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

learn more… | top users | synonyms (1)

3
votes
1answer
37 views

Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $ (n - 1) $-sphere in the $ n $-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
0
votes
0answers
38 views

Proving two spaces are homotopy equivalent

We are given a topologic space X, defined as: $$X= \mathbb{S}^2 \cup \mathbb{D}_2 \cup \mathbb{I} \subset \mathbb{R}^3$$ Where $$\mathbb{S}^1=\{ (x,y,z) \in \mathbb{R}^3 | x^2+y^2+z^2=1 \} $$ ...
2
votes
1answer
46 views

Brouwer's fixed-point theorem and the intermediate value theorem?

In one dimension, Brouwer's fixed-point theorem (BPFT) can be proved easily based on the Intermediate Value Theorem (IVT). Is the inverse also true? I.e, is it possible to prove the IVT directly from ...
5
votes
2answers
49 views

Even number of points in the preimage of a regular value of a map $f:M^n \to \mathbb{R}P^n$

Let $M^n$ be a closed manifold (I think we can also assume it is connected, though it isn't explicitly stated). Let $f:M^n \to \mathbb{R}P^n$ be a smooth map, and let be $a \in \mathbb{R}P^n$ be a ...
0
votes
1answer
68 views

homology groups of a torus

How can I find the homology group of a torus without using cellular homology and the CW complex ? in other words , how can i calculate the homology groups of a torus using only relative homology ? I ...
-1
votes
0answers
15 views

degree of orientation-preserving map

Let's consider $f:X\rightarrow Y$ to be a $m$-covering space, with $X$ and $Y$ compact connected and oriented $n$-topological manifolds. Let $\alpha_{x}$ be a generator of $H_{n}(X,X\setminus\lbrace ...
4
votes
1answer
61 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
2
votes
1answer
50 views

Show that the comb space has fixed-point property

By "comb space", I mean the space $X=([0,1] \times \{0\}) \cup (K \times [0,1])$, where $K=\{ \frac{1}{n} : n \in \mathbb{N}^+ \}$, without the leftmost vertical line segment. How to prove that this ...
1
vote
1answer
47 views

Sufficient condition for $\mathbb{Z}$-orientability

Let $X$ be a topological $n$-manifold. Let's define a R-orientation on $X$ as a choice of generators $\alpha_{x}\in H_{n}(X,X\setminus\lbrace x\rbrace;R)$ that is consistent. Suppose that $X$ is ...
2
votes
0answers
56 views

Assignment - Euler characteristic constant under barycentric subdivision

I am working on an assignment and I am stuck, mostly I have no clue how to quite attack it. I don't want the answer or anything just advice on angels at which I can go about this. For an n-dimensional ...
6
votes
3answers
130 views

$S^m * S^n \approx S^{m+n+1}$

I'm interested in showing that $S^m * S^n \approx S^{m+n+1} $, as discussed in exercise 0.18 of Hatcher's Algebraic Topology. One way to show it would be to show that $X * Y \approx \Sigma(X \wedge ...
0
votes
0answers
35 views

Algorithm for finding zero of an odd function from n-sphere -> R^n

There is a well-known Borsuk-Ulam theorem stating that each continuous mapping $f : S^n \rightarrow \mathbb{R}^n$ that is odd in sence of $f(v) = -f(-v)$ for each $v \in S^n$ (where $-v$ denotes the ...
5
votes
1answer
55 views

Does every even-dimensional sphere admit an almost complex structure?

We know that there is an almost complex structure on $S^6$ which is not integrable. Is it always possible to find almost complex structures on $S^{2n}$? In particular does $S^4$ admit one?
0
votes
0answers
26 views

Where can I find Kan's paper “On c.s.s. categories” from 1957.

Does anyone know how I can find the following article by Daniel Kan: Kan, D. M. On c.s.s. categories Bull. Soc. Math. Mexicana (1957), 82-94. Quillen lists it as a reference in his paper Rational ...
1
vote
1answer
35 views

Understanding the generator of $H_1 (S^1 \times I)$

I'm trying to work with the space $X=S^1\times I$. It is obvious that $X\simeq S^1$ and therefore $H_1(X)=H_1(S^1)=\mathbb Z$, but I want the properties of $X$ itself. I would assume that a ...
0
votes
0answers
30 views

Homology groups of the complex projective plane of dimension 2 - an affine line and a point not in the line

This is a question from a problem sheet we had in class and the solution says the following : We first note that the complex projective plane of dimension 2 minus an affine line is isomorphic to ...
1
vote
1answer
36 views

The annulus with with antipodal points on the outer circle identified gives a mobius strip

I ve been told that the real projective plane of dimension two can be expresses as the union of a disk and a mobius strip. The only way that this makes sense to me is that if an annulus with with ...
0
votes
0answers
24 views

Pontryagin classes of a tensor product of bundles

This question is related to " How to calculate characteristic classes of tensor products? " but interested in the Pontryagin classes instead of $c_1$. Specifically, given two real vector bundles $E$, ...
5
votes
2answers
105 views

Concatenating countably many homotopies

On page 15 of Hatcher's Algebraic Topology, he discusses constructing a homotopy $X \times I \to X \times \{0\} \cup A \times I$, where $(X,A)$ is a CW pair. He does so by concatenating homotopies ...
1
vote
1answer
46 views

Let Y and Z subspaces of X such that Y deformation retarcts to Z are their relative homology groups isomorphic?

