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

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3
votes
2answers
74 views

Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?
6
votes
0answers
102 views

Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...
8
votes
2answers
199 views

Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...
8
votes
0answers
84 views

Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces $$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction. Richard ...
6
votes
1answer
146 views

Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
-10
votes
0answers
94 views

Please help me for answers to question. best regards [on hold]

Find the definition of a locale and its dual (i.e. a frame) and consider the definition of a Grothendieck topology. Discuss the differences between this concept and an ordinary topology on a set ...
9
votes
2answers
268 views

Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.
2
votes
1answer
101 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
8
votes
1answer
126 views

Schur multiplier of $Sp(2g, \mathbb{Z}_2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}_2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}_2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}_2$. With respect to ...
0
votes
0answers
30 views

to find topological properties under a metric on a set [on hold]

we define a metric d on a set of composition operators on L2. I would like to find connected component and path connected component and other topological properties by d . Is there any book or paper ...
8
votes
1answer
208 views

$K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism $$ K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}), $$ which is induced by the Chern character. What is the analogous statement for $KO(X)$? ...
2
votes
0answers
95 views

The homology of $\varinjlim SO(p,q)$

Is there a way to explicitly compute the homology of the space $$ \varinjlim_{(p,q)} SO(p,q)^+, $$ where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...
2
votes
1answer
86 views

Lifting a differential operator

Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...
24
votes
0answers
254 views

What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...
3
votes
0answers
90 views

Intersection patterns of loops on surfaces

Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on ...
3
votes
1answer
207 views

Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...
1
vote
1answer
253 views

research articles in topology/geometry [closed]

There is a saying "Do you read the masters?" I want to read some basic papers in Topology/geometry... I can not clearly state what is basic as of now... My back ground includes course in ...
4
votes
1answer
180 views

The fibration map $Diff(M) \rightarrow Emb(N,M)$

Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map $$ Diff(M) \rightarrow ...
2
votes
0answers
71 views

A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces. The Künneth-Theorem which I ...
3
votes
0answers
101 views

Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...
10
votes
2answers
466 views

Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows. Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...
2
votes
0answers
79 views

Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [closed]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
3
votes
2answers
342 views

Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...
1
vote
0answers
131 views

Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?
4
votes
1answer
218 views

Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question. In the knot theory, the Reidemeister moves play fundamental roles. ...
4
votes
0answers
123 views

Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
2
votes
0answers
103 views

Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$ If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...
6
votes
0answers
685 views

Ordinary cohomology groups of $(\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ that I need for my string theory research

Let $X=(B^6\times T^2)/\mathbb{Z}_k\subset (\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ where $k=2,3,4,6$, where the generator of $\mathbb{Z}_k$ acts on $\mathbb{C}^3$ by the multiplication by a primitive ...
0
votes
0answers
119 views

Is there a t-structure on the homotopy category of spectra that has the sphere spectrum in its heart?

Maybe the heart of such a t-structure should be the category of abelian groups, and the t-homology functor should be given by taking usual homology groups of the spectrum. Is it impossible?
10
votes
2answers
464 views

A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms ...
4
votes
0answers
57 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
0
votes
1answer
159 views

group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram If $\xi$ is a trivial bundle, ...
7
votes
1answer
142 views

(Non)-equivariant equivalence in $G$-spectra

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in ...
8
votes
1answer
267 views

Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
0
votes
1answer
59 views

Conjugation Cells [equivariant cohomology]

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5) ...
3
votes
0answers
153 views

some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus ...
9
votes
1answer
319 views

Wild half-line in a Euclidean space

Is there an $m$-dimensional simplicial complex $S$ with the following properties: The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...
30
votes
1answer
660 views

A dictionary of Characteristic classes and obstructions

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians. In an effort to ...
16
votes
2answers
963 views

Why do people say DG-algebras behave badly in positive characteristic?

It seems to be a common wisdom in derived algebraic geometry that commutative DG-algebras are not, in general, a good model for derived rings, with the stated reason that they behave badly in positive ...
6
votes
0answers
92 views

Detection tools for (reduced) suspension

I'm learning about loop spaces and the work of Stasheff on $A_{\infty}$-spaces. The broad idea that I'm getting is the following. Given a space $Y$, we want to know under which conditions there exists ...
6
votes
3answers
758 views

Does anyone know the classification of fourth order surfaces?

Does anyone know the classification of fourth order surfaces? By "fourth order surface" I mean a surface defined by an equation of the form $$f(x, \, y, \, z)=0,$$ where $f$ is a polynomial of degree ...
4
votes
0answers
101 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
4
votes
1answer
133 views

The principal bundle of embeddings

In a paper of P. Michor, it was shown that Emb(M,N) is a smooth principal diff(M)-bundle, M and N are smooth locally compact manifolds provided dim M < dim N. My question is why there is a ...
3
votes
0answers
98 views

Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...
7
votes
1answer
346 views

higher algebraic homotopy groups for schemes?

I think I understand how to define the algebraic fundamental group $\pi^{alg}_{1}(X)$ of a scheme and I think I understand the relation between $\pi^{alg}_{1}(X)$ and $\pi_{1}(X(\mathbb{C}))$, where ...
6
votes
0answers
201 views

Two different Thom diagonals in recent literature?

Taking the point of view that a Thom spectrum functor should be a map $Top_{/BGL_1(R)}\to LMod_R$, for $R$ some $\mathbb{E}_n$-ring spectrum, there seem to be two morphisms (in $Top_{/BGL_1(R)}$) that ...
1
vote
0answers
207 views

Can one prove the poincare duality for projective scheme by proving it for projective space?

It's well known the relationship between Poincare duality and Thom isomorphism(I mean cohomology purity $R^q i^! F=0$ if $q\neq c $ ) $\quad $ $Rf_!Ri_!=R(f|_Z)_!$ where f is $P_k^n\rightarrow k$ ...
13
votes
0answers
177 views

Uniqueness of connected cover of Morava K-theory

Let $k(n)$ denote the connected cover of Morava $K$-theory $K(n)$ at the prime $2$ and in particular $n=2$. It is known that $$ H^*(k(n)) = A//E(Q_n), $$ where $A$ is the Steenrod algebra and $Q_n$ is ...
10
votes
1answer
285 views

Dimension in CW-approximation

The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question. Let $X$ be a topological space, and let $\tilde{X}\to X$ be a ...
18
votes
5answers
920 views

Book recommendation for cobordism theory

I am planning to organize a seminar on cobordism theory and I'm looking for a reference. Such a reference is preferably a book, but I'm open to other ideas. The audience is familiar with ...