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

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4
votes
1answer
139 views

Semidirect product of torus with cyclic group: representations/cohomology?

Let $p$ be prime and let $T^p$ be the $p$-torus and $\mathbb{Z}/p$ the cyclic group of order $p$ generated by $(12\ldots p)$. Consider the semidirect product $T^p\rtimes\mathbb{Z}/p$ with the natural ...
3
votes
1answer
90 views

Relative flasqueness?

It is known that a flasque sheaf on a topological space has trivial cohomology. Suppose that we are in a relative situation of a smooth fibration $\pi: X \to S$ and $F$ is a sheaf on $X$. Is there ...
7
votes
1answer
282 views

Two questions about sphere bundles

I would like to better understand the relationship between different notions of orientable sphere bundle. Let me say that a locally trivial fiber bundle $\pi\colon E\to M$ with fiber $S^n$ and ...
2
votes
1answer
429 views

A question about the universal coefficients theorems

This seemingly simple question stands unanswered on math.stackexchange.com for a couple of days (http://math.stackexchange.com/questions/680211/a-question-about-the-universal-coefficient-theorem), so ...
11
votes
1answer
288 views

Are totally degenerate chains null-homologous?

Let $X$ be a CW complex. Suppose $\gamma\in C_n(X)$ is a cycle which is a sum of maps $\sigma:\Delta^n\to X$ which factor as $\Delta^n\to\mathbb R^{n-1}\to X$. Does it follow that $\gamma$ is ...
21
votes
5answers
1k views

(Short) Exact sequences with no commutative diagram between them

This question was asked by a student (in a slightly different form), and I was unable to answer it properly. I think it's quite interesting. The problem is to produce an example of the following ...
11
votes
0answers
333 views

Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
2
votes
1answer
124 views

Seifert fiberable manifolds with several Seifert fiberings

I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds: Hatcher: Notes on Basic 3-Manifold Topology. Regarding the class d), it follows from Proposition 2.1 on page 31, that ...
0
votes
0answers
61 views

Problem on infinite dimensional metric space, with rigidity assumption

By inspiring from this answer of S. Ivanov, here is a specialization with a rigidity assumption. Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces verifying : ...
1
vote
0answers
74 views

PL or projective PL map on the links of a PL manifold

Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
20
votes
0answers
470 views

On a homological finiteness condition

Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated. Question: does there exist a finite CW complex $Y$ and a map ...
1
vote
0answers
47 views

Twisted calibrations and sufficient conditions on homology of sub-manifolds

I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...
7
votes
2answers
265 views

Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex

Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence $$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to ...
9
votes
0answers
230 views

Cohomology and impossible figures

In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": ...
1
vote
2answers
243 views

Equivalence of homotopy categories and model structure theory

I apologize if this is to basic for MO, but it just seems a bit to advanced for SE. Let $\text{Top}$ be the category of topological spaces and $\text{sSet}$ the category of simplicial sets. There is ...
8
votes
0answers
157 views

Homology of compact symmetric spaces

Could anyone point me to a table giving the homology of all compact symmetric spaces, i.e. $G/U$ where $G$ is a compact Lie group and $U$ the fixed points of an involution of $G$? I'd be happy even ...
2
votes
0answers
152 views

Homotopy groups of the classifying spaces BO and BTOP

I was reading a proof and it concludes, from the fact that the rational Pontryagin classes are topological invariants, that the map $\pi_{4n}(BO)\rightarrow\pi_{4n}(BTOP)$ induced by the inclusion ...
9
votes
1answer
284 views

Are these two notions of “dualizable” spectra equivalent?

A spectrum $X$ is dualizable if the natural map $$Map(X,\mathbb S) \wedge X \rightarrow Map(X,X)$$ is an equivalence of spectra. This is equivalent to having evaluation and coevaluation maps in the ...
2
votes
0answers
135 views

Stacks and Maurer-Cartan elements

One can associate to any deformation problem a dg Lie or $L_{\infty}$-algebra $g$. For instance, in algebraic deformation theory, let's say the deformation theory of algebras over a Koszul operad $P$, ...
1
vote
1answer
181 views

Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$

Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
1
vote
1answer
111 views

A formula for isotropy group $\pi_1(G_a)$

Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
5
votes
2answers
201 views

computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...
2
votes
0answers
194 views

Invariants of solutions of systems of equations

What can be said about invariants of zero set of a function that don't change under small enough continuous perturbations of the functions? I define an $\epsilon$-perturbation $g$ of $f$ to be any ...
10
votes
0answers
215 views

Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$. Nondegenerate here means ...
3
votes
0answers
175 views

Bordism and complex $K$-theory

In work related to positive scalar curvature, Stephan Stolz (Ann. Math.), but later Stolz-Kreck ($HP2$ bundles and Elliptic cohomology) introduced a version of Real connective $K$-homology by ...
10
votes
3answers
455 views

Space of embeddings of circle in a surface

Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$. Question : what is the ...
12
votes
1answer
255 views

Equivariant classifying spaces from classifying spaces

Given compact Lie groups $G$ and $\Pi$, there is a notion of "$G$-equivariant principal $\Pi$-bundle", and a corresponding notion of classifying space, often denoted $B_G\Pi$, so that $G$-equivariant ...
4
votes
1answer
228 views

What's a good reference for the following obstruction theory yoga?

Fix a colored operad, which I will leave implicit, and a field $\mathbb K$ of characteristic $0$. By algebra in this post I will mean a dg algebra over $\mathbb K$ for the given colored operad. I ...
0
votes
0answers
92 views

fixed simplicial complex under group action

I have found in an article dealing with combinatorial manifolds the following definition: Let $C$ be a finite simplicial complex, and let $G$ be a finite group acting by automorphisms of $C$. The ...
3
votes
1answer
175 views

Cohomologically minimal spaces

Let $X$ be a compact connected Hausdorff topological space. We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property: "For every proper subset ...
13
votes
1answer
446 views

Adams' theorems on the Hopf-Whitehead J-homomorphism

The J-homomorphism is a well-known and classical map $\pi_n (O(k)) \to \pi_{n+k} (S^k)$, or after stabilizing with respect to $k$, a map $J_n:\pi_n (O) \to \pi_{n}^{st}$, from the stable homotopy of ...
20
votes
4answers
1k views

origin of spectral sequences in algebraic topology

I have the following somewhat vague question. I am not sure if it is appropriate for this forum, please feel free to close (or migrate to stackexchange). I have been "brought up" as an algebraic ...
7
votes
2answers
236 views

Reference Request: Compact manifolds with boundary have the homotopy type of a CW-complex

Let $M$ be a compact manifold (possibly non-smooth) manifold with boundary $\partial M$. Is the inclusion $\partial M\hookrightarrow M$ homotopy equivalent to the inclusion of a subcomplex into a ...
1
vote
1answer
212 views

What's the relationship between B Aut(G) and B Aut(BG) for a (discrete) group G?

(Some of the notational choices I"m about to make might be iffy; I'm happy to take suggestions for improvements.) Let $G$ be a (discrete) group. Think of it as an object in the $2$-category of small ...
-1
votes
1answer
150 views

A closed manifold with a subset with the same ring cohomology

Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies? In this question ...
2
votes
0answers
141 views

Why is a spectrum (in topology) called a spectrum? [duplicate]

The title says it all, I guess. Thank you for your answers.
9
votes
1answer
199 views

Equivariant homotopy, simplicially

It is a classic result of Kan that the homotopy categories (with appropriate model structures) of simplicial sets and of topological spaces (in fact, one could only care about CW-complexes) are ...
1
vote
1answer
135 views

comparison between two monadic definitions for an operad

According to May, an operad $\mathcal{C}$ valued in sets is equivalent to a monad in Cat on the endofunctor $C\colon X\mapsto \coprod_i \mathcal{C}(i)\times X^i.$ According to Leinster, an operad is ...
9
votes
1answer
309 views

Stable moduli interpretation of $\mathbb{R}\mathrm{P}^\infty_{-1}$

I attended a talk recently which closed with the following tantalizing facts: there is a naturally occurring map of spectra $$K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1},$$ which can be ...
1
vote
0answers
82 views

Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...
5
votes
1answer
227 views

Image of J splitting

The image of J space is defined for $p$ odd as the homotopy fibre $J_{(p)}$ of the self map $$\psi^k - 1: BU_{(p)} \to BU_{(p)}.$$ Here, $\psi^k$ is an Adams operation, and $k \in \mathbb{N}$ ...
34
votes
2answers
896 views

Is there an analog of Sperner's lemma for the Hopf invariant?

Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic." My question is, does there exist ...
1
vote
0answers
100 views

When is equivariant cohomology generated by equivariant Euler classes?

Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map ...
4
votes
0answers
68 views

Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin

Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...
3
votes
0answers
118 views

Robust Invariants of a solution of systems of equations

Let $X$ be a $k$-dimensional triangulated manifold and $A:=\partial X$ be its boundary. Let $f:X\to R^d$ be a continuous function such that $g:=f|_{A\cup X^{(i-1)}}$ avoids zero and let $z_g\in Z^{i} ...
1
vote
1answer
97 views

Image of the map induced on homology by a covering

I asked this question on math.se (http://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering), but it have not attracted much of attention. Let $X$ and $Y$ are ...
3
votes
0answers
197 views

Homotopy category of groupoids

The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description: iso-classes of functors. formally invert equivalence functors (i.e. ...
4
votes
1answer
135 views

Sullivan's $H$-space equivalence between $G/PL[1/2]$ and $BO[1/2]$

There is a theorem by Sullivan of the following form: Theorem: There is an equivalence of $H$-spaces $$ G/PL[\tfrac{1}{2}] \simeq BO_{\otimes}[ \tfrac{1}{2} ]\ . $$ It can be found for example ...
3
votes
0answers
115 views

Topological interpretation for groups of type $FP_2$

A group $G$ is of type $FP_2$ if it admits a partial projective resolution of $\mathbb{Z}G$-modules $$ P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0$$ with each $P_i$ being ...
1
vote
0answers
62 views

Homology of a higher-dimensional full torus with a certain disk removed

Let $X=S^2\times B^5$ and let us consider an embedding of an open $5$-disk in $X$ with its boundary glued by a map of degree $2$ to some $\{ s_0\} \times S^{4}\subseteq \partial X$ . The embedding ...