Determining the rational homotopy type of the component of inclusion in the space of continuous mappings from gr(k, n) to gr(k , n + r)
Date
2020
Authors
Otieno, Paul Antony
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Publisher
Strathmore University
Abstract
The complex Grassmann manifold Gr(k, n) is the space of k dimensional sub spaces of en. Fork= 1, one gets epn-l, the space of lines in en. There is a natural embedding G(k , n) <-+ G(k, n + r). Moreover, any complex manifold can be embedded in some projective space epN. In particular, there is an embedding Gr(k, n) <-+ epN-l where N = (~) 0 To a simply connected topological space, Sullivan associates in a functional way a commutative differential graded algebra (cdga) henceforth (/\ V, d) which encodes the rational homotopy type of X. This is called a Sullivan model of X . Given that H *(ePn, Q) is the truncated polynomial algebra 1\xj(xn+l ), one gets a Sullivan model of the form (1\(x, y), d) where lxl = 2, jyj = 2n + 1 and dx = 0, dy = xn+l . For k 2: 1, one might use the homeo morphism G(k, n) = U(n)/(U(k) x U(n- k)) to find a Sullivan model. Moreover, iff :X --+ Y is a continuous mapping between CW-complexes, then there is a commutative differential graded algebra ( cdga) morphism ¢ : (1\Vy, d) --+ (1\Vx, d) between Sullivan models of X andY. This is called a Sullivan model of f. In this thesis, we use a Sullivan model of the inclusion Gr(k, n) ~ Gr(k, n+ r) to compute the rational homotopy type of the component of the inclusion in the space of mappings from Gr(k , n) to Gr(k, n + r). Further, we will compute an L00-model of the component of the inclusion i and deduce its Sullivan model, using the generalised cochain Quillen functor. We seek to define a model both Sullivan and Quillen for the component of the inclusion and from it obtain the cohomology algebra and even attempt to determine whether the space is formal or not.
Description
Submitted in total fulfillment of the requirements for the degree of Doctor of Philosophy in Pure Mathematics at Strathmore University