Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, fundamental groups, covering spaces, and beyond.

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5
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228 views

Smith normal form of graded modules. (Major edit)

Ok, this is a major rewriting of my previous entry which no one answered. Let us have two graded $F[t]$-modules M and N with bases $m_1, \ldots, m_m$ and $n_1, \ldots, n_n$, respectively, and $F$ is ...
5
votes
0answers
170 views

Invariance of Wall's self-intersection under the regular homotopy

For an immersion $f\colon N^n\to M^{2n}$ with fixed lift $\tilde{f}\colon N\to \tilde{M}$ and $N,M,\tilde{M}$ are oriented, first we define a unordered double point set $S_2[f]=\lbrace(x_1,x_2)\in ...
4
votes
0answers
39 views

a topological space with finite integer first cohomology group?

One more problem preparing for a PhD exam! It states "describe a space such that $H^1(X,Z)=Z_5$." I thought this was impossible by the universal coefficient theorem since $H_0(X;Z)$ is always free, ...
4
votes
0answers
79 views

Different Proof for $\mathbb{R}^m\cong \mathbb{R}^n$ if and only if $n=m$.

Using homotopy it is easy to prove that (in topology) $\mathbb{R}^n\cong \mathbb{R}^m$ if and only if $n=m$. This result seems intuitively true, but, as realized very earlier and almost everyone who ...
4
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25 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces ...
4
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0answers
51 views

group actions of fundamental groups on homotopy groups

Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to ...
4
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0answers
36 views

Source request for $H^*(B\mathrm{TOP},\mathbb{Q})\cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
4
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53 views

Homotopy of boundary paths

Let $G$ be a bounded, simply connected, open set in $\mathbb{R}^2$, and let $\gamma_1$ and $\gamma_1$ denote two paths such that $\gamma_0(0)=\gamma_1(0)$, $\gamma_0(1)=\gamma_1(1)$, and the following ...
4
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34 views

If $M = \partial W$, with $W$ parallelizable, then an embedding $\iota : M \to S^{n+k}$ extends to an embedding $W \to D^{n+k+1}$

Suppose $M$ is an $n$-dimensional $s$-parallelizable manifold which is the boundary of the parallelizable compact manifold $W$. It is claimed in Milnor & Kervaire's Groups of Homotopy Spheres ...
4
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102 views

Surjectivity of Edge morphism in A-H cohomology Spectral Sequence

As the title suggests, I'm interested in proving the following claim: Recall the AH-spectral sequence:$$ E_2^{pq}=H^p(X,\mathcal{H}^q(\ast)) \Longrightarrow \mathcal{H}^{p+q}(X)$$ and since ...
4
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50 views

Examples of calculating perverse sheaves on algebraic varieties with easy stratification.

This question is also asked in mathoverflow http://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati I have been learning intersection ...
4
votes
0answers
80 views

Lie-group existence on universal covering manifold

Let $X$ be an n-dimensional smooth manifold with Lie group $G$ acting transitively on $X$, i.e. $X$ is a homogeneous space. Let $\tilde{X}$ be the associated universal covering space. To what extent ...
4
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45 views

Characterisation of Stable Cohomological Operations: $\Sigma (\tau_n (\imath_{A,n}))=\rho_{n+1}^*(\tau_{n+1}(\imath_{A,n+1}))$

I've started studying (stable) cohomological operations on my lecture notes, and I was given that an equivalent definition for a family of cohomological operations to be a stable cohomological ...
4
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129 views

“Global” dimension for topological spaces // Geometric interpretation for global dimension of rings

The global dimension of a ring $R$ is the supremum of the projective dimensions of it's $R$-modules. $$\dim (R)=\sup \{\dim_\mathrm{proj}(M):M \in R\text{-mod} \}$$ I'd like to have some geometric ...
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108 views

Learning Galois theory geometrically?

Recently I started poking at algebraic geometry and commutative algebra. My background is basic category theory and basic algebraic topology. I don't know a lot of other mathematics. I noticed Galois ...
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59 views

Curves Knotted in the Torus

I've been struggling with the following problem. I suspect it's actually not so hard, but I'm missing something relatively obvious about the proof. The attached image will help. Suppose $K$ and $L$ ...
4
votes
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51 views

Computing homology of square with all vertices identified.

I'm trying to compute the homology of $X = (I \times I)/\sim$, where $(0,0)\sim (0,1) \sim (1,0) \sim (1,1)$. I want to do this via cellular homology, using degrees, etc, but I don't got that very ...
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72 views

Direct Summand and Intersection Homology

Let $X$ the projective cone of $C$, here $C$ a curve of genus $g$. We can compute its intersection cohomology : $IH^0(X) = \mathbb Q, IH^1(X) = \mathbb{Q}^{2g}, IH^2(X) = \mathbb 0, IH^3(X) = ...
4
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57 views

Can a disconnected surface have a (negative) genus?

