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1
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1answer
358 views

Can a function from R^n to R^n fail to be one-to-one, and also lack critical points? [closed]

I feel like there should either be a straightforward example of such a thing, or a homological reason why it can't exist. The reason why I ask is that if the Hessian matrix of a Lagrangian is ...
11
votes
1answer
914 views

Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1

So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$. For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
4
votes
2answers
634 views

Homology of Covering Spaces

Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...
0
votes
1answer
266 views

Simple Equivariant homology [no borel-Moore]

Hey. I'm working with Bredon's equivariant cohomology. At some point I need to compute the $4$th equivariant cohomology group of $S^1 \x D^3$ relatively to its boundary for the antipodal action of ...
0
votes
1answer
266 views

What is a quasi-isomorphism of two crossed modules

Could you tell me how are two crossed modules quasi-isomorphic. And I have known a result: Let $\mu: M \rightarrow G$ and $\mu': M' \rightarrow G'$ are isomorphic, then the integral homology of them ...
14
votes
2answers
611 views

Massey Products vs. $A_\infty$-Structures

Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More ...
11
votes
1answer
637 views

Does the bordism homology theory satisfy the weak equivalence axiom?

There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is ...
2
votes
1answer
384 views

Rational Homology of a Covering Space

I have heard that the rational homology of a covering space is easy to compute, compared with the ordinary homology. However, I don't know any details about that. Can anyone help me? Any reference ...
1
vote
1answer
397 views

Computing the homology groups of spaces in a fibration

Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...
1
vote
1answer
265 views

Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of the coboundary map on the cochain complex of abelian cosimplicial groups. What I have in mind is something similar to the ...
12
votes
2answers
596 views

Are acyclic subcomplexes of finite contractible 2-complexes contractible?

Let $Y$ be a contractible finite simplicial 2-complex. Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$). Is $X$ contractible? (Equivalently, is $\pi_1(X)$ ...
22
votes
3answers
979 views

Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain $G_0 > G_1 > ... > ...
18
votes
3answers
2k views

Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...
0
votes
0answers
57 views

weak divisble module [duplicate]

hi, all! I want to know what is the definition of reduced module over general ring. I remember that: a module $B$ always have smallest submodule $D(B)$ satisfy $B/D(B)$ is divisible module. If ...
5
votes
4answers
2k views

Examples of non-simply connected manifolds with trivial H^1

It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can ...
8
votes
0answers
293 views

Singular chains as an HZ-module spectrum

For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of ...
6
votes
1answer
306 views

Third bordism group of BG, where G is an arbitrary compact Lie group.

Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...
2
votes
1answer
395 views

Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let $$ ...
1
vote
0answers
257 views

connection between non-orientable manifolds and torsion in 1D (co) homology

I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex. Lets say you have a ...
3
votes
0answers
115 views

Uniform distribution of special homology classes mod-p

Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here ...
1
vote
1answer
660 views

Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...
3
votes
3answers
325 views

Reference needed: Isomorphism on pi_1 and homology gives weak equivalence

Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi_1(X) \to \pi_1(Y)$ is an isomorphism, and $H_n(f) : H_n(X, G) \to H_n(Y, G)$ is an isomorphism for all $n \ge ...
13
votes
1answer
416 views

Ordinal-indexed homology theory?

Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ...
6
votes
3answers
530 views

Can homologous submanifolds be connected by an immersed manifold with boundary?

Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional ...
2
votes
2answers
334 views

Algorithm that decreases the size of the simplicial triangulation

Hello! Let X be a topological space. We are considering only abstract simplicial complexes, i.e. a finite list of finite lists of integers. Is there any algorithm (more or less efficient?), that ...
0
votes
0answers
186 views

map of inductive systems of distinguished triangles

I have for any $n$ some distinguished triangles $$X_n->Y_n->Z_n$$ $$X'_n->Y'_n->Z'_n$$ The families $(X_n), (Y_n), (X'_n), (Y'_n)$ are inductive systems, $(X_n)$ is quasi-isomorphic to ...
25
votes
3answers
3k views

Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
1
vote
1answer
191 views

Equivariant homology of Hilb and torus stable curves

The torus equivariant homology of a compact homogeneous variety $G/B$ may be described by using the "skeleton" consisting of torus fixed points and the torus invariant curves that connect them. ...
5
votes
2answers
631 views

Proof of the ''trangression theorem''

Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
7
votes
2answers
912 views

Deformation theory of co-$A_\infty$ structures.

The following question is related to my previous post on co-$A_\infty$ spaces (co-$A_\infty$ spaces), but goes in a somewhat different direction. Some Background: In trying to classify $A_\infty$ ...
4
votes
4answers
2k views

How to triangulate real projective spaces (as simplicial complexes in Mathematica)?

Hello! I have written a program in Mathematica 7, which calculates for a (finite abstract) simplicial complex all its homology groups. I would really like to test it on the projective spaces, but ...
6
votes
2answers
681 views

Simple examples of equivariant homology and bordism

I'm looking for simple examples of calculations of equivariant homology and of equivariant bordism. I have a finite group G acting on an CW-complex X. I would like to calculate the equivariant ...
15
votes
2answers
1k views

Is there a map of spectra implementing the Thom isomorphism?

A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: ...
15
votes
2answers
968 views

Simple curves on non-orientable surfaces.

Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...
2
votes
1answer
143 views

The growthrate of the homology of $H_*(M^{\otimes_A n})$ for a DG-bimodule $M$

Suppose you have an DG-algebra $A$, and a DG-bimodule $M$ over $A$. Under which conditions will the rank of the bimodules $H_*(M^{\otimes_A n})$ will grow exponentially in terms of $n$? Here ...
1
vote
0answers
231 views

Homology of nice planar sets

Is there a quick and simple proof of the fact that the homology group of a nice (say with piecewise smooth boundary) planar domain is free abelian with a basis corresponding to the holes in the ...
3
votes
1answer
750 views

Homology is computable because it is stable under suspension

I've heard it said that the reason why the homology groups of a space are a computable invariant is because they are a stable invariant in the sense that they are stable under suspension. I'm ...
1
vote
1answer
356 views

When do submanifolds lie in the same homology class? [closed]

Hello, this may be a trivial question, but I am not very familiar with the topic. Let (M,g) be a Riemannian Manifold. (In fact, we don't need the metric here.) What exactly does it take for two ...
6
votes
3answers
530 views

Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$: ...
3
votes
0answers
356 views

Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?

In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes ...
17
votes
2answers
1k views

What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...
7
votes
2answers
775 views

Are the homology and cohomology Serre spectral sequences dual to each other?

If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...
4
votes
1answer
596 views

Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is $H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except ...
10
votes
1answer
524 views

What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map ...
3
votes
1answer
531 views

Computation of homology groups of $M_{g,n}$

First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, ...
2
votes
2answers
871 views

Proving homotopy invariance of cellular homology by constructing a chain homotopy

I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy ...
5
votes
1answer
1k views

quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?
3
votes
3answers
930 views

When are the homology and cohomology Hopf algebras of topological groups equal?

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology ...
7
votes
2answers
692 views

The word “torsion” and its connection to geometry and homology

In an $R$-module $M$, an element $m \in M$ is said to be torsion if $am = 0$ for some $a \in R$ with $a \neq 0$. Also, for a non-orientable (closed) surface such as the projective plane or the Klein ...
4
votes
2answers
2k views

Singular homology of a graph.

By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ ...