# Can we 'build' spinor structure not only from a Riemann Manifold but 'extract it' also from another algebraic structures?

I want to understand what type of structures are Spin Structure: are a monoids, ringoids, groups?
Can we build spinor structure find also from another structures not 'extract it' only from a Riemann Manifold?
Can you help me to understand what are structure and other ways to 'find' spinor structures

In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M has a spin structure.

1. This is not always possible since there is potentially a topological obstruction to the existence of spin structures.

2. Spin structures will exist if and only if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes.

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For example, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors but can we start from a Pseudo-Riemannian manifold to see if admit the so-called pseudospinors?

• What is your definition of a spinor structure? – Michael Albanese Feb 9 at 21:20
• I follow this definition. I wish an example where spin structure 'exit' as another application, not only wired to complex vector field because then we link it to hilbert space and spin notion of quantum phys. No, I read a more technical definition that give me idea that we can use spin structure as a model that can carry another context: from the point of view of the theory of G-structures, a spinor structure is a generalized G-structure with structure group G=Spin together with a non-faithful representation ρ:Spin→SOn – Jenn Feb 10 at 10:38
• If that's your definition, then such a structure exists on a smooth manifold $M$ if and only if $M$ is orientable and $w_2(M) = 0$. – Michael Albanese Feb 10 at 19:51