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Questions tagged [algebraic-topology]

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

Algebraic topology is a mathematical subject that associates algebraic objects to topological spaces which are invariant under homeomorphism or homotopy equivalence. Often, these associations are functorial so that continuous maps induce morphisms between appropriate algebraic objects.

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21356 questions
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Bordism as a generalized cohomology theory.

Can you tell me, where I can find a proof of the following fact: The bordism functor is a generalized cohomology theory, i.e. we can find suitable connecting homomorphisms to obtain long exact cohomology sequences.
nick
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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 constructed on $X \times [1/2^{n+1}, 1/2^n]$. About…
Eric Auld
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homotopic maps?

For cell complexes, Whitehead's theorem says that a weak homotopy equivalence is an actual homotopy equivalence. More generally, if I have two maps between cell complexes which agree on homotopy groups, are they homotopic?
010110111
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When the Induced Homomorphism on the $n$-th Cohomology is an Isomorphism

I am trying to show that when you're given a continuous map $f:M\rightarrow N$ between compact orientable $n$-dimensional manifolds and $f^*:H^n(N)\rightarrow H^n(M)$ is an isomorphism, then $f^*:H^k(N)\rightarrow H^k(M)$ is injective for every…
TinaBelcher
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The Fundamental group of Klein Bottle

My question is if $$\pi_{1}(KB)\cong\frac{\mathbb{Z}(a)}{\langle a^{2} \rangle}*\frac{\mathbb{Z}(b)}{\langle b^{2}\rangle}\cong\frac{\mathbb{Z}(a)*\mathbb{Z}(b)}{\langle a^{2}b^{2} \rangle}$$ and $$\dfrac{\mathbb{Z}(c)}{\langle c^{2} \rangle} \cong…
altairsot
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Homology of a space obtained from $S^n$ by attaching a cell $e^{n+1}$ by a map of degree $m$

I am trying to understand how to use cellular homology on this simple example: let $X$ be a space obtained from $S^n$ by attaching a cell $e^{n+1}$ by a map of degree $m$. I understand that the generators of the group $H_k(X^{(k)},X^{(k-1)})$ are in…
user198182
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Hatcher Exercise 2.2.13

In this exercise we have $X$, a 2-complex obtained from $S^1$ by attaching two 2-cells by maps of degree 2 and 3, respectively. I am trying to see why $X \simeq S^2$, in part b) of the exercise. I can't seem to construct a homotopy equivalence map.…
user198182
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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.
Danny
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Isomorphism between Homology Groups

Let $A$ be a retract of $X$. Show that if $i:A\to X$ is the inclusion operator , then $i_*:H_n(A) \to H_n(X)$ is an isomorphism (It's easy to see that it's a monomorphism. But why is it an epimorphism?) Thanks!
joshua
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How do I prove that a finite covering space of a compact space is compact?

Let $C$ be a finite sheeted covering space of compact space $X$. How do I prove that $C$ is compact? Someone please give me a proof sketch.. Let $p:C\rightarrow X$ be a covering map. Let $\mathscr{A}$ be an open cover of $C$. Since $p$ is open,…
cococomi
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Product of infinite covering maps

If $p_i:E_i\rightarrow B_i, \ i\in I$ are covering maps, then is it true that $$\prod_{i\in I} p_i:\prod_{i\in I}E_i \rightarrow \prod_{i\in I}B_i$$ is a covering map ? It is true if $\mbox{card}\ I <\infty$, but I don't have any idea to prove it if…
mikis
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Given a space what spaces can it cover?

I was thinking about my previous question and thought about going the other way around. Assume we are given a space $Y$ and $Y$ covers $X$, then how much can be said about $X$? The most trivial property would be that if $Y$ is compact, then $X$…
pki
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Covering spaces and Euler characteristc obstruction

Let $X$ and $Y$ be surfaces. If we have a covering $X\to Y$, then $\chi(Y)\mid \chi(X)$. I assume that the condition on Euler characteristics is not enough to guarantee that a surface covers another. Are there any trivial examples of a surface $X$…
pki
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$n$-skeleta of a CW-complex and the cup product

Hatcher doesn't really say anything about defining the cup product for cellular cohomology, instead he gives the definition in terms of singular and simplicial cohomology, and my interpretation is that his definition would not generalize in a nice…
asdf
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Reduced suspension and mapping cone.

Reading about Steenrod squares and a result regarding the Hopf invariant the following homeomorphism is used in a proof without being proved $$\Sigma^k(C_f) = C_{\Sigma^kf}.$$ Here $\Sigma$ is the reduced suspension and $C_f$ the mapping cone. I…
M.B.
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