This question is rather about a convention. Is it possible (and conventional) to asign to, say the disconnected sum of two connected surfaces $X=\Sigma_h \sqcup \Sigma_g$ a genus? Since one has ...
4
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0answers
61 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
4
votes
0answers
92 views

Question about Gysin map (pushforward in cohomology)

The setting is the following: $X \subset \mathbb{CP}^r$ is an algebraic subvariety of dimension $n$, codimension $e=r-n$, and degree $d$. Call $j$ the inclusion. Then, Poincaré duality induces a map ...
4
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127 views

How to classify this surface

I know that it should be either a sphere, torus, Klein bottle, real projective plane, or a connect sum of any combination of these, but I don't know the steps in identifying what kind of surface ...
4
votes
0answers
38 views

Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...
4
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79 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
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votes
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41 views

The homology groups of an infinite product of spaces

Suppose $I$ is some index set and let $\{X_i \}_{i\in I}$ be a collection of topological spaces (as nice as you like them to be). What is known about the (say) singular homology of $X := \prod_{i\in ...
4
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0answers
29 views

Properties of spectra

I've been trying to understand spectra better, and I was wondering what would be a good source to learn from. My issue with the classical treatments (e.g., Adams, Switzer) is that people keep telling ...
4
votes
0answers
54 views

Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$

I would like to compute the first Chern class of $\mathbb C\mathbb P^2 \# 6\overline{\mathbb C\mathbb P^2}$ in terms of the generators of $\mathbb C\mathbb P^2$ and $\overline{\mathbb C\mathbb P^2}$. ...
4
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0answers
79 views

Cohomology of $K(\mathbb{Z}_2, n)$

Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy ...
4
votes
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66 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
4
votes
0answers
61 views

3-manifolds with certain fundamental group

Suppose $M$ is a 3-manifold with $\pi_1(M)=\mathbb{Z}/p\mathbb{Z}$. Is it true that $M$ is diffeomorphic to a lens space $L(p,q)$?
4
votes
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105 views

Mapping $f: (S^n, pt) \to (S^n, pt)$ induces multiplication by $deg(f)$ on $H_*(S^n \wedge X, pt)$.

I'm studying for my final exam in Algebraic Topology and got stuck with somewhat straightforward looking preparation problem. Let $\tilde H$ be any reduced homology theory and $f: (S^n, pt) \to ...
4
votes
0answers
43 views

Definitions of the ring $R/(x_0^\infty,\ldots,x_{n-1}^\infty)$

Let $R$ be a ring, and let $I = (x_0,\ldots,x_{n-1})$ be a finitely-generated ideal inside of $R$, generated by a regular sequence. In algebraic topology one often encounters a ring, usually denoted ...
4
votes
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87 views

Does naturality for characteristic classes imply the classifying space is universal for them?

Let $G$ be a Lie group, $\mathfrak g$ its Lie algebra, $K$ its maximal compact subgroup. To every flat $G$-bundle $P$ over a smooth manifold $M$ I can associate a homomorphism $w_P: H^*(\mathfrak g, ...
4
votes
0answers
86 views

When the free loop space fibration splits?

Let $X$ be a (nice) connected topological space. Let $LX=Map(S^1,X)$ be the free loop space and $\Omega X = Map_*(S^1,X)$ the subspace of based loops (with some choice of base point for X). Now, there ...
4
votes
0answers
130 views

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
4
votes
0answers
82 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
4
votes
0answers
77 views

Proof of the isomorphism $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2).$

I want to prove the following: $[X, \mathbb{R} P^\infty] \cong H^1(X; \mathbb{Z}/2)$ via the map $[f] \mapsto f^* w_1(\gamma^1)$ where $\gamma^1$ denotes the tautological line bundle over ...
4
votes
0answers
153 views

Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general situation. Especially I do not work in any "convenient" category of spaces. Wikipedia ...
4
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0answers
46 views

Inductively Constructing Chain Homotopies

Let $f,g:C_\bullet \rightarrow D_\bullet$ be chain maps. Let $T$ be a generator (possibly not the only one) of $C_n$ and assume $H_n(C)=0$. I'm trying to prove that for every $T$ there's a solution ...
4
votes
0answers
85 views

Kan's loop group construction

I'm looking for a good place to read about the loop group construction $G : \mathbf{sSet_0} \to \mathbf{sGrp}$ taking a reduced simplicial set $X$ and producing a simplicial group $GX$. I would also ...
4
votes
0answers
154 views

Smash and join products of spheres

How to prove rigorously that $\mathbb S^n \wedge \mathbb S^m =\mathbb S^{n+m}$ and $\mathbb S^n \ast \mathbb S^m = \mathbb S^{n+m+1}$? And what intuition should i have for compute $\wedge,\ast$ for ...
4
votes
0answers
154 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
4
votes
0answers
56 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
4
votes
0answers
159 views

On the path-connectedness of $X_{i} \setminus X_{i-2}$

Suppose $X$ is an $n$-dimensional regular CW-space (a space with a regular CW decomposition). What are the weakest sufficient conditions required on $X$ to ensure that all regular CW decompositions ...
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45 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
4
votes
0answers
67 views

Minimal resolutions of singularities in higher dimensions and their cobordism class.

For a given singular algebraic variety over $\mathbb{C}$, how many minimal resolutions can exist? More specifically, consider a projective quadratic cone $Q$ in $\mathbb{C}P^{4}$ (with homogeneous ...
4
votes
0answers
92 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
4
votes
0answers
226 views

The unit square has dimension two?

Show that the Lebesgue Covering Dimension of the unit square $I^2$ with $I=[0,1]$ is two. I know that a compact subset of Euclidean space $\Bbb R^n$ has Lebesgue dimension at most $n$. So it ...
4
votes
0answers
38 views

pontriyagin class of quaternionic vector bundle

Let $\xi^{\mathbb{H}}$ be a quaternionic vector bundle over $X$. How to define the Pontriyagin class of $\xi^{\mathbb{H}}$ efficiently? Of course we can let $(\xi^{\mathbb{H}})_{\mathbb{R}}$ be the ...
4
votes
0answers
125 views

Prove that the quotient map $P:G \to G/H$ is a covering space.

Let $G$ be a topological group and $H$ is a subgroup of $G$. Suppose that the subspace topology on $H$ is the discrete topology. Prove that the quotient map $P:G \to G/H$ is a covering space. My ...