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        Questions tagged [cobordism]

        Cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold.

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        4
        votes
        1answer
        144 views

        Cobordism modelling fibration over $S^1$

        Let $X$ be a closed oriented manifold which is a fibration over $S^1$ whose fiber $F$ is connected, i.e. $X\cong F\times[0,1]/\sim h$, for an $h\in \mathrm{Diff}(F)$. Suppose that $b_1(X)=1$. ...
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        1answer
        120 views

        Diffeomorphism type of the added sphere in simply connected surgery

        A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...
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        votes
        1answer
        706 views

        Cobordism Theory of Topological Manifolds

        Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ...
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        1answer
        257 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 ...
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        1answer
        276 views

        Third differential in the homology AHSS

        I need some guidance in identifying the third differential in the homology AHSS for $\Omega_{\ast}^{\text{Spin}^c}(X)$ in degrees $\leq 4$. Remember that $\pi_0(M\text{Spin}^c)=\Bbb Z$, $\pi_2(M\...
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        1answer
        194 views

        Unoriented bordism with twisted orientation

        The computation of the unoriented bordism group of the point $N_*=\Omega_*^O$ is a classic result. I would like to know a related bordism group, where we specify the twisted fundamental class $[M]\in ...
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        1answer
        118 views

        Decompose $MT(E(d)\times_{\mathbb Z_2} SU(2))$ as the wedge sum or smash product of spectra

        Consider the extension $$1\to SU(2)\to X\to O\to1,$$ there are 4 possibilities for $X$: $X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\...
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        1answer
        102 views

        Basic question on the cobordism spectrum

        I am reading a little about cobordism and I have a basic question, which makes sense both in the topological and motivic setting. Let $\mathrm{Gr}_{n,\infty}$ denote the infinite $n$-Grassmanian and ...
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        208 views

        What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

        I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...
        11
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        1answer
        330 views

        Reference on complex cobordism

        I am trying to study a little of algebraic cobordism and I lack background from the classic setting. Hence, I am looking for a textbook or expository writing covering the basics of complex cobordism. ...
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        96 views

        Conformal group and cobordism

        In this post, I am exploring my thoughts on the implementation of conformal symmetry group structure and cobordism relations. Namely, I like to know what has been done and explored in the past? on ...
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        0answers
        77 views

        Relate two different mod 2 indices: $\eta$ invariant and the number of zero modes of Dirac operator, associated to SU(2)

        My major question in this post here is that: How can we relate the following two mod 2 indices: $\eta$ invariant, the number of the zero modes of the Dirac operator $N_0'$ mod 2, associated to ...
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        0answers
        196 views

        Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

        In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ ...
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        0answers
        123 views

        Twisted spin-bordism invariant and a possible Postnikov square from $d=2$ to $d=5$

        This is a follow up more advanced question following Twisted spin bordism invariants in 5 dimensions. We follow the definitions in the earlier post. I had discussed my computation of $$ \Omega_5^{...
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        1answer
        241 views

        Twisted spin bordism invariants in 5 dimensions

        [Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...

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