I was wondering the following let $Y$ a subspace of a space $X$ and suppose there exists another subspace $Z$ of $X$ such that $Y$ deformation retracts to $Z$ does it then follow that ...
0
votes
2answers
15 views

Proving that the number of elements in inversed sets are equal

Define a cover mapping $f:Y\to X$ so that for all $x\in X$ the set $f^{-1}(x)$ is finite. Define a function $g:X\to \mathbb{Z}$ with $g(x) = \# (f^{-1}(x))$, as in: the number of elements in the set ...
1
vote
3answers
38 views

Let Y and Z be homotopy equivalent subspaces of X, are their relative homology groups isomorphic?

I was wondering the following let $Y$ a subspace of a space $X$ and suppose there exists another subspace $Z$ of $X$ such that $Y$ and $Z$ are homotopy equivalent does it then follow that $H_n(X,Y) ...
2
votes
0answers
43 views

References for equivariant cohomology

I am studying the paper An introduction to equivariant cohomology and homology, follwing Goresky, Kottwitz, and Macpherson - Julianna S. Tymoczko but there are too many gaps. I can't link most of ...
0
votes
1answer
42 views

Coherent Topology and Open Covers

Let $X$ be a topological space, and let $\mathcal{A}$ be an open cover for $X$. To say that $Open(X)$ is coherent with $\mathcal{A}$ means that $$B\in Open(X) \Leftrightarrow B\cap A\in ...
-1
votes
0answers
80 views

Is a line with all points 'doubled" a differentiable manifold?

The line with two origins is $ X=\mathbb{R}∖\{0\}∪\{0',0''\}$, that is X is the union of the reals minus 0, and two points. Let, $$U_a=(−a,0)∪{0'}∪(0,a)$$ $$V_a=(−a,0)∪{0''}∪(0,a)$$ where $a>0$. ...
2
votes
1answer
20 views

Sphere bundle of the tangent bundle of 2-dim sphere

Let sph($\tau S^2$) be the sphere bundle of the tangent bundle of 2-dim$^l$ sphere. Could someone tell me why sph($\tau S^2$)=$\mathbb{R}$P$^2$$\cup$$e^3$ holds? Where $e^3$ is a 3-dim$^l$ cell.
0
votes
1answer
31 views

About the Chern class of the determinant line bundle

Is it true that the first Chern class of a rank $k$ complex line bundle is equal to the first Chern class of its determinant line bundle?
2
votes
1answer
30 views

Covering Map of Torus

how can I show that the following map is a covering map of $T:=$ $S^1$ x $S^1$? $\pi: T\rightarrow T$ with $(x,y)$ $\mapsto$ $(x^ay^b, x^cy^d)$, where $a,b,c,d \in \mathbb{Z}$ and $ad-bc=m\neq 0$. ...
0
votes
1answer
20 views

Definition of linear chains

I am reading Hatcher books in algebraic topology and he defined by $LC_n(Y)$ as the subgroup of $C_n(Y)$ generated by linear maps $\Delta^n$ $\to Y$ where $C_n(Y)$ is the abelian group of the $n$ ...
1
vote
1answer
86 views

Prove the long line is not contractible.

Given the following definition of the long line: Let $\omega_1$ be the first uncountable ordinal and consider $[0,1)$ as an ordinary set. Define the long ray to be the ordered set $\omega_1 \times ...
5
votes
1answer
68 views

How general $ [X,[Y,Z]] \cong [X \times Y, Z] $ is?

In some algebraic categories (abelian groups, modules on a ring) and in pointed spaces categories the following holds: $$ [X,[Y,Z]] \cong [X \times Y, Z] $$ In wich generality this lemma holds?
0
votes
0answers
30 views

Injectivity of precomposition with the suspension of the Hopf map

Let $h_2$ denote the usual hopf map and $h_n$ the $(n-2)$-th suspension of the hopf map i.e $h_n \colon S^{n+1} \to S^n $. Does anyone of you know a proof or a source of a proof of the fact that for ...
4
votes
2answers
50 views

Torsion of homology group

If $U$ is an open connected subset of $\mathbb{R}^n$ where $n\ge 2$,is it true that $H_1(U,\mathbb{Z})$ is torsion free?Or in general,$H_i(U)$ is free?I am thinking whether it has deformation retract ...
3
votes
1answer
56 views

Induced map on homology by $f\colon S^4 \to S^2 \times S^2$

Show that $$f_* \colon H_4(S^4) \to H_4(S^2 \times S^2)$$ is the zero map for any $f\colon S^4 \to S^2 \times S^2$. We are working with integral coefficients. I tried applying the naturality of ...
0
votes
0answers
30 views

The complement of the image of the zero section is still a $\mathbb{G}_m$-torsor?

This came up while doing some reading Schneps text on Galois Groups and Fundamental groups, but it's glossed over. In any case, suppose that you have a line bundle over a scheme $L\to X$, with zero ...
2
votes
1answer
65 views

Higher homology groups of infinite cyclic cover

Prove that all homology groups of the infinite cyclic cover of a knot complement are trivial except $H_1$. I've posted an answer below using Mayer-Vietoris. If you know of other arguments, please ...
0
votes
1answer
37 views

Small doubt about the connecting homomorphism on the long homology sequence

When you consider the long homology sequence (of spaces $A,X$ , with $A$ subspace of $X$) you need to define an homomorphism from $H_q(X,A)$ to $H_{q-1}(A)$ to obtain the long homology sequence from ...
2
votes
1answer
28 views

degree of a self map on the sphere

could you help me with this one? How do I determine the degree of the continous map $\mathbb{S}^n\to \mathbb{S}^n$ induced by multiplication with an orthogonal matrix $A$? I think it should be ...
0
votes
2answers
51 views

What is the homeomorphism type of the surface given by the polygonal presentation $aaa$?

More precisely, I am interested in the mapping cone of the map $S^1 \to S^1,$ $z \mapsto z^3.$ It seems like it should yield a "surface" with the following polygonal presentation: What is this ...
0
votes
2answers
53 views

Does the singular homology functor preserve injectivity and surjectivity?

I was wondering if the singular homology functor preserve injectivity and surjectivity? I've been trying to figure out a proof or counterexample for ages now and I just can't. This came up when I was ...
5
votes
0answers
47 views

No retract $X \wedge \mathbb{R}P^2 \to X \wedge \mathbb{R}P^1$

Let $X$ be a finite CW complex, and suppose $\Sigma X \cong X \wedge \mathbb{R}P^1$ is not contractible. By considering the fundamental group or otherwise, it is easy to see that there can be no ...
5
votes
1answer
49 views

Tautological line bundle over $\mathbb{RP}^n$ isomorphic to normal bundle? Also “splitting” of transition functions

Hallo fellow mathematicians. I try to understand why the normal bundle of $\mathbb{PR}^n$ is isomorphic (in the category of vector bundles) to the tautological line Bundle. More aptly, why ...
3
votes
1answer
42 views

almost complex structures on $R^4$

How should I see that the set of almost complex structures on $R^4$ preserving the positive orientation, namely $\{J\in GL^{+}(4,R), J^2=-I\}$ is homotopy equialent to $S^2$. There is a similar ...
1
vote
0answers
28 views

$\beta_{q}=\dim H_{q}(X,\mathbb{Q})$

Let's define $\beta_{q}$ to be $q^{th}$-Betti number of X, i.e. the rank of of $H_{q}(X,\mathbb{Z})$, the $q^{th}$-homology of $X$. How can I see that $\beta_{q}=\dim H_{q}(X,\mathbb{Q})$, where ...
3
votes
0answers
34 views

Cobordism of two manifolds [closed]

Is $\mathbb RP^4 \times \mathbb RP^{12} \times \mathbb RP^{15}$ cobordant to $\mathbb RP^6 \times \mathbb RP^{9} \times \mathbb RP^{9} \times \mathbb RP^{7}$?
-4
votes
1answer
83 views

Homology groups of retracts of algebraic topological spaces

Sup, recently I had an extremely vivid dream about algebraic topology. In it, I computed the homology group of the Klien Bottle $K$ to be $$H_1(K) \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$ ...
0
votes
0answers
52 views

cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring ...
5
votes
1answer
41 views

What is the relation between cohomology of $K(\mathbf{Z}_2, n)$ and $K(\mathbf{Z}_2, n+1)$?

I'm reading a proof to a theorem, "Suppose a, b: $H^*(-, \mathbf{Z}_2)\to H^{*+k}(-, \mathbf{Z}_2)$ are two stable (commuting with suspension isomorphism) cohomology operation of degree k. If ...
3
votes
1answer
33 views

Relation between cohomology of Eilenberg- MacLane space and product of projective space

In an article, it says that "Consider the map $\mathbf{RP}^\infty\times\cdots\times\mathbf{RP}^\infty$(n copies) $\to$ $K(\mathbf{Z}_2,n)$", I think this map is the map related to killing homotopy to ...
1
vote
1answer
47 views

There is no equivariant map $f:S^2 \to S^1$

To fix some notation, let $n \geq 2$ and let $p:S^n \to P^n$ be the canonical double cover. Let $\gamma:I \to S^n$ be a lift of a representative of a nontrivial element in $\pi_1(P_n) \cong ...