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        Questions tagged [at.algebraic-topology]

        Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

        3
        votes
        1answer
        101 views

        Cohomogy of local systems over CW-complexes

        Let $M$ be a finite CW-complex. Let $F$ be a finite rank local system over $M$ with coefficients in any field. Is it true that $\dim(H^k(M,F))$ is at most the number of $k$-cells times $\operatorname{...
        6
        votes
        1answer
        191 views

        Orientable with respect to complex cobordism?

        I have learned that an orientation of a manifold $M$ with respect to ordinary cohomology is an ordinary orientation, that an orientation with respect to complex K-theory is a Spin$^c$ structure, and ...
        -1
        votes
        2answers
        119 views

        Directed colimit and homology

        I am looking for a reference or a proof of the following fact: Let $X_{1}\subset X_{2}\subset\dots $ be a sequence of (hausdorff) topological spaces indexed by natural numbers such that each $X_{i}\...
        3
        votes
        1answer
        114 views

        A pair of spaces equivalent to a pair of CW-complexes

        Suppose that $X$ is a CW-complex and $Y$ a CW-subcomplex of $X$. Let $A$ be a closed subspace of $Z$ such that $Z-A$ is homeomorhic to $X-Y$ and $Z/A$ homeomorphic to $X/Y$ and The closure of $Z-A$ ...
        1
        vote
        0answers
        210 views

        “a result of Atiyah implies that the top cell of an orientable manifold splits off stably”

        I am looking for a reference for the following sentence “a result of Atiyah implies that the top cell of an orientable manifold splits off stably” Thank you very much for helping me find a reference ...
        5
        votes
        0answers
        103 views

        Closed embedding of CW-complexes

        Suppose that $i: X\rightarrow Y$ is a closed embedding such that $X$ and $Y $ are (retracts) of CW-complexes. Does it follow that $i$ is a cofibration ? Remark: There is a similar question here, ...
        15
        votes
        2answers
        807 views

        What is the generator of $\pi_9(S^2)$?

        This is more or less the same question as [ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way not only topologists can ...
        8
        votes
        1answer
        160 views

        Terminology about G- simplicial complexes

        For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...
        1
        vote
        0answers
        115 views

        Space of biholomorphic maps into a Riemann surface

        Let $F$ be a Riemann surface and $Q\in F$. Consider $U:=(\mathbb{C}\cup\infty)\setminus [-1,1]^2$. I am interested in the space $$X:=\{f:U\to F;\,\text{$f:U\to f(U)$ biholomorphic and $f(\infty)=Q$}\},...
        10
        votes
        0answers
        188 views

        Interpretation of determinants on commutative rings

        In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map. This interpretation conceptually depends ...
        6
        votes
        0answers
        235 views

        Restriction of a cofibration to closed subspaces

        Let $i: X\rightarrow Y$ be a cofibration between CW-complexes, more precisely a cellular embedding. Let $A$ be a closed subspace of $Y$ and $Z=i^{-1}(A)$. Let $$j: Z\rightarrow A$$ be the restriction ...
        5
        votes
        0answers
        83 views

        Analog of cellular approximation theorem for $CW_0$-complexes ($CW_\mathcal P$-complexes)

        $CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a ...
        -1
        votes
        0answers
        87 views

        homotopy VS isotopy classes of embeddings

        Let $X$ any compact set in $R^n$, not necessarily a manifold. Let $f,g:X \longrightarrow M^k $ be two homotopic PL embeddings of $X$. When $X$ is an $m$-manifold then the two embeddings are also ...
        11
        votes
        2answers
        381 views

        Realizing cohomology classes by submanifolds

        In "Quelques propriétés globales des variétés différentiables", Thom gives conditions for a class in singular homology of a compact manifold to be realized by a smooth oriented submanifold (see e.g. ...
        2
        votes
        0answers
        74 views

        How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones?

        There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ...

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