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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Relative Homology and Quotients

Are there any topological spaces $X$ with subspaces $A$ such that $H_n(X,A)$ is not isomorphic to $H_n(X/A)$? I've been trying some familiar spaces, but everything seems to be me an isomorphism via ...
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187 views

diagonal image of a primitive homology class

Let $X$ be a topological space. A class $a\in H_n(X;\mathbb Z)$ is said to be primitive if $a\not = m b$ for every integer $m>1$ and $b \in H_n(X;\mathbb Z)$. Let $$\Delta_*:H_n(X;\mathbb Z)\to H_n(...
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428 views

Cohomology of fiber bundle with a section

Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map $$ f^{*}:H^{*}(B,\...
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215 views

tensor product of two chain homotopic maps are again chain homotopic?

Let $C$, $C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps. How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to C'...
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747 views

understanding the torus described as a product of two circles

I would like to ask two (related) things about the torus: The torus can be described as the cartesian product $S^1 \times S^1$ of two circles in $\mathbb{R}^3$. Then one can talk about meridional ...
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824 views

Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.
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Decomposition of vector bundles over a CW complex

Let $X$ be CW complex having only cells up to dimension $n$. I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a $n$-...
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119 views

Fundamental group of the complement of a linear subspace

Let $m<n-1$ be two positive integers. Consider $\mathbb{R}^m$ as a subspace of $\mathbb{R}^n$ via $\mathbb{R}^m\times \{(0,0,...0)\}$. Any suggestions on how to compute $\pi_1(\mathbb{R}^n\...
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341 views

Definition of Cobordism

I am currently trying to read Quillen's Cobordism paper - On the formal group laws of unoriented and complex cobordism theory Bull. Amer. Math. Soc, 75 (1969) pp. 1293–1298. Still on the first page ...
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233 views

Properties of Surgery on Manifolds?

I am trying to give a brief explanation in which I make use of the concept of surgery on an $m$-manifold $M$. This is along the lines of (and taken generously from) the Wikipedia entry on Surgery; ...
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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 ...
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171 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 N\...
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71 views

what is a (co)homology theory?

There are different axioms for a (co)homology theory: Homotopy invariance is always there, but for the rest there is or isn't: long exact sequence for pairs of topological spaces exact sequence for ...
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69 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
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52 views

Cohomology of local systems on spaces with the same homotopy type

Suppose we have a homotopy equivalence $f: X \to Y$ with homotopy inverse $g: Y \to X$, and a local system (i.e. a locally constant sheaf) $\mathcal{V}$ on $Y$. In this situation, are any of the ...
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44 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, ...
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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 ...
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55 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 ...
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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 ...
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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 ...
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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 ...
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Computing fundamental group. Using Van Kampen? Visualization of a space

I'm trying to compute the fundamental group (using Van Kampen) of a space which appears when identifying the disjoint boundaries of a 3-manifold with boundaries. My knowledge of 3-manifolds is none ...
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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 ...
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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 ...
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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 ...
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28 views

Does the sum of weights in Kirchhoff’s construction equal the Gram determinant?

Background: An electrical network is modeled by a complex. Branch current distributions $\mathbf I\in C_1$ are represented by $1$-chains; branch voltage drop distributions $\mathbf V\in C^1$ are $1$-...
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130 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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121 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 $L$ ...
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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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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) = \...
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63 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 $\...
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63 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 ...
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109 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 $...
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129 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 this ...
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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 ...
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$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

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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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 I}...
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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 ...
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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 $\overline{\mathbb C\mathbb P^2}$. ...
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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 answer,...
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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 ...
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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 $L(p,q)$?
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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 (S^...
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44 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 ...
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88 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, ...
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90 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 ...
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134 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: ...
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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: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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178 views

A question on Hawaiian earring

Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group $$\...