem: Let ...
11
votes
0answers
193 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be, it seems to me, a close relation between p-adic string theory and the refinement of ...
29
votes
11answers
3k views

Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter. Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...
4
votes
2answers
246 views

Fundamental group of a manifold with an $S^1$-action

Let $M$ be a compact connected manifold with an $S^1$-action. Suppose that $S^1$ has a fixed point in $M$. Is it true that $\pi_1(M)=\pi_1(M/S^1)$? I is there some reference or a short proof of this ...
2
votes
1answer
336 views

Topological degree and polynomial degree

Let $F:\mathbb{C}^n\to \mathbb{C}^n$ be a homeomorphism homogeneous of degree 1 (i.e., $F(tx)=tF(x)$, $t>0$) and $g:\mathbb{C}^n\to \mathbb{C}$ a homogeneous polynomial of degree $k$. Let $L$ ...
3
votes
2answers
134 views

Nielsen-Thurston classification of homeomorphisms for open surfaces?

In Proposition 3.1. in this article by John Franks, he applies the Nielsen-Thurston classification of surface homeomorphisms to a homeomorphism $ \ f:M \rightarrow M$ of an open surface $M$ which is ...
2
votes
2answers
185 views

Infinite loop of a p-completed specta vs p-completion of infinite loop of the spectra

Assume that we have a connective spectrum $X$, and denote the $p$-completion of this spectrum in the sense of Bousfield by $X^{\wedge}_p$ (which is given by the function spectrum ...
13
votes
5answers
1k views

Are rational varieties simply connected?

Is it true that every smooth rational variety X is simply connected? How is the proof? Would it be still true if X has mild (for example orbifold) singularities?
65
votes
21answers
5k views

Occurrences of (co)homology in other disciplines and/or nature

I am curious if the setup for (co)homology theory appears outside the realm of pure mathematics. The idea of a family of groups linked by a series of arrows such that the composition of consecutive ...
3
votes
4answers
294 views

What does it mean to speak of a homotopy fibration sequence?

I'm reading a paper in which the following is done. We have a certain particular map of spaces $f:X\to Y$ and then it is said something along the lines of "let $Z_f$ denote the space whose defining ...
14
votes
0answers
300 views

What is to tmf as KR is to KO?

The $E_\infty$-ring spectrum $KU$ of complex K-theory carries a canonical involution induced from complex conjugation of complex vector bundles. The homotopy fixed points of this $\mathbb{Z}_2$-action ...
11
votes
3answers
365 views

rationalization of classifying spaces

This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway: Let $G$ be a simply-connected topological group. In particular, it is an ...
1
vote
2answers
155 views

How can I prove that Hopf fibrations are the only ones with fiber, total space and base space homeomorphic to spheres?

I know that Hopf fibrations (the four ones) are the only ones that have the form $S^k \to S^m \to S^n$, but I never seen a proof. Could anyone link me a paper or text where this is proved, or prove it ...
1
vote
0answers
153 views

Units of a ring spectrum

Is there a good notion of the spectrum of units $R^\ast$ in a (possibly non-connective) $E_\infty$-ring spectrum $R$? A standard definition (see section 1.2 in http://arxiv.org/abs/0810.4535) seems ...
4
votes
1answer
191 views

Equivalent fomulations of Bott periodicity

Is there an easy way to see the equivalence of the two statements of Bott periodicity. $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and $$K(X)\otimes K(S^2) \cong K(X\times S^2)$$
2
votes
1answer
341 views

Two questions about the grassmannian

There are two statements about the grassmannian (of complex k-planes in n-space embedded via Plucker coordinates) that I have encountered in several places never accompanied with a proof or reference. ...
4
votes
0answers
124 views

Third cohomology of mapping class group

I would like to know the third cohomology with coefficients in $U(1)$ or $\mathbb{C}^\ast$ of the mapping class group of a surface of genus at least one. I found many results on the rational ...
-1
votes
0answers
36 views

connected components of a real algebraic variety and its hyperplane section

Let $X$ be a smooth projective variety of dimension at least $2$ over the real numbers $\mathbb{R}$ and $H \subset X$ a smooth hyperplane section. Assume that the set of real points is non-empty for ...
5
votes
3answers
701 views

Fake projective spaces

I'm looking for examples of "fake" projective spaces. Question: Are there smooth manifolds other than $\mathbb{C}\mathbb{P}^n$ whose cohomology ring is the truncated polynomial ring ...
-1
votes
1answer
210 views

Topological degree of homogeneous function of degree k [closed]

Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k? This is true if F is ...
0
votes
0answers
134 views

Symplectic submanifolds in $\mathbb{R}^4$

Which symplectic submanifolds can be realized in $\mathbb{R}^4$ with standard ($\text{d}\,\boldsymbol{p} \wedge \text{d}\,\boldsymbol{q}$) symplectic structure? It's easy to show that such ...
3
votes
0answers
85 views

Special representations for morphisms of spectra from a smash product

I follow the definitions of spectra, function, morphism, found on Switzer, chapter 8. After definition 8.15 where he defines homotopies of spectra, he says: In terms of cofinal subspectra we can ...
21
votes
6answers
1k views

Down-to-earth expositions of Hodge theory

What are nice expositions of Hodge theory not using advanced language of algebraic geometry or category theory? Of course, since I haven't found a (for me) readable introduction, I don't know what I ...
-2
votes
1answer
215 views

Can you overcome the 6th degree obstruction?

I read and am still thinking about a 3-year old paper from the Danish-Norwegian "Niels Abel Journal". Two authors, named Somethingson (not Jacobson) and another Somethingelseson (still not Jacobson), ...
98
votes
7answers
7k views

Proofs of Bott periodicity

K-theory sits in an intersection of a whole bunch of different fields, which has resulted in a huge variety of proof techniques for its basic results. For instance, here's a scattering of proofs of